Iterated digits squaring
You are encouraged to solve this task according to the task description, using any language you may know.
If you add the square of the digits of a Natural number (an integer bigger than zero), you always end with either 1 or 89:
15 -> 26 -> 40 -> 16 -> 37 -> 58 -> 89 7 -> 49 -> 97 -> 130 -> 10 -> 1
An example in Python:
>>> step = lambda x: sum(int(d) ** 2 for d in str(x))
>>> iterate = lambda x: x if x in [1, 89] else iterate(step(x))
>>> [iterate(x) for x in xrange(1, 20)]
[1, 89, 89, 89, 89, 89, 1, 89, 89, 1, 89, 89, 1, 89, 89, 89, 89, 89, 1]
- Task
- Count how many number chains for integers 1 <= n < 100_000_000 end with a value 89.
Or, for much less credit - (showing that your algorithm and/or language is slow):
- Count how many number chains for integers 1 <= n < 1_000_000 end with a value 89.
This problem derives from the Project Euler problem 92.
For a quick algorithm for this task see the talk page
- Related tasks
11l
F next_step(=x)
V result = 0
L x > 0
result += (x % 10) ^ 2
x I/= 10
R result
F check(number)
V candidate = 0
L(n) number
candidate = candidate * 10 + n
L candidate != 89 & candidate != 1
candidate = next_step(candidate)
I candidate == 89
V digits_count = [0] * 10
L(d) number
digits_count[d]++
V result = factorial(number.len)
L(c) digits_count
result I/= factorial(c)
R result
R 0
V limit = 100000000
V cache_size = Int(ceil(log10(limit)))
assert(10 ^ cache_size == limit)
V number = [0] * cache_size
V result = 0
V i = cache_size - 1
L
I i == 0 & number[i] == 9
L.break
I i == cache_size - 1 & number[i] < 9
number[i]++
result += check(number)
E I number[i] == 9
i--
E
number[i]++
L(j) i + 1 .< cache_size
number[j] = number[i]
i = cache_size - 1
result += check(number)
print(result)- Output:
85744333
Ada
with Ada.Text_IO;
procedure Digits_Squaring is
function Is_89 (Number : in Positive) return Boolean
is
Squares : constant array (0 .. 9) of Natural :=
(0, 1, 4, 9, 16, 25, 36, 49, 64, 81);
Sum : Natural := Number;
Acc : Natural;
begin
loop
Acc := Sum;
Sum := 0;
while Acc > 0 loop
Sum := Sum + Squares (Acc mod 10);
Acc := Acc / 10;
end loop;
if Sum = 89 then return True; end if;
if Sum = 1 then return False; end if;
end loop;
end Is_89;
use Ada.Text_IO;
Count : Natural := 0;
begin
for A in 1 .. 99_999_999 loop
if Is_89 (A) then
Count := Count + 1;
end if;
if A = 999_999 then
Put_Line ("In range 1 .. 999_999: " & Count'Image);
end if;
end loop;
Put_Line ("In range 1 .. 99_999_999: " & Count'Image);
end Digits_Squaring;
- Output:
In range 1 .. 999_999: 856929 In range 1 .. 99_999_999: 85744333
ALGOL 68
Brute-force with some caching.
# count the how many numbers up to 100 000 000 have squared digit sums of 89 #
# compute a table of the sum of the squared digits of the numbers 00 to 99 #
[ 0 : 99 ]INT digit pair square sum;
FOR d1 FROM 0 TO 9 DO
FOR d2 FROM 0 TO 9 DO
digit pair square sum[ ( d1 * 10 ) + d2 ] := ( d1 * d1 ) + ( d2 * d2 )
OD
OD;
# returns the sum of the squared digits of n #
PROC squared digit sum = ( INT n )INT:
BEGIN
INT result := 0;
INT rest := n;
WHILE rest /= 0 DO
INT digit pair = rest MOD 100;
result PLUSAB digit pair square sum[ digit pair ];
rest OVERAB 100
OD;
result
END # squared digit sum # ;
# for values up to 100 000 000, the largest squred digit sum will be that of 99 999 999 #
# i.e. 81 * 8 = 648, we will cache the values of the squared digit sums #
INT cache max = 81 * 8;
[ 1 : cache max ]INT cache;
FOR i TO cache max DO cache[ i ] := 0 OD;
INT count 89 := 0;
# fill in the cache #
FOR value FROM 2 TO cache max DO cache[ value ] := squared digit sum( value ) OD;
# we "know" that 89 and 1 are the terminal values #
cache[ 1 ] := 1;
cache[ 89 ] := 89;
FOR value FROM 2 TO cache max DO
INT sum := cache[ value ];
WHILE sum /= 1 AND sum /= 89 DO
sum := cache[ sum ]
OD;
cache[ value ] := sum
OD;
FOR value FROM 1 TO 100 000 000 DO
IF cache[ squared digit sum( value ) ] = 89 THEN count 89 +:= 1 FI
OD;
print( ( "Number of values whose squared digit sum is 89: ", whole( count 89, -10 ), newline ) )- Output:
Number of values whose squared digit sum is 89: 85744333
Arturo
gen: function [n][
result: n
while [not? in? result [1 89]][
s: new 0
loop digits result 'd ->
's + d*d
result: s
]
return result
]
chainsEndingWith89: function [ndigits][
[prevCount,currCount]: #[]
loop 0..9 'i -> prevCount\[i*i]: 1
res: new 0
loop 2..ndigits 'x [
currCount: #[]
loop prevCount [val,cnt][
v: to :integer val
loop 0..9 'newDigit [
mm: v + newDigit*newDigit
if not? key? currCount mm -> currCount\[mm]: 0
currCount\[mm]: currCount\[mm] + cnt
]
]
prevCount: currCount
]
loop currCount [val,cnt][
v: to :integer val
if and? [v <> 0] [89=gen v] ->
'res + cnt
]
return res
]
print [
"Number chains for integers <100000000 that end with an 89 value:"
chainsEndingWith89 8
]
- Output:
Number chains for integers <100000000 that end with an 89 value: 85744333
AWK
We use a brute-force approach with buffering for better performance. Numbers are assumed to be double precision floats, which is true for most implementations. It runs in about 320 s on an Intel i5.
# Usage: GAWK -f ITERATED_DIGITS_SQUARING.AWK
BEGIN {
# Setup buffer for results up to 9*9*8
for (i = 1; i <= 648; i++) {
k = i
do {
k = squaredigitsum(k)
} while ((k != 1) && (k != 89))
if (k == 1) # This will give us 90 entries
buffer[i] = ""
}
# Check sequence for every number
pow10 = 1
for (i = 1; i <= 100000000; i++) {
count += (squaredigitsum(i) in buffer) ? 0 : 1
if (i == pow10) {
printf("1->10^%d: %d\n", length(i) - 1, count)
pow10 *= 10
}
}
}
function squaredigitsum(n, r) {
while (n) {
r += (n % 10) ^ 2
n = int(n / 10)
}
return r
}
- Output:
1->10^0: 0 1->10^1: 7 1->10^2: 80 1->10^3: 857 1->10^4: 8558 1->10^5: 85623 1->10^6: 856929 1->10^7: 8581146 1->10^8: 85744333
BBC BASIC
Three versions timed on a 2.50GHz Intel Desktop.
REM Version 1: Brute force
REM ---------------------------------------------------------
T%=TIME
N%=0
FOR I%=1 TO 100000000
J%=I%
REPEAT
K%=0:REPEAT K%+=(J%MOD10)^2:J%=J%DIV10:UNTIL J%=0
J%=K%
UNTIL J%=89 OR J%=1
IF J%>1 N%+=1
NEXT
PRINT "Version 1: ";N% " in ";(TIME-T%)/100 " seconds."
REM Version 2: Brute force + building lookup table
REM ---------------------------------------------------------
T%=TIME
DIM B% 9*9*8,H%(9)
N%=0
FOR I%=1 TO 100000000
J%=I%
H%=0
REPEAT
K%=0:REPEAT K%+=(J%MOD10)^2:J%=J%DIV10:UNTIL J%=0
H%(H%)=K%:H%+=1
J%=K%
IF B%?J%=1 EXIT REPEAT
UNTIL J%=89 OR J%=1
IF J%>1 N%+=1:WHILE H%>0:H%-=1:B%?H%(H%)=1:ENDWHILE
NEXT
PRINT "Version 2: ";N% " in ";(TIME-T%)/100 " seconds."
REM Version 3: Calc possible combinations (translation of C)
REM ---------------------------------------------------------
T%=TIME
DIM B%(9*9*8):B%(0)=1
FOR N%=1 TO 8
FOR I%=9*9*N% TO 1 STEP -1
FOR J%=1 TO 9
S%=J%*J%
IF S%>I% EXIT FOR
B%(I%)+=B%(I%-S%)
NEXT
NEXT
NEXT
N%=0
FOR I%=1 TO 9*9*8
J%=I%
REPEAT
K%=0:REPEAT K%+=(J%MOD10)^2:J%=J%DIV10:UNTIL J%=0
J%=K%
UNTIL J%=89 OR J%=1
IF J%>1 N%+=B%(I%)
NEXT
PRINT "Version 3: ";N% " in ";(TIME-T%)/100 " seconds."
END
- Output:
Version 1: 85744333 in 1447.08 seconds. Version 2: 85744333 in 718.04 seconds. Version 3: 85744333 in 0.02 seconds.
Befunge
This is just a brute force solution, so it's not very fast. A decent interpreter will probably take a minute or two for a 1,000,000 iterations. If you want to test with 100,000,000 iterations, change the ::** (100³) near the end of the first line to :*:* (100²²). With that many iterations, though, you'll almost certainly want to be using a compiler, otherwise you'll be waiting a long time for the result.
1-1\10v!:/+55\<>::**>>-!|
v0:\+<_:55+%:*^^"d":+1$<:
>\`!#^ _$:"Y"-#v_$\1+\:^0
>01-\0^ @,+55.<>:1>-!>#^_
>,,,$." >=",,,^ >>".1">#<
- Output:
1..1000000 => 856929
1..100000000 => 85744333
BQN
A simple solution is to compute all square-digit sums in the desired range as an addition table, then repeatedly select from this list using itself as an index so that all values that end at 1 converge (those that reach 89 will find some point in the cycle, but not always the same one).
+´1?1? ?˜?(?2???) ?+?´6?<ט?10
856929
It will take a lot of memory and many seconds to compute the count under 1e8 this way. The following program computes the count for numbers below 10??? by using dynamic programming to determine how many numbers have each possible digit sum. Then it finds the fate of each number in this greatly reduced set. This gives an exact result for inputs up to 16, taking a fraction of a millisecond for each.
DigSq ? {
d ? ט ?10 # Digit values
m ? 1+81×2??? # One plus maximum digit sum
c ? (+´ d ??0?»¨ <)??? m??1 # Count of numbers having each sum
s ? m ? ? d +??(?10??m) 0 # Sum for each sum
e ? 1??˜?(?2???)s # Which sums end at 89
¯1 +´ c×e # Total up; subtract 1 to exclude 0
}
>??DigSq¨ 1+?16
+-
? 1 7
2 80
3 857
4 8558
5 85623
6 856929
7 8581146
8 85744333
9 854325192
10 8507390852
11 84908800643
12 850878696414
13 8556721999130
14 86229146720315
15 869339034137667
16 8754780882739336
+
C
C99, tested with "gcc -std=c99". Record how many digit square sum combinations there are. This reduces numbers to , and the complexity is about . The 64 bit integer counter is good for up to , which takes practically no time to run.
#include <stdio.h>
typedef unsigned long long ull;
int is89(int x)
{
while (1) {
int s = 0;
do s += (x%10)*(x%10); while ((x /= 10));
if (s == 89) return 1;
if (s == 1) return 0;
x = s;
}
}
int main(void)
{
// array bounds is sort of random here, it's big enough for 64bit unsigned.
ull sums[32*81 + 1] = {1, 0};
for (int n = 1; ; n++) {
for (int i = n*81; i; i--) {
for (int j = 1; j < 10; j++) {
int s = j*j;
if (s > i) break;
sums[i] += sums[i-s];
}
}
ull count89 = 0;
for (int i = 1; i < n*81 + 1; i++) {
if (!is89(i)) continue;
if (sums[i] > ~0ULL - count89) {
printf("counter overflow for 10^%d\n", n);
return 0;
}
count89 += sums[i];
}
printf("1->10^%d: %llu\n", n, count89);
}
return 0;
}
- Output:
1->10^1: 7 1->10^2: 80 1->10^3: 857 1->10^4: 8558 1->10^5: 85623 1->10^6: 856929 1->10^7: 8581146 1->10^8: 85744333 1->10^9: 854325192 1->10^10: 8507390852 1->10^11: 84908800643 1->10^12: 850878696414 1->10^13: 8556721999130 1->10^14: 86229146720315 1->10^15: 869339034137667 1->10^16: 8754780882739336 1->10^17: 87975303595231975 1->10^18: 881773944919974509 1->10^19: 8816770037940618762 counter overflow for 10^20
Fast C implementation (<1 second my machine), which performs iterated digits squaring only once for each unique 8 digit combination. The cases 0 and 100,000,000 are ignored since they don't sum to 89:
#include <stdio.h>
const int digits[] = { 0,1,2,3,4,5,6,7,8,9 };
// calculates factorial of a number
int factorial(int n) {
return n == 0 ? 1 : n * factorial(n - 1);
}
// returns sum of squares of digits of n
unsigned int sum_square_digits(unsigned int n) {
int i,num=n,sum=0;
// process digits one at a time until there are none left
while (num > 0) {
// peal off the last digit from the number
int digit=num % 10;
num=(num - digit)/10;
// add it's square to the sum
sum=sum+digit*digit;
}
return sum;
}
// builds all combinations digits 0-9 of length len
// for each of these it will perform iterated digit squaring
// and for those which result in 89 add to a counter which is
// passed by pointer.
long choose_sum_and_count_89(int * got, int n_chosen, int len, int at, int max_types, int *count89)
{
int i;
long count = 0;
int digitcounts[10];
for (i=0; i < 10; i++) {
digitcounts[i]=0;
}
if (n_chosen == len) {
if (!got) return 1;
int sum=0;
for (i = 0; i < len; i++) {
int digit=digits[got[i]];
digitcounts[digit]++;
sum=sum + digit * digit;
}
if (sum == 0) {
return 1;
}
if ((sum != 1) && (sum != 89)) {
while ((sum != 1) && (sum != 89)) {
sum=sum_square_digits(sum);
}
}
if (sum == 89) {
int count_this_comb=factorial(len);
for (i=0; i<10; i++) {
count_this_comb/=factorial(digitcounts[i]);
}
(*count89)+=count_this_comb;
}
return 1;
}
for (i = at; i < max_types; i++) {
if (got) got[n_chosen] = i;
count += choose_sum_and_count_89(got, n_chosen + 1, len, i, max_types, count89);
}
return count;
}
int main(void)
{
int chosen[10];
int count=0;
// build all unique 8 digit combinations which represent
// numbers 0-99,999,999 and count those
// whose iterated digit squaring sum to 89
// case 0, 100,000,000 are ignored since they don't sum to 89
choose_sum_and_count_89(chosen, 0, 8, 0, 10, &count);
printf("%d\n",count);
return 0;
}
- Output:
85744333
C#
The largest sum possible for any number is 9*9*9, so the first 730 numbers are calculated and stored in an array.
The rest is then looked up. A limit of 100 million takes about 6 seconds. int.MaxValue takes about 2 and a half minutes.
using System;
public static class IteratedDigitsSquaring
{
public static void Main() {
Console.WriteLine(Count89s(1_000_000));
Console.WriteLine(Count89s(100_000_000));
}
public static int Count89s(int limit) {
if (limit < 1) return 0;
int[] end = new int[Math.Min(limit, 9 * 9 * 9 + 2)];
int result = 0;
for (int i = 1; i < end.Length; i++) {
for (end[i] = i; end[i] != 1 && end[i] != 89; end[i] = SquareDigitSum(end[i])) { }
if (end[i] == 89) result++;
}
for (int i = end.Length; i < limit; i++) {
if (end[SquareDigitSum(i)] == 89) result++;
}
return result;
int SquareDigitSum(int n) {
int sum = 0;
while (n > 0) {
int digit = n % 10;
sum += digit * digit;
n /= 10;
}
return sum;
}
}
}
- Output:
856929 85744333
BigInteger version
Translation of the first C version, with BigIntegers. This can get pretty far in six seconds, even on Tio.run.
using System;
using System.Numerics;
class Program {
const int MaxPow = 301;
static int [] sq = {1, 4, 9, 16, 25, 36, 49, 64, 81};
static BigInteger [] sums;
static bool is89(int x) {
while (true) {
int s = 0, t;
do if ((t = (x % 10) - 1) >= 0) s += sq[t]; while ((x /= 10) > 0);
if (s == 89) return true;
if (s == 1) return false;
x = s;
}
}
static BigInteger count89(int n) {
BigInteger result = 0;
for (int i = n * 81; i > 0; i--) {
foreach (int s in sq) { if(s > i) break; sums[i] += sums[i - s]; }
if (is89(i)) result += sums[i];
}
return result;
}
static void Main(string[] args) {
BigInteger [] t = new BigInteger[2] {1, 0}; sums = new BigInteger[MaxPow * 81]; Array.Copy(t, sums, t.Length);
DateTime st = DateTime.Now;
for (int n = 1; n < MaxPow; n++) {
Console.Write("1->10^{0,-3}: {1}\n", n, count89(n));
if ((DateTime.Now - st).TotalSeconds > 6) break;
}
Console.WriteLine("{0} seconds elapsed.", (DateTime.Now - st).TotalSeconds);
}
}
- Output:
1->10^1 : 7 1->10^2 : 80 1->10^3 : 857 1->10^4 : 8558 1->10^5 : 85623 1->10^6 : 856929 1->10^7 : 8581146 1->10^8 : 85744333 1->10^9 : 854325192 1->10^10 : 8507390852 1->10^11 : 84908800643 1->10^12 : 850878696414 1->10^13 : 8556721999130 1->10^14 : 86229146720315 1->10^15 : 869339034137667 1->10^16 : 8754780882739336 1->10^17 : 87975303595231975 1->10^18 : 881773944919974509 1->10^19 : 8816770037940618762 1->10^20 : 87994965555707002706 1->10^21 : 877214809753814412449 1->10^22 : 8740475212714948184620 1->10^23 : 87086767569032964273481 1->10^24 : 867912763131207135645491 1->10^25 : 8652685884347431487002838 1->10^26 : 86292591735549905389544085 1->10^27 : 860834491746260610360036431 1->10^28 : 8589383648492973833587962133 1->10^29 : 85719021282987319689186339605 1->10^30 : 855551075003449256539175506135 1->10^31 : 8539846767881104092122936276127 1->10^32 : 85245373514507207808857201531419 1->10^33 : 850921798797738318678358430121498 1->10^34 : 8493602724656082624921256124945709 1->10^35 : 84775765928320499747460839463166887 1->10^36 : 846127234701773214379999133850790428 1->10^37 : 8445101119798901092741398494615146552 1->10^38 : 84297231641833173945386163054551847907 1->10^39 : 841596309978956515337376882969248454407 1->10^40 : 8404688192812158407616126296428757287918 1->10^41 : 83966751636707267524727665346136900559808 1->10^42 : 839249062380369832617111284115323596416189 1->10^43 : 8392404334111393647768734710144578436411820 1->10^44 : 83963458265257975880706035079312646291089162 1->10^45 : 840390620671402119260216748725664301844515595 1->10^46 : 8414380030090502032224993998030998898525187113 1->10^47 : 84268378296544752164579356732419005387066100619 1->10^48 : 844021806190251380758758476585216084473498054164 1->10^49 : 8453427257465803796850958549692384862623307213954 1->10^50 : 84654382110763756920355712358557288888652143589824 1->10^51 : 847537750217936548550698726085731005366031699187697 1->10^52 : 8482595213704622541116090344851904585191448008008698 1->10^53 : 84867114171087369978017651353669784240040553506347863 1->10^54 : 848763596449838290475849513610494144653829069301555744 1->10^55 : 8485560484449848898784875907345401899210439410548661905 1->10^56 : 84809241613331707710051455489300240267084096119421192555 1->10^57 : 847435762855526547824875506635678396375585724580676401281 1->10^58 : 8466611716350744168054316461343227422117005648835357501042 1->10^59 : 84584794275749872157313978459784905712596125963065261887087 1->10^60 : 845072003706634444132974487900963836188835216550117810064042 1->10^61 : 8443993493344896883975002240891650144660444520675597442455846 1->10^62 : 84388114632696697235622301117847815843833846024601175739193818 1->10^63 : 843550873686677877815689986525235580589881417969543147823955468 1->10^64 : 8434235773893302085490040550865199018569738474571414542759091420 1->10^65 : 84349704267294170985441634483996250787548886013951222960902661326 1->10^66 : 843754473866041852258692025296354310048924258957916675280270848383 1->10^67 : 8441681459520956437757926334685945498961397097395903879157454648255 1->10^68 : 84470346186515545447015226246395087364669380975589204241143094714939 1->10^69 : 845322904478358163301387625325558514367250680088175647529431667836984 1->10^70 : 8459894886866423548013102102792807954026974401198186041921374671243280 1->10^71 : 84666946651672233790199404593444235036570835062615369577576893503846807 1->10^72 : 847334186689171226140326334304533885325304121282085041331744831976064350 1->10^73 : 8479611353077227882670645671814119173057689480878540635598355377828682030 1->10^74 : 84853427154299528465645782338465354008595064980299709100384597454288492601 1->10^75 : 849043816798387454378741585441510256257380507579571260789337785619852662347 1->10^76 : 8494856623561856764693867992374031564067482173292744174563186543401165633828 1->10^77 : 84986053947449161776274594245379439290411038180766518290148488651233497246177 1->10^78 : 850173060522433967868762132659331655304647077578564191004961947049948444228365 1->10^79 : 8504307025056582630550030771632855466797975344124296918801658700334809780702180 1->10^80 : 85064277463177819389470781000622284519150163101058649012431344960771964694651571 1->10^81 : 850819394401164074041611461274974545538571046366755739155832724445684685332968364 1->10^82 : 8509704926350516030551921703403722812551632988675827388709515859909259694851647834 1->10^83 : 85110487385572993176128742106677248626395367895834794577983294930763809503319428014 1->10^84 : 851229369385484114012959335791260147114430529630506351180704055826749898923602659534 1->10^85 : 8513481307977862280230328507187688326498927519035854456240791004629338052316252944962 1->10^86 : 85146211618619610403764635439165512633290370187486987569410348341895803492714570487527 1->10^87 : 851568818184158929602427279369759555161328403821517914312791180281061345268793327734747 1->10^88 : 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84999508275236126020381830508721543111473067554541966121589517605650398866751293320363799608712233989549184897359325724581354866372415617540113985299280733097725364579425684134507256488872 1->10^189: 849306296274651882683597422387001923066889384195634499192702671890502375676555190136845726720160747215701213796501622133175006452307225424987441822517508280951544748121555031660346304639001 1->10^190: 8487095911796817623434988696115120825517986522671534265823536015609746863774499420587993803271710140150813684221232159026871093033542365679393034583857094925097687013070985086109793273080806 1->10^191: 84821370246558618609839073367520567864334999557370002647757230099277664105830851234474210899445947000757007737070796386462591548949557365070963572571938326301912601377555644042593381481793406 1->10^192: 847825310704237766880070980506466839658817888477084199818418242818651205859175872441362805453796299246427513786781377975949558744142534256078247658163358081519312966146099874702063865152192028 1->10^193: 8475489555842905339842177447569741889165387333475856347815246130384309138300825424276838743453380208632107646061552066006714782109537135813874993527510102283805650143962763452053420700390547699 1->10^194: 84738701485690450024204013592451605689013445522452704386773936803484296887639018283654199502249150235096649421520132242000546408214958227307607245837895669350870541210854287264223165212086998226 1->10^195: 847339745670078204744803652316933944651393806640268021489971965448240790293322017482045797037713314415845471889231439866706574167693441163081038944023000406430511708854956075209156416422852179500 1->10^196: 8474053834141387286935964031508774220009550200077496813045300456052204100457807211493137607068260269409818097530282626325036633749271383972601174815602843212760330743419406501022962731951260401758 1->10^197: 84758029161467432826119523898351898166282436501022577393790061641703167401164409394596608786688863136133007710604265705419119450632223820509239634351960138619310640443557197024847616720893834274556 1->10^198: 847859155127420710222264067465116270942345054705199086260046652911576677856574523726808872321246548759971208827865770218551918373067777291678034921219960068578854130134085087015093149046394684100436 1->10^199: 8482352079297050413772331207215755280136129216773296221264447386959877587192996245987959911290858363977880814068455839545348484912691954109930202230242025230487816060905500273627824722575046074286281 1->10^200: 84870048617336420197389727058863637424671320482647512598431088512958837365267475929180277480415786149971408009142371369019940386605860817912162814180167542797887574063677218639448400450114606809880493 1->10^201: 849246096864711958382140828789832341037541146449645765069443078515013899970754673189319764864878892878621617974941584582320988065756019066241199502894793046393859712076059928423517853448438863067863226 1->10^202: 8498624925606251826356755549843759984710649122272275679601866626755415511646464540638291701107135708359249734273392831194179172090092563707579355305268689407169443037588913674810901899690254156949367275 1->10^203: 85053974280141213857917842958407824057059580960693420355947997850134934176999635011238282707984566250378973275121221997840304574485093785863913446713844175190847992565857356412233852383266765346485977576 1->10^204: 851267793619122872630161782552104405308466360210406324429007428807568945258070114676614779203675989708980801172282855929544686979305743506670689733623543666749958573824475094895872235404030549992394708150 1->10^205: 8520367073359573551884982418477013557872986932518885258980976615694060910595076784527929984289747991467246665285419316072687197970688845733149905180272321105942958007079330817594754267934705067270350499956 1->10^206: 85283687596660830853502342927042854500321341431910436337225191217936420987092196489885047475301196582960695295636335734026004076930907280217960505685464052960893990657933932574032561361873483568439847208942 1->10^207: 853659202819987541247160120887834521284760886339623274004394773066680299538480440121545335974469636403243323858304782951228852284398676153012813966360440529328380943838264684632008844950545647229432408419763 1->10^208: 8544952483138354272479452096326871415134170280533803383323382424225457971898051136208471099688648406254320867700931409630655569469573762186473633746768288944756293766713800079402839068636993722539464464014343 1->10^209: 85533733751997391022351921893082947709535591919068528290332852883967864327556645461585949789541151338595934256435388217939077881109820952288423214884499900454917289123598440879211618935608105485422232130901905 1->10^210: 856178628495280232427926168171704244059740501955488094383750755874097867668689843542569526012394028099851590049094767507865596216897633456922557606621505076739639542878814850925266187250164585329019683542629923 1->10^211: 8570131419693618760966628804173079377257638567195555240493311797353797176489667120305647330718511690292693777674170531508951053735086796114378779311384689107558641543989958869638931709314824117850451000380143021 1->10^212: 85783575440047109414461631493979767235315177615543985312544659159056610182875530781537114391430809057845669077275617329819355730217516493005290650519511006772034807212815853844586588988998227020982705473135728765 1->10^213: 858642157725030348223249509804854637186812832528125424148240160404301905107061460749243511335395491708297289251008381529398185566019895635547713725083517506594429645460660280089932247894935780442351898303313140945 1->10^214: 8594287902664891352913658993716294858626490086433812797669386361846753520746735854191540089415236757792436360765472598893490276479705293275725004645785633334970346652528315678317371791476569985195862855380417401542 1->10^215: 86019274419595998866603848298462306888943095199436257368964632161654631977139765349618577697513221456293489609436575083556837195728003126429131084775756767513323846022083954060095590650961801996674310643811350499369 1->10^216: 860931655618474553063989578082933761682457487457489431016612890143003640365600635223968806347174143526789184001346670039335478957198921437817108109746919660278785057686592225736631570790200460497279046021138265882633 1->10^217: 8616435837729622438277275972924774728615421337934287445283316837787898004557496433610326251817310796184207726389126647218963569950986844300319174321919640256132751161493356727133356946791128243709978104039611221929848 1->10^218: 86232688597037059510650496264090044650582001854485728865997396326431091394758611347974329812789632648546842598755241789989941130068748668444311151669575147062603211288550024198424668424830168061712646711077510494361276 1->10^219: 862980094753456140909290085044350220214633094993933779019549067941866017572800599010552481547901256759922208947964802924199156525319895138745088257779570795905170294076748630599291864563865074304312620545350757654949167 1->10^220: 8636018033559847409410591107268210323615373240747163973441778682144682091271457867694794590752142037466191114879096481073228275415949941743493243536215269195973060634460249426462226091308923903368436341962027769078827045 1->10^221: 86419056489171874993789903489060892796455772565320723803637734001174467305715397474011039116397479926656101081127323729931224029537185192310507718579127400181923630610893970171709931933408787373551201437680240028108283958 1->10^222: 864744809847891662279169612523698598292488130571716275496525629546147011808378355197408294680827690818106146785428930909260093817332410616456120307490986571108036081906704392578762581319448395342866886631949686317950079685 1->10^223: 8652627854354814142021372274236655285412150520596931780504767483938734335443732380812376622197005225828947530209843225214814025134173901180081812700123059474516130142427254496259962223594226063501914926447312107963923255816 1->10^224: 86574251920745435851516366426371437236500297164193572646840936487152446338309838340335914152667764515000963423743096085244510673418048367379825329583910629778255879869131857325574114859632491805222223301052145753142373402038 1->10^225: 866181808148306233153035741070162251220868553540999914228623747312278276373367606857723265686649243714013350573228296681022846168327651205292941587217634862968933831862883106033509203213506996294508078581405379889390718900884 1->10^226: 8665782332173490493624385682733920863071548417814598549460323221230417435701507196058986103580683262649034387808689514519733914026030490404939317647480340690773539215255948949250489348489232493792231877166169779541100947510992 1->10^227: 86692919814710675783589991859619703285754546114420206472621039807364718363865261702270566986525194821577798634423416768340074070191330534224131123412508988375899005313698308076230449814177068560441644180880325653197748136923185 1->10^228: 867231990206473247052871536000445102504192081962389666619381949130498077603334870783904602961079237586504849633880397962186289424840869477113707180313165669320870472468535571056811158182157681436281093030894644992830298477981244 1->10^229: 8674838596801362677923547970017972903037995901012236697795083419765821418590394410295854079897879829098141309577554554785416139269074776459368918266410283727698808695686584095897235693981445795558726984321580750493111552025864339 1->10^230: 86768211402812128806590576564537513494737520987736487082881857738963221877281843731844788716420658593474347727365894819526796319707828593251356370569187398794672340428112756386987781701631240923503544557476729747177320351749598558 6.0396929 seconds elapsed.
It doesn't always get to 10^230 in six seconds at Tio.run, sometimes it only gets to 10^201 or so.
C++
Slow (~10 seconds on my machine) brute force C++ implementation:
#include <iostream>
// returns sum of squares of digits of n
unsigned int sum_square_digits(unsigned int n) {
int i,num=n,sum=0;
// process digits one at a time until there are none left
while (num > 0) {
// peal off the last digit from the number
int digit=num % 10;
num=(num - digit)/10;
// add it's square to the sum
sum+=digit*digit;
}
return sum;
}
int main(void) {
unsigned int i=0,result=0, count=0;
for (i=1; i<=100000000; i++) {
// if not 1 or 89, start the iteration
if ((i != 1) || (i != 89)) {
result = sum_square_digits(i);
}
// otherwise we're done already
else {
result = i;
}
// while we haven't reached 1 or 89, keep iterating
while ((result != 1) && (result != 89)) {
result = sum_square_digits(result);
}
if (result == 89) {
count++;
}
}
std::cout << count << std::endl;
return 0;
}
- Output:
85744333
Ceylon
shared void run() {
function digitsSquaredSum(variable Integer n) {
variable value total = 0;
while(n > 0) {
total += (n % 10) ^ 2;
n /= 10;
}
return total;
}
function lastSum(variable Integer n) {
while(true) {
n = digitsSquaredSum(n);
if(n == 89 || n == 1) {
return n;
}
}
}
variable value eightyNines = 0;
for(i in 1..100M - 1) {
if(lastSum(i) == 89) {
eightyNines++;
}
}
print(eightyNines);
}
Clojure
Direct Method
(ns async-example.core
(:require [clojure.math.numeric-tower :as math])
(:use [criterium.core])
(:gen-class))
(defn sum-sqr [digits]
" Square sum of list of digits "
(let [digits-sqr (fn [n]
(apply + (map #(* % %) digits)))]
(digits-sqr digits)))
(defn get-digits [n]
" Converts a digit to a list of digits (e.g. 545 -> ((5) (4) (5)) (used for squaring digits) "
(map #(Integer/valueOf (str %)) (String/valueOf n)))
(defn -isNot89 [x]
" Returns nil on 89 "
(cond
(= x 0) 0
(= x 89) nil
(= x 1) 0
(< x 10) (recur (* x x))
:else (recur (sum-sqr (get-digits x)))))
;; Cached version of isNot89 (i.e. remembers prevents inputs, and returns result by looking it up when input repeated)
(def isNot89 (memoize -isNot89))
(defn direct-method [ndigits]
" Simple approach of looping through all the numbers from 0 to 10^ndigits - 1 "
(->>
(math/expt 10 ndigits)
(range 0) ; 0 to 10^ndigits
(filter #(isNot89 (sum-sqr (get-digits %)))) ; filters out 89
(count) ; count non-89
(- (math/expt 10 ndigits)))) ; count 89 (10^ndigits - (count 89))
(time (println (direct-method 8)))
- Output:
85744333 Time: 335 seconds
Using Combinations
(def DIGITS (range 0 10))
(defn -factorial [n]
(apply * (take n (iterate inc 1))))
; Cached version of factorial
(def factorial (memoize -factorial))
(defn -combinations [coll k]
" From http://rosettacode.org/wiki/Combinations_with_repetitions#Clojure "
(when-let [[x & xs] coll]
(if (= k 1)
(map list coll)
(concat (map (partial cons x) (-combinations coll (dec k)))
(-combinations xs k)))))
; Cached version of combinations
(def combinations (memoize -combinations))
(defn comb [n r]
" count of n items select r "
(/ (/ (factorial n) (factorial r)) (factorial (- n r))))
(defn count-digits [digit-list]
" count nunmber of occurences of digit in list "
(reduce (fn [m v] (update-in m [v] (fnil inc 0))) {} digit-list))
(defn count-patterns [c]
" Count of number of patterns with these digits "
(->>
c
(count-digits)
(reduce (fn [accum [k v]]
(* accum (factorial v)))
1)
(/ (factorial (count c)))))
(defn itertools-comb [ndigits]
(->>
ndigits
(combinations DIGITS)
(filter #(is89 (sum-sqr %))) ; items which are not 89 (i.e. 1 since lower count)
(reduce (fn [acc c]
(+ acc (count-patterns c)))
0)
(- (math/expt 10 ndigits))))
(println (itertools-comb 8))
;; Time obtained using benchmark library (i.e. (bench (itertools-comb 8)) )
{
- Output:
85744333
Time: 78 ms (i.e. using combinations was over 4,000 times faster
both tested on i7 CPU 920@2.67GHZ)
Common Lisp
(defun square (number)
(expt number 2))
(defun list-digits (number)
"Return the `number' as a list of its digits."
(loop
:for (rest digit) := (multiple-value-list (truncate number 10))
:then (multiple-value-list (truncate rest 10))
:collect digit
:until (zerop rest)))
(defun next (number)
(loop
:for digit :in (list-digits number)
:sum (square digit)))
(defun chain-end (number)
"Return the ending number after summing the squaring of the digits of
`number'. Either 1 or 89."
(loop
:for next := (next number) :then (next next)
:until (or (eql next 1)
(eql next 89))
:finally (return next)))
(time
(loop
:with count := 0
:for candidate :from 1 :upto 100000000
:do (when (eql 89 (chain-end candidate))
(incf count))
:finally (return count)))
- Output:
Evaluation took: 1128.773 seconds of real time 1126.231095 seconds of total run time (1117.296987 user, 8.934108 system) [ Run times consist of 56.419 seconds GC time, and 1069.813 seconds non-GC time. ] 99.77% CPU 2,815,545,509,836 processor cycles 580,663,356,272 bytes consed *
Craft Basic
for i = 1 to 1000
let j = i
do
let k = 0
do
let k = int(k + (j % 10) ^ 2)
let j = int(j / 10)
wait
loop j <> 0
let j = k
loopuntil j = 89 or j = 1
if j > 1 then
let n = n + 1
endif
print "iterations: ", i
next i
print "count result: ", n
end
- Output:
857
D
A simple memoizing partially-imperative brute-force solution:
import std.stdio, std.algorithm, std.range, std.functional;
uint step(uint x) pure nothrow @safe @nogc {
uint total = 0;
while (x) {
total += (x % 10) ^^ 2;
x /= 10;
}
return total;
}
uint iterate(in uint x) nothrow @safe {
return (x == 89 || x == 1) ? x : x.step.memoize!iterate;
}
void main() {
iota(1, 100_000_000).filter!(x => x.iterate == 89).count.writeln;
}
- Output:
85744333
The run-time is about 10 seconds compiled with ldc2.
A fast imperative brute-force solution:
void main() nothrow @nogc {
import core.stdc.stdio: printf;
enum uint magic = 89;
enum uint limit = 100_000_000;
uint[(9 ^^ 2) * 8 + 1] lookup = void;
uint[10] squares;
foreach (immutable i, ref x; squares)
x = i ^^ 2;
foreach (immutable uint i; 1 .. lookup.length) {
uint x = i;
while (x != magic && x != 1) {
uint total = 0;
while (x) {
total += squares[(x % 10)];
x /= 10;
}
x = total;
}
lookup[i] = x == magic;
}
uint magicCount = 0;
foreach (immutable uint i; 1 .. limit) {
uint x = i;
uint total = 0;
while (x) {
total += squares[(x % 10)];
x /= 10;
}
magicCount += lookup[total];
}
printf("%u\n", magicCount);
}
The output is the same. The run-time is less than 3 seconds compiled with ldc2.
A more efficient solution:
import core.stdc.stdio, std.algorithm, std.range;
enum factorial = (in uint n) pure nothrow @safe @nogc
=> reduce!q{a * b}(1u, iota(1u, n + 1));
uint iLog10(in uint x) pure nothrow @safe @nogc
in {
assert(x > 0);
} body {
return (x >= 1_000_000_000) ? 9 :
(x >= 100_000_000) ? 8 :
(x >= 10_000_000) ? 7 :
(x >= 1_000_000) ? 6 :
(x >= 100_000) ? 5 :
(x >= 10_000) ? 4 :
(x >= 1_000) ? 3 :
(x >= 100) ? 2 :
(x >= 10) ? 1 : 0;
}
uint nextStep(uint x) pure nothrow @safe @nogc {
typeof(return) result = 0;
while (x > 0) {
result += (x % 10) ^^ 2;
x /= 10;
}
return result;
}
uint check(in uint[] number) pure nothrow @safe @nogc {
uint candidate = reduce!((tot, n) => tot * 10 + n)(0, number);
while (candidate != 89 && candidate != 1)
candidate = candidate.nextStep;
if (candidate == 89) {
uint[10] digitsCount;
foreach (immutable d; number)
digitsCount[d]++;
return reduce!((r, c) => r / c.factorial)
(number.length.factorial, digitsCount);
}
return 0;
}
void main() nothrow @nogc {
enum uint limit = 100_000_000;
immutable uint cacheSize = limit.iLog10;
uint[cacheSize] number;
uint result = 0;
uint i = cacheSize - 1;
while (true) {
if (i == 0 && number[i] == 9)
break;
if (i == cacheSize - 1 && number[i] < 9) {
number[i]++;
result += number.check;
} else if (number[i] == 9) {
i--;
} else {
number[i]++;
number[i + 1 .. $] = number[i];
i = cacheSize - 1;
result += number.check;
}
}
printf("%u\n", result);
}
The output is the same. The run-time is about 0.04 seconds or less. This third version was ported to D and improved from: mathblog.dk/project-euler-92-square-digits-number-chain/
A purely functional version, from the Haskell code. It includes two functions currently missing in Phobos used in the Haskell code.
import std.stdio, std.typecons, std.traits, std.typetuple, std.range, std.algorithm;
auto divMod(T)(T x, T y) pure nothrow @safe @nogc {
return tuple(x / y, x % y);
}
auto expand(alias F, B)(B x) pure nothrow @safe @nogc
if (isCallable!F &&
is(ParameterTypeTuple!F == TypeTuple!B)
&& __traits(isSame, TemplateOf!(ReturnType!F), Nullable)
&& isTuple!(TemplateArgsOf!(ReturnType!F)[0])
&& is(TemplateArgsOf!(TemplateArgsOf!(ReturnType!F)[0])[1] == B)) {
alias NAB = ReturnType!F;
alias AB = TemplateArgsOf!NAB[0];
alias A = AB.Types[0];
struct Expand {
bool first;
NAB last;
@property bool empty() pure nothrow @safe @nogc {
if (first) {
first = false;
popFront;
}
return last.isNull;
}
@property A front() pure nothrow @safe @nogc {
if (first) {
first = false;
popFront;
}
return last.get[0];
}
void popFront() pure nothrow @safe @nogc { last = F(last.get[1]); }
}
return Expand(true, NAB(AB(A.init, x)));
}
//------------------------------------------------
uint step(uint x) pure nothrow @safe @nogc {
Nullable!(Tuple!(uint, uint)) f(uint n) pure nothrow @safe @nogc {
return (n == 0) ? typeof(return)() : typeof(return)(divMod(n, 10u).reverse);
}
return expand!f(x).map!(x => x ^^ 2).sum;
}
uint iter(uint x) pure nothrow @nogc {
return x.recurrence!((a, n) => step(a[n - 1])).filter!(y => y.among!(1, 89)).front;
}
void main() {
iota(1u, 100_000u).filter!(n => n.iter == 89).count.writeln;
}
With a small back-porting (to run it with the Phobos of LDC2 2.065) it runs in about 15.5 seconds.
Delphi
Takes 16 seconds to complete on an a Ryzen 7 Win11 machine
function SumSquaredDigits(N: integer): integer;
{Sum the squares of the digits in a number}
var T: integer;
begin
Result:=0;
repeat
begin
T:=N mod 10;
N:=N div 10;
Result:=Result+T*T;
end
until N<1;
end;
function TestNumber(N: integer): integer;
{Sum the squares of the digits of number, and do it again}
{with tne new number until the result is either 89 or 1}
begin
Result:=N;
repeat Result:=SumSquaredDigits(Result);
until (Result=89) or (Result=1);
end;
procedure TestSquareDigitsSum(Memo: TMemo);
{Count the number of square digit sums end up 89}
var I,Cnt: integer;
begin
Cnt:=0;
for I:=1 to 100000000 do
if TestNumber(I)=89 then Inc(Cnt);
Memo.Lines.Add(IntToStr(Cnt));
end;
- Output:
85744333
EasyLang
fastfunc ends89 n .
repeat
s = 0
while n > 0
d = n mod 10
s += d * d
n = n div 10
.
n = s
if n = 89
return 1
.
until n = 1
.
return 0
.
k = 8
arrbase sums[] 0
len sums[] k * 81 + 1
sums[0] = 1
for n = 1 to k
for i = n * 81 downto 1
for j = 1 to 9
s = j * j
if s > i
break 1
.
sums[i] += sums[i - s]
.
.
.
cnt = 0
for i = 1 to k * 81
if ends89 i = 1
cnt += sums[i]
.
.
print cnt
- Output:
85744333
ERRE
PROGRAM ITERATION
BEGIN
PRINT(CHR$(12);) ! CLS
INPUT(N)
LOOP
N$=MID$(STR$(N),2)
S=0
FOR I=1 TO LEN(N$) DO
A=VAL(MID$(N$,I,1))
S=S+A*A
END FOR
IF S=89 OR S=1 THEN PRINT(S;) EXIT END IF
PRINT(S;)
N=S
END LOOP
PRINT
END PROGRAMThis program verifies a number only. With a FOR..END FOR loop it's possible to verify a number range.
Factor
A brute-force approach with some optimizations. It uses the fact that the first digit-square-sum of any number < 100,000,000 is, at most, 648. These few chains are rapidly memoized as the results for all hundred-million numbers are calculated for the first time or looked up.
USING: kernel math math.ranges math.text.utils memoize
prettyprint sequences tools.time ;
IN: rosetta-code.iterated-digits-squaring
: sum-digit-sq ( n -- m ) 1 digit-groups [ sq ] map-sum ;
MEMO: 1or89 ( n -- m )
[ dup [ 1 = ] [ 89 = ] bi or ] [ sum-digit-sq ] until ;
[
0 1
[
dup sum-digit-sq 1or89 89 = [ [ 1 + ] dip ] when
1 + dup 100,000,000 <
] loop
drop .
] time
- Output:
85744333 Running time: 55.76544594 seconds
Forth
Tested for VFX Forth and GForth in Linux
\ To explain the algorithm: Each iteration is performed in set-count-sumsq below.
\ sum square of digits for 1 digit numbers are
\ Base 1 2 3 4 5 6 7 8 9
\ Sumsq: 1 4 9 16 25 36 49 54 81
\ Adding 10 to the base adds 1 to the sumsq,
\ Adding 20 to the base adds 4
\ ||
\ Adding 90 adds 81
\ Similarly for n00, n000 etc..
\ Worked example for base 3 ( to keep the lists short ).
\ The base 10 version performs 1.1 .. 1.9 with shifts of 1, 4, 9 .. 81 cells
\
\ Ix 0 1 2 3 4 5 6 7 8
\ 0 [ 1 ]
\ 1.1 [ 1 ] Previous result shifted 1 cell ( 1**2 )
\ 1.2 [ 1 ] Previous result shifted 4 cells ( 2** 2 )
\ ------------------------------
\ Sum [ 1, 1, 0, 0, 1 ]
\ 2.1 [ 1, 1, 0, 0, 1 ] Previous result shifted 1 cell ( 1**2 )
\ 2.2 [ 1, 1, 0, 0, 1 ] Previous result shifted 4 cells ( 2** 2 )
\ --------------------------------------------
\ Sum [ 1, 2, 1, 0, 2, 2, 0, 0, 1 ] Number of integers with ix as first iteration sum of digits sq
CELL 8 * 301 * 1000 / CONSTANT max-digits \ 301 1000 / is log10( 2 )
\ 19 for a 64 bit Forth and 9 for a 32 bit one.
\ **********************************
\ **** Create a counted array ****
\ **********************************
: counted-array \ create: #elements -- ; does> -- a ;
CREATE
HERE SWAP 1+ CELLS DUP ALLOT ERASE
DOES> ;
\ ***********************************
\ **** Array manipulation words. ****
\ ***********************************
: arr-copy \ a-src a-dest -- ; \ Copy array array at a-src to array at a-dest
OVER @ 1+ CELLS CMOVE ;
: arr-count \ a -- a' ct ;
\ Fetch the count of cells in the array and shift addr to point to element 0.
DUP CELL+ SWAP @ ;
: th-element \ a ix -- a' ; \ Leave address of the ix th element of array at a on the stack
1+ CELLS + ;
: arr-empty \ a -- ; \ Sets all array elements to zero and zero length
dup @ 1+ CELLS ERASE ;
: arr+ \ a-src a-dest count -- ;
\ Add each cell from a-src to the cells from a-dest for count elements
\ Storing the result in a-dest
CELLS 0 DO
OVER I + @ OVER I + +! \ I is a byte count offset into either array
CELL +LOOP
2DROP ; \ DROP the two base addresses
: arr. \ a -- ; \ Print the array. Used to debug.
." [ " arr-count CELLS BOUNDS ?DO i @ . CELL +LOOP ." ]" ;
\ ***********************************
\ **** Sum digit squared words ****
\ ***********************************
: sum-digit-sq \ n -- n' ;
0 SWAP
BEGIN DUP WHILE
10 /MOD >R DUP * + R>
REPEAT DROP ;
: 89or1<> \ n -- f ; \ True if n not equal to 89 or 1.
DUP 89 <> AND 1 > ;
: iterated-89= \ n -- f ; \ True if n iterates to 89, false once it iterates to 1 ( or 0 ).
BEGIN DUP 89or1<> WHILE
sum-digit-sq
REPEAT 89 = ;
\ *****************************************************
\ **** Create `count-sumsq` and `sumsq-old` arrays ****
\ *****************************************************
max-digits 81 * 1+ counted-array count-sumsq
max-digits 1- 81 * 1+ counted-array sumsq-old
: init-count-sumsq \ -- ; \ Initialise the count-sumsq to [ 1 ]
count-sumsq arr-empty \ Ensure all zero
1 count-sumsq ! \ Set the length of the count-sumsq to 1 cell.
1 count-sumsq 0 th-element ! ; \ Store 1 in the first element.
: set-count-sumsq \ #digits -- ; \ The main work. Only called with valid #digits
init-count-sumsq
0 ?DO
count-sumsq sumsq-old arr-copy \ copy count-sumsq to sumsq-old
81 count-sumsq +! \ Extend count-sumsq by 81 (9*9) cells
10 1 DO
sumsq-old arr-count ( a-sumsq-old' len )
count-sumsq I DUP * th-element SWAP arr+
LOOP
LOOP ;
: count-89s \ #digits -- n ;
DUP max-digits U> IF
." Number of digits must be between 0 and " max-digits .
DROP 0
ELSE
set-count-sumsq
0 count-sumsq @ 0 DO
count-sumsq I th-element @ ( cum ith-count )
I iterated-89= \ True if the index delivers 89.
AND + \ True is -1 ( all bits set ) AND with the count and add to the cum.
LOOP
THEN ;
: test \ #digits :
CR max-digits min 1+ 1 ?DO
I 5 .r 2 SPACES I count-89s . CR
LOOP ;
- Output:
19 test
1 7
2 80
3 857
4 8558
5 85623
6 856929
7 8581146
8 85744333
9 854325192
10 8507390852
11 84908800643
12 850878696414
13 8556721999130
14 86229146720315
15 869339034137667
16 8754780882739336
17 87975303595231975
18 881773944919974509
19 8816770037940618762
FreeBASIC
' FB 1.05.0 Win64
' similar to C Language (first approach)
' timing for i3 @ 2.13 GHz
Function endsWith89(n As Integer) As Boolean
Dim As Integer digit, sum = 0
Do
While n > 0
digit = n Mod 10
sum += digit * digit
n \= 10
Wend
If sum = 89 Then Return True
If sum = 1 Then Return False
n = sum
sum = 0
Loop
End Function
Dim As Double start = timer
Dim sums(0 To 8 * 81) As UInteger
sums(0) = 1
sums(1) = 0
Dim s As Integer
For n As Integer = 1 To 8
For i As Integer = n * 81 To 1 Step -1
For j As Integer = 1 To 9
s = j * j
If s > i Then Exit For
sums(i) += sums(i - s)
Next j
Next i
If n = 8 Then
Dim As UInteger count89 = 0
For i As Integer = 1 To n * 81
If Not endsWith89(i) Then Continue For
count89 += sums(i)
Next i
Print "There are";count89; " numbers from 1 to 100 million ending with 89"
End If
Next
Print "Elapsed milliseconds ="; Int((timer - start) * 1000 + 0.5)
Print
Print "Press any key to quit"
Sleep
- Output:
There are 85744333 numbers from 1 to 100 million ending with 89 Elapsed milliseconds = 2
Frink
total = 0
d = new dict
var sum
for n = 1 to 100 million - 1
{
sum = n
do
{
if sum < 1000 and d@sum != undef
{
sum = d@sum
break
}
c = sum
sum = 0
for digit = integerDigits[c]
sum = sum + digit^2
} while (sum != 89) and (sum != 1)
if (n < 1000)
d@n = sum
if (sum == 89)
total = total + 1
}
println[total]- Output:
85744333
Fōrmulæ
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.
Programs in Fōrmulæ are created/edited online in its website.
In this page you can see and run the program(s) related to this task and their results. You can also change either the programs or the parameters they are called with, for experimentation, but remember that these programs were created with the main purpose of showing a clear solution of the task, and they generally lack any kind of validation.
Solution
The following function returns the end value (either 1 or 89) of a given number:
The following function calculates the number of 89 endings, from 1 to a given number:
Test case
Go
It's basic. Runs in about 30 seconds on an old laptop.
package main
import (
"fmt"
)
func main() {
var d, n, o, u, u89 int64
for n = 1; n < 100000000; n++ {
o = n
for {
u = 0
for {
d = o%10
o = (o - d) / 10
u += d*d
if o == 0 {
break
}
}
if u == 89 || u == 1 {
if u == 89 { u89++ }
break
}
o = u
}
}
fmt.Println(u89)
}
- Output:
85744333
Haskell
Basic solution that contains just a little more than the essence of this computation. This runs in less than eight minutes:
import Data.List (unfoldr)
import Data.Tuple (swap)
step :: Int -> Int
step = sum . map (^ 2) . unfoldr f where
f 0 = Nothing
f n = Just . swap $ n `divMod` 10
iter :: Int -> Int
iter = head . filter (`elem` [1, 89]) . iterate step
main = do
print $ length $ filter ((== 89) . iter) [1 .. 99999999]
- Output:
85744333
J
Here's an expression to turn a number into digits:
digits=: 10&#.inv
And here's an expression to square them and find their sum:
sumdigsq=: +/"1@:*:@digits
But note that while the task description claims "you always end with either 1 or 89", that claim is somewhat arbitrary.
- But only somewhat the loop is 89 ? 145 ? 42 ? 20 ? 4 ? 16 ? 37 ? 58 ? 89, so it only ends with 1 or one of the numbers in this loop. 42 is of course far more significant and the one I would choose!!--Nigel Galloway (talk) 10:12, 16 September 2014 (UTC)
sumdigsq^:(i.16) 15
15 26 40 16 37 58 89 145 42 20 4 16 37 58 89 145
Here, after the initial three values, the final digit repeats with a period of 8 (so it does not end).
10 8$sumdigsq^:(i.80) 15
15 26 40 16 37 58 89 145
42 20 4 16 37 58 89 145
42 20 4 16 37 58 89 145
42 20 4 16 37 58 89 145
42 20 4 16 37 58 89 145
42 20 4 16 37 58 89 145
42 20 4 16 37 58 89 145
42 20 4 16 37 58 89 145
42 20 4 16 37 58 89 145
42 20 4 16 37 58 89 145
You could just as easily claim that you always end with either 1 or 4. So here's a routine which repeats the sum-square process until the sequence converges, or until it reaches the value 4:
itdigsq4=:4 = sumdigsq^:(0=e.&4)^:_"0
But we do not actually need to iterate. The largest value after the first iteration would be:
sumdigsq 99999999
648
So we could write a routine which works for the intended range, and stops after the first iteration:
itdigsq1=:1 = sumdigsq^:(0=e.&4)^:_"0
digsq1e8=:(I.itdigsq1 i.649) e.~ sumdigsq
In other words, if the result after the first iteration is any of the numbers in the range 0..648 which converges to 1, it's not a result which would converge to the other loop. This is considerably faster than trying to converge 1e8 sequences, and also evades having to pick an arbitrary stopping point for the sequence which loops for the bulk computation.
And this is sufficient to find our result. We don't want to compute the entire batch of values in one pass, however, so let's break this up into 100 batches of one million each:
+/+/@:-.@digsq1e8"1(1+i.100 1e6)
85744333
Of course, there are faster ways of obtaining that result. The fastest is probably this:
85744333
85744333
This might be thought of as representing the behavior of a highly optimized compiled program. We could abstract this further by using the previous expression at compile time, so we would not have to hard code it.
Java
import java.util.stream.IntStream;
public class IteratedDigitsSquaring {
public static void main(String[] args) {
long r = IntStream.range(1, 100_000_000)
.parallel()
.filter(n -> calc(n) == 89)
.count();
System.out.println(r);
}
private static int calc(int n) {
while (n != 89 && n != 1) {
int total = 0;
while (n > 0) {
total += Math.pow(n % 10, 2);
n /= 10;
}
n = total;
}
return n;
}
}
85744333
jq
The algorithm presented here caches the results for 1 ... D*81 (where D is the relevant number of digits) and uses the combinatorial approach, but to keep things relatively brief, the factorials themselves are not cached.
Part 1: Foundations
def factorial: reduce range(2;.+1) as $i (1; . * $i);
# Pick n items (with replacement) from the input array,
# but only consider distinct combinations:
def pick(n):
def pick(n; m): # pick n, from m onwards
if n == 0 then []
elif m == length then empty
elif n == 1 then (.[m:][] | [.])
else ([.[m]] + pick(n-1; m)), pick(n; m+1)
end;
pick(n;0) ;
# Given any array, produce an array of [item, count] pairs for each run.
def runs:
reduce .[] as $item
( [];
if . == [] then [ [ $item, 1] ]
else .[length-1] as $last
| if $last[0] == $item then (.[0:length-1] + [ [$item, $last[1] + 1] ] )
else . + [[$item, 1]]
end
end ) ;Part 2: The Generic Task
Count how many number chains beginning with n (where 0 < n < 10^D) end with a value 89.
def terminus:
# sum of the squared digits
def ssdigits: tostring | explode | map(. - 48 | .*.) | add;
if . == 1 or . == 89 then .
else ssdigits | terminus
end;
# Count the number of integers i in [1... 10^D] with terminus equal to 89.
def task(D):
# The max sum of squares is D*81 so return an array that will instantly
# reveal whether n|terminus is 89:
def cache:
reduce range(1; D*81+1) as $d ([false]; . + [$d|terminus == 89]);
# Compute n / (i1! * i2! * ... ) for the given combination,
# which is assumed to be in order:
def combinations(n):
runs | map( .[1] | factorial) | reduce .[] as $i (n; ./$i);
cache as $cache
| (D|factorial) as $Dfactorial
| reduce ([range(0;10)] | pick(D)) as $digits
(0;
($digits | map(.*.) | add) as $ss
| if $cache[$ss] then . + ($digits|combinations($Dfactorial))
else .
end) ;Part 3: D=8
task(8)- Output:
$ jq -M -n -f Iterated_digits_squaring_using_pick.jq
85744333
# Using jq>1.4:
# user 0m2.595s
# sys 0m0.010s
# Using jq 1.4:
# user 0m3.942s
# sys 0m0.009s
Julia
Brute force solution:
function iterate(m::Integer)
while m != 1 && m != 89
s = 0
while m > 0 # compute sum of squares of digits
m, d = divrem(m, 10)
s += d ^ 2
end
m = s
end
return m
end
itercount(k::Integer) = count(x -> iterate(x) == 89, 1:k)
More clever solution:
using Combinatorics
function itercountcombos(ndigits::Integer)
cnt = 0
f = factorial(ndigits)
# loop over all combinations of ndigits decimal digits:
for comb in combinations(1:(10+ndigits-1), ndigits)
s = 0
perms = 1
prevd = -1
rep = 1
for k = eachindex(comb) # sum digits ^ 2 and count permutations
d = comb[k] - k
s += d ^ 2
# accumulate number of permutations of repeated digits
if d == prevd
rep += 1
perms *= rep
else
prevd = d
rep = 1
end
end
if s > 0 && iterate(s) == 89
cnt += f ÷ perms # numbers we can get from digits
end
end
return cnt
end
Benchmarks
@time itercount(100_000_000)
@time itercountcombos(8)
@time itercountcombos(17)
- Output:
8.866063 seconds (4.32 k allocations: 232.908 KiB) 0.053470 seconds (101.05 k allocations: 8.729 MiB) 1.588977 seconds (12.50 M allocations: 1.536 GiB, 16.94% gc time)
Kotlin
// version 1.0.6
fun endsWith89(n: Int): Boolean {
var digit: Int
var sum = 0
var nn = n
while (true) {
while (nn > 0) {
digit = nn % 10
sum += digit * digit
nn /= 10
}
if (sum == 89) return true
if (sum == 1) return false
nn = sum
sum = 0
}
}
fun main(args: Array<String>) {
val sums = IntArray(8 * 81 + 1)
sums[0] = 1
sums[1] = 0
var s: Int
for (n in 1 .. 8)
for (i in n * 81 downTo 1)
for (j in 1 .. 9) {
s = j * j
if (s > i) break
sums[i] += sums[i - s]
}
var count89 = 0
for (i in 1 .. 8 * 81)
if (endsWith89(i)) count89 += sums[i]
println("There are $count89 numbers from 1 to 100 million ending with 89")
}
- Output:
There are 85744333 numbers from 1 to 100 million ending with 89
Lua
squares = {}
for i = 0, 9 do
for j = 0, 9 do
squares[i * 10 + j] = i * i + j * j
end
end
for i = 1, 99 do
for j = 0, 99 do
squares[i * 100 + j] = squares[i] + squares[j]
end
end
function sum_squares(n)
if n < 9999.5 then
return squares[n]
else
local m = math.floor(n / 10000)
return squares[n - 10000 * m] + sum_squares(m)
end
end
memory = {}
function calc_1_or_89(n)
local m = {}
n = memory[n] or n
while n ~= 1 and n ~= 89 do
n = memory[n] or sum_squares(n)
table.insert(m, n)
end
for _, i in pairs(m) do
memory[i] = n
end
return n
end
counter = 0
for i = 1, 100000000 do
if calc_1_or_89(i) == 89 then
counter = counter + 1
end
end
print(counter)
- Output:
85744333
Mathematica / Wolfram Language
sumDigitsSquared[n_Integer] := Total[IntegerDigits[n]^2]
stopValues = Join[{1}, NestList[sumDigitsSquared, 89, 7]];
iterate[n_Integer] :=
NestWhile[sumDigitsSquared, n, Intersection[stopValues, {#}] == {} &]
numberOfDigits = 8;
maxSum = numberOfDigits 9^2;
loopVariables =
ToExpression@Table["i" <> ToString[n], {n, numberOfDigits}];
iteratesToOne = Cases[Range@maxSum, _?(iterate[#] == 1 &)];
allIterators =
Flatten[{Reverse@#, 9}] & /@ Partition[loopVariables, 2, 1];
maxCombinations = numberOfDigits!;
ssd =
SparseArray[Table[n^2 -> numberOfDigits, {n, 9}], {maxSum}];
Do[
variables = loopVariables[[;; digitCount]];
iterators = allIterators[[;; digitCount - 1]];
Do[ssd +=
SparseArray[
Total[variables^2] ->
maxCombinations/
Times @@ (Tally[PadRight[variables, numberOfDigits]][[All,
2]]!), {maxSum}], {i, 9}, Evaluate[Sequence @@ iterators]],
{digitCount, 2, numberOfDigits}];
onesCount =
Total[Cases[
ArrayRules[ssd] /.
HoldPattern[{a_} -> b_] :> {a,
b}, {_?(MemberQ[iteratesToOne, #] &), _}][[All, 2]]];
(10^numberOfDigits - 1) - onesCount
- Output:
85744333
Nim
An extremely fast version which computes how many numbers gives a sum (starting with one digit and adding digits one by one). If a sum ends with 89, we adds the associated count to the result. As we have no need to deal with the numbers, but only with the sums of square of digits, there is no need to use big numbers.
We provide the result for 8 digits and also for 50 digits. The result is obtained in 7 ms.
import tables
iterator digits(n: int): int =
## Yield the digits starting from the unit.
var n = n
while true:
yield n mod 10
n = n div 10
if n == 0:
break
func gen(n: int): int =
## Compute the chain.
result = n
while result notin [1, 89]:
var s = 0
for d in digits(result):
inc s, d * d
result = s
func chainsEndingWith89(ndigits: Natural): Natural =
## Compute the number of chains ending with 89.
# Initialize the count table with values for one digit numbers.
var prevCount, currcount: CountTable[int]
for i in 0..9: prevcount[i * i] = 1
# Add next digits.
for _ in 2..ndigits:
# Create the next generation count array.
currcount.clear()
for val, count in prevcount:
for newdigit in 0..9:
# As 0 is included, "currcount" includes "prevcount".
currcount.inc(newdigit * newdigit + val, count)
prevcount = currcount
for val, count in currcount:
if val != 0 and gen(val) == 89:
inc result, count
echo "For 8 digits: ", chainsEndingWith89(8)
echo "For 50 digits: ", chainsEndingWith89(15)
- Output:
For 8 digits: 85744333 For 50 digits: 869339034137667
Oberon-2
MODULE DigitsSquaring;
IMPORT
Out;
VAR
i,hits89: LONGINT;
PROCEDURE Squaring(n: LONGINT): LONGINT;
VAR
d, sum: LONGINT;
BEGIN
LOOP
sum := 0;
WHILE n > 0 DO
d := n MOD 10;
INC(sum,d * d);
n := n DIV 10
END;
IF (sum = 1) OR (sum = 89) THEN EXIT END;
n := sum;
END;
RETURN sum
END Squaring;
BEGIN
hits89 := 0;
FOR i := 1 TO 100000000 DO
IF Squaring(i) = 89 THEN INC(hits89) END
END;
Out.LongInt(hits89,0);Out.Ln
END DigitsSquaring.
- Output:
85744333 real 0m12.201s user 0m12.179s sys 0m0.001s
Objeck
class Abbreviations {
function : Main(args : String[]) ~ Nil {
Count89s(1000000)->PrintLine();
Count89s(100000000)->PrintLine();
}
function : Count89s(limit : Int) ~ Int {
if(limit < 1) {
return 0;
};
result := 0;
ends := Int->New[Int->Min(limit, 9 * 9 * 9 + 2)];
for(i := 1; i < ends->Size(); i++;) {
ends[i] := i;
while(ends[i] <> 1 & ends[i] <> 89) {
ends[i] := SquareDigitSum(ends[i]);
};
if(ends[i] = 89) {
result++;
};
};
for(i := ends->Size(); i < limit; i++;) {
if(ends[SquareDigitSum(i)] = 89) {
result++;
};
};
return result;
}
function : SquareDigitSum(n : Int) ~ Int {
sum := 0;
while(n > 0) {
digit := n % 10;
sum += digit * digit;
n /= 10;
};
return sum;
}
}Output:
856,929 85,744,333
Oforth
Brute force implementation
: sq_digits(n)
while (n 1 <> n 89 <> and ) [
0 while(n) [ n 10 /mod ->n dup * + ]
->n
] n ;
: iterDigits | i | 0 100000000 loop: i [ i sq_digits 89 &= + ] . ;- Output:
85744333
PARI/GP
ssd(n)=n=digits(n); sum(i=1, #n, n[i]^2);
happy(n)=while(n>6, n=ssd(n)); n==1;
ct(n)=my(f=n!,s=10^n-1,d); forvec(v=vector(9,i,[0,n]), d=vector(9,i, if(i>8,n,v[i+1])-v[i]); if(happy(sum(i=1,9,d[i]*i^2)), s-=f/prod(i=1,9,d[i]!)/v[1]!), 1); s;
ct(8)- Output:
%1 = 85744333
Pascal
A limited, but fast implementation (up to 10e14). It calculates first all the possible sums up to cM= sqrt(MAX), with the drawback that cM must be 10^n and can only count from 10^n to 10^(2*n). Runtime: n-> 100*n => t(100*n)-> ~10*t(n) O(n) sqrt(n)
1E8 -> runtime 0..4 ms // not really measureable 1E12-> runtime 0.22 secs 1E14 -> runtime 2,7 secs 1E16 -> runtime 31,0 secs 1E18 -> runtime 354 secs // 2GByte
Tested with freepascal.
program Euler92;
const
maxdigCnt = 14;
//2* to use the sum of two square-sums without access violation
maxPoss = 2* 9*9*maxdigCnt;// every digit is 9
cM = 10*1000*1000;// 10^(maxdigCnt div 2)
IdxSqrSum = cM;//MaxPoss;//max(cM,MaxPoss);
type
tSqrSum = array[0..IdxSqrSum] of Word;
tEndsIn = array[0..maxPoss]of Byte;
tresCache = array[0..maxPoss]of Uint64;
var
aSqrDigSum : tSqrSum;
aEndsIn: tEndsIn;
aresCache : tresCache;
procedure CreateSpuareDigitSum;
var
i,j,k,l : integer;
begin
For i := 0 to 9 do
aSqrDigSum[i] := sqr(i);
k := 10;
l := k;
while k < cM do
begin
For i := 1 to 9 do
For j := 0 to k-1 do
begin
aSqrDigSum[l]:=aSqrDigSum[i]+aSqrDigSum[j];
inc(l);
end;
k := l;
end;
aSqrDigSum[l] := 1;
end;
function InitEndsIn(n:LongWord):longWord;
{fill aEndsIN recursive}
var
d,s:LongWord;
begin
IF n in [0..1] then
begin
InitEndsIn := n;
EXIT;
end;
s := aSqrDigSum[n];
{if unknown}
IF aEndsIN[s] = byte(-1) then
begin
d := InitEndsIn(s);
aEndsIN[s]:= d;
InitEndsIn := d;
end
else
InitEndsIn := aEndsIN[s];
end;
function CntSmallOnes(s:longWord;
n:longWord=cM-1):NativeUint;
var
i: longword;
begin
result := 0;
For i := cM-1 downto 0 do
result := result+aEndsIN[aSqrDigSum[i]+s];
end;
procedure Init;
var
i,j,cnt : integer;
begin
CreateSpuareDigitSum;
fillchar(aEndsIN,Sizeof(aEndsIN) ,#255);
aEndsIN[0] := 0;
aEndsIN[1]:= 1;
aEndsIN[89]:= 0;// no need to use 89
For i := 1 to maxPoss do
aEndsIN[i]:= InitEndsIN(i);
cnt := 0;
fillchar(aresCache,SizeOf(aresCache),#0);
For i := Low(tSqrSum) to high(tSqrSum) do
begin
j := aSqrDigSum[i];
If aresCache[j] = 0 then
begin
// write(i,',');
aresCache[j] := CntSmallOnes(j);
inc(cnt);
end;
end;
// writeln; writeln(cnt,' small counts out of ',cM);
end;
{
function EndsIn(n:LongWord):Word;
var
d,s:LongWord;
begin
d := n;
s := 0;
while d > High(tSqrSum) do
begin
s := s+aSqrDigSum[d Mod cM];
d := d Div cM
end;
s :=s+aSqrDigSum[d];
EndsIn := aEndsIN[s];
end;
}
function CntOnes(s: longWord;n:Int64):Int64;
var
i : Int64;
begin
writeln;
result := 0;
i := n div cM;
repeat
result := result+aresCache[s+aSqrDigSum[i]];
dec(i)
until i < 0
end;
const
upperlimit = cM*cM ;
var
Res : Int64;
begin
Init;
Res := CntOnes(0,upperlimit-1)+1;
writeln('there are ',res,' 1s ');
writeln('there are ',upperlimit-res,' 89s ');
end.
output i3 3.5 Ghz
10e18 //658 small counts out of 1000000000 there are 118226055080025491 1s there are 881773944919974509 89s real 5m54.431s user 5m53.977s 10e14 there are 13770853279685 1s there are 86229146720315 89s real 0m2.699s user 0m2.693s
PascalABC.NET
function digits(n: integer): sequence of integer;
begin
result := new List<integer>;
repeat
result := result + n mod 10;
n := n div 10;
until n = 0;
end;
function gen(n: integer): integer;
begin
result := n;
while (result <> 1) and (result <> 89) do
begin
var s := 0;
foreach var d in digits(result) do s += d * d;
result := s;
end;
end;
function chainsEndingWith89(ndigits: integer): int64;
begin
var prevcount := new Dictionary<integer, int64>;
var currcount := new Dictionary<integer, int64>;
for var i := 0 to 9 do prevcount[i * i] := 1;
loop ndigits - 1 do
begin
currcount.Clear;
foreach var prev in prevcount do
for var newdigit := 0 to 9 do
begin
var nextgen := newdigit * newdigit + prev.key;
currcount[nextgen] := currcount.get(nextgen) + prev.value;
end;
prevcount := new Dictionary<integer, int64>(currcount);
end;
foreach var curr in currcount do
if (curr.key <> 0) and (gen(curr.key) = 89) then
result += curr.value;
end;
begin
println('For 8 digits: ', chainsEndingWith89(8));
println('For 18 digits: ', chainsEndingWith89(18));
println(milliseconds, 'ms');
end.
- Output:
For 8 digits: 85744333 For 18 digits: 881773944919974509 16 ms
Perl
use warnings;
use strict;
my @sq = map { $_ ** 2 } 0 .. 9;
my %cache;
my $cnt = 0;
sub Euler92 {
my $n = 0 + join( '', sort split( '', shift ) );
$cache{$n} //= ($n == 1 || $n == 89) ? $n :
Euler92( sum( @sq[ split '', $n ] ) )
}
sub sum {
my $sum;
$sum += shift while @_;
$sum;
}
for (1 .. 100_000_000) {
++$cnt if Euler92( $_ ) == 89;
}
print $cnt;
85744333
Phix
with javascript_semantics constant MAXINT = power(2,iff(machine_bits()=32?53:64)) procedure main(integer limit) sequence sums = repeat(0,limit*81+1) sums[1] = 1 for n=1 to limit do for i=n*81 to 1 by -1 do for j=1 to 9 do integer s = j*j, i1= i+1, i1ms = i1-s if s>i then exit end if sums[i1] += sums[i1ms] end for end for atom count89 = 0 for i=1 to n*81 do integer r, digit, w = i, i1 = i+1 while w!=1 do r = 0 while w!=0 do digit = mod(w,10) r += digit*digit w = floor(w/10) end while if r=89 then count89 += sums[i1] if count89>MAXINT then printf(1,"counter overflow for 10^%d\n",n) return end if exit end if w = r end while end for printf(1,"There are %d numbers from 1 to 10^%d ending with 89\n",{count89,n}) end for end procedure atom t0 = time() main(20) ?time()-t0
- Output:
on 32 bit:
There are 7 numbers from 1 to 10^1 ending with 89 There are 80 numbers from 1 to 10^2 ending with 89 There are 857 numbers from 1 to 10^3 ending with 89 There are 8558 numbers from 1 to 10^4 ending with 89 There are 85623 numbers from 1 to 10^5 ending with 89 There are 856929 numbers from 1 to 10^6 ending with 89 There are 8581146 numbers from 1 to 10^7 ending with 89 There are 85744333 numbers from 1 to 10^8 ending with 89 There are 854325192 numbers from 1 to 10^9 ending with 89 There are 8507390852 numbers from 1 to 10^10 ending with 89 There are 84908800643 numbers from 1 to 10^11 ending with 89 There are 850878696414 numbers from 1 to 10^12 ending with 89 There are 8556721999130 numbers from 1 to 10^13 ending with 89 There are 86229146720315 numbers from 1 to 10^14 ending with 89 There are 869339034137667 numbers from 1 to 10^15 ending with 89 There are 8754780882739336 numbers from 1 to 10^16 ending with 89 counter overflow for 10^17 0.031
same on 64-bit, but ending
There are 87975303595231975 numbers from 1 to 10^17 ending with 89 There are 881773944919974509 numbers from 1 to 10^18 ending with 89 There are 8816770037940618762 numbers from 1 to 10^19 ending with 89 counter overflow for 10^20 0.109
Combinatorics version
Following the steps outlined on the talk page.
I realised I needed to do this in two stages.
Phase 1. Make sure we can count.
with javascript_semantics function comb(sequence res, set, integer n, at=1, sequence chosen={}) if length(chosen)=n then sequence digits = repeat(0,10) for i=1 to length(chosen) do integer idx = chosen[i]+1 digits[idx]+=1 end for atom p = factorial(length(chosen)) for i=1 to 10 do if digits[i] then p /= factorial(digits[i]) end if end for res = sq_add(res,{p,1}) else for i=at to length(set) do res = comb(res,set,n,i,append(deep_copy(chosen),set[i])) end for end if return res end function constant nums = {0,1,2,3,4,5,6,7,8,9} for i=1 to 8 do printf(1,"%V\n",{comb({0,0},nums,i)}) end for
Starting with the combinations method from http://rosettacode.org/wiki/Combinations_with_repetitions#Phix converted to a function, make sure we are covering all the numbers correctly by checking that we have indeed found power(10,n) of them, and show we are looking at significantly fewer combinations.
- Output:
{10,10}
{100,55}
{1000,220}
{10000,715}
{100000,2002}
{1000000,5005}
{10000000,11440}
{100000000,24310}
Phase 2. Add in the rest of the logic, as suggested count 1's in preference to 89's and subtract from 10^n to get the answer.
[PS There is an eerie similarity between this and the 2nd C version, but I swear it is not a copy, and I noticed that later.]
with javascript_semantics sequence is1 function comb(atom res, sequence set, integer n, at=1, sequence chosen={}) if length(chosen)=n then sequence digits = repeat(0,10) atom sumsq = 0 for i=1 to length(chosen) do integer ci = chosen[i] sumsq += ci*ci ci += 1 digits[ci] += 1 end for if sumsq=0 or is1[sumsq] then atom perms = factorial(length(chosen)) for i=1 to 10 do if digits[i] then perms /= factorial(digits[i]) end if end for res += perms end if else chosen = deep_copy(chosen)&0 for i=at to length(set) do chosen[$] = set[i] res = comb(res,set,n,i,chosen) end for end if return res end function procedure setis1(integer n) is1 = repeat(0,n*81) for i=1 to length(is1) do integer r, digit, w = i while 1 do r = 0 while w!=0 do digit = mod(w,10) r += digit*digit w = floor(w/10) end while if r=89 then exit end if if r=1 then is1[i] = 1 exit end if w = r end while end for end procedure constant nums = {0,1,2,3,4,5,6,7,8,9} atom t00 = time() for i=1 to 16 do atom t0 = time() setis1(i) printf(1,"There are %d numbers from 1 to 10^%d ending with 89 (%3.2fs)\n",{power(10,i)-comb(0,nums,i),i,time()-t0}) end for ?elapsed(time()-t00)
- Output:
Sadly, while still very much faster than brute force, several times slower than the translated from C version.
There are 7 numbers from 1 to 10^1 ending with 89 (0.00s) There are 80 numbers from 1 to 10^2 ending with 89 (0.00s) There are 857 numbers from 1 to 10^3 ending with 89 (0.00s) There are 8558 numbers from 1 to 10^4 ending with 89 (0.02s) There are 85623 numbers from 1 to 10^5 ending with 89 (0.00s) There are 856929 numbers from 1 to 10^6 ending with 89 (0.02s) There are 8581146 numbers from 1 to 10^7 ending with 89 (0.02s) There are 85744333 numbers from 1 to 10^8 ending with 89 (0.03s) There are 854325192 numbers from 1 to 10^9 ending with 89 (0.08s) There are 8507390852 numbers from 1 to 10^10 ending with 89 (0.16s) There are 84908800643 numbers from 1 to 10^11 ending with 89 (0.31s) There are 850878696414 numbers from 1 to 10^12 ending with 89 (0.61s) There are 8556721999130 numbers from 1 to 10^13 ending with 89 (1.16s) There are 86229146720315 numbers from 1 to 10^14 ending with 89 (2.16s) There are 869339034137667 numbers from 1 to 10^15 ending with 89 (3.77s) There are 8754780882739336 numbers from 1 to 10^16 ending with 89 (6.59s) "15s"
PicoLisp
Brute force with caching
(de *Idx1or89 (89 . 89) ((1 . 1)))
(de 1or89 (N)
(let L (mapcar format (chop N))
(if (lup *Idx1or89 (setq N (sum * L L)))
(cdr @)
(prog1
(1or89 N)
(idx '*Idx1or89 (cons N @) T) ) ) ) )Test:
(let Ones 0
(for I 100000000
(and (=1 (1or89 I)) (inc 'Ones)) )
(println (- 100000000 Ones)) )Output:
85744333
PL/I
test: procedure options (main, reorder); /* 6 August 2015 */
declare (m, n) fixed decimal (10);
declare (i, j, p, s, tally initial (0) ) fixed binary (31);
declare d fixed binary (7);
declare (start_time, finish_time, elapsed_time) float (15);
start_time = secs();
do m = 1 to 1000000;
n = m;
do until ((n = 1) | (n = 89));
p = n; s = 0;
do while (p > 0);
d = mod(p, 10);
p = p/10;
s = s + d*d;
end;
n = s;
end;
if n = 89 then tally = tally + 1;
end;
finish_time = secs();
put skip edit (Tally, ' numbers iterated to 89') (f(10), A);
elapsed_time = finish_time - start_time;
put skip edit ('Elapsed time=', elapsed_time, ' secs') (A, F(10,3));
end test;Output:
856929 numbers iterated to 89 Elapsed time= 39.280 secs
PureBasic
OpenConsole()
Procedure is89(x)
Repeat
s=0
While x : s+ x%10*x%10 : x/10 : Wend
If s=89 : ProcedureReturn 1 : EndIf
If s=1 : ProcedureReturn 0 : EndIf
x=s
ForEver
EndProcedure
Procedure main()
Dim sums(32*81+1) : sums(0)=1 : sums(1)=0
For n=1 To n+1
For i=n*81 To 1 Step -1
For j=1 To 9
s=j*j : If s>i : Break : EndIf
sums(i)+sums(i-s)
Next
Next
count89=0
For i=1 To n*81+1
If Not is89(i) : Continue : EndIf
If sums(i)>9223372036854775807-count89
PrintN("counter overflow for 10^"+Str(n))
ProcedureReturn 0
EndIf
count89+sums(i)
Next
PrintN("1->10^"+LSet(Str(n),2,Chr(32))+": "+Str(count89))
Next
EndProcedure
start=ElapsedMilliseconds()
main()
Print("elapsed milliseconds= "+Str(ElapsedMilliseconds()-start))
Input()
- Output:
1->10^1 : 7 1->10^2 : 80 1->10^3 : 857 1->10^4 : 8558 1->10^5 : 85623 1->10^6 : 856929 1->10^7 : 8581146 1->10^8 : 85744333 1->10^9 : 854325192 1->10^10: 8507390852 1->10^11: 84908800643 1->10^12: 850878696414 1->10^13: 8556721999130 1->10^14: 86229146720315 1->10^15: 869339034137667 1->10^16: 8754780882739336 1->10^17: 87975303595231975 1->10^18: 881773944919974509 1->10^19: 8816770037940618762 counter overflow for 10^20 elapsed milliseconds= 9
OpenConsole()
Procedure sum_square_digits(n)
num=n : sum=0
While num>0
digit=num%10
num=(num-digit)/10
sum+ digit*digit
Wend
ProcedureReturn sum
EndProcedure
Procedure main()
i=0 : result=0 : count=0
For i=1 To 1e8
If Not i=1 Or Not i=89
result=sum_square_digits(i)
Else
result=i
EndIf
While Not result=1 And Not result=89
result=sum_square_digits(result)
Wend
If result=89 : count+1 : EndIf
Next
PrintN(Str(count))
EndProcedure
start=ElapsedMilliseconds()
main()
Print("elapsed milliseconds: "+Str(ElapsedMilliseconds()-start))
Input()
- Output:
85744333 elapsed milliseconds: 65553
Python
Combinatorics
Translation of D
from math import ceil, log10, factorial
def next_step(x):
result = 0
while x > 0:
result += (x % 10) ** 2
x /= 10
return result
def check(number):
candidate = 0
for n in number:
candidate = candidate * 10 + n
while candidate != 89 and candidate != 1:
candidate = next_step(candidate)
if candidate == 89:
digits_count = [0] * 10
for d in number:
digits_count[d] += 1
result = factorial(len(number))
for c in digits_count:
result /= factorial(c)
return result
return 0
def main():
limit = 100000000
cache_size = int(ceil(log10(limit)))
assert 10 ** cache_size == limit
number = [0] * cache_size
result = 0
i = cache_size - 1
while True:
if i == 0 and number[i] == 9:
break
if i == cache_size - 1 and number[i] < 9:
number[i] += 1
result += check(number)
elif number[i] == 9:
i -= 1
else:
number[i] += 1
for j in xrange(i + 1, cache_size):
number[j] = number[i]
i = cache_size - 1
result += check(number)
print result
main()
- Output:
85744333
The run-time is less than half a second.
Translation of Ruby
from itertools import combinations_with_replacement
from array import array
from time import clock
D = 8
F = [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000]
def b(n):
yield 1
for g in range(1,n+1):
gn = g
res = 0
while gn > 0:
gn,rem = divmod(gn,10)
res += rem**2
if res==89:
yield 0
else:
yield res
N = array('I',b(81*D))
for n in range(2,len(N)):
q = N[n]
while q>1:
q = N[q]
N[n] = q
es = clock()
z = 0
for n in combinations_with_replacement(range(10),D):
t = 0
for g in n:
t += g*g
if N[t] == 0:
continue
t = [0,0,0,0,0,0,0,0,0,0]
for g in n:
t[g] += 1
t1 = F[D]
for g in t:
t1 /= F[g]
z += t1
ee = clock() - es
print "\nD==" + str(D) + "\n " + str(z) + " numbers produce 1 and " + str(10**D-z) + " numbers produce 89"
print "Time ~= " + str(ee) + " secs"
- Output:
D==8 14255667 numbers produce 1 and 85744333 numbers produce 89 Time ~= 0.14 secs D==11 15091199357 numbers produce 1 and 84908800643 numbers produce 89 Time ~= 1.12 secs D==14 13770853279685 numbers produce 1 and 86229146720315 numbers produce 89 Time ~= 7.46 secs D==17 12024696404768025 numbers produce 1 and 87975303595231975 numbers produce 89 Time ~= 34.16 secs
Python: Simple caching
>>> from functools import lru_cache
>>> @lru_cache(maxsize=1024)
def ids(n):
if n in {1, 89}: return n
else: return ids(sum(int(d) ** 2 for d in str(n)))
>>> ids(15)
89
>>> [ids(x) for x in range(1, 21)]
[1, 89, 89, 89, 89, 89, 1, 89, 89, 1, 89, 89, 1, 89, 89, 89, 89, 89, 1, 89]
>>> sum(ids(x) == 89 for x in range(1, 100000000))
85744333
>>>
This took a much longer time, in the order of hours.
Python: Enhanced caching
Notes that the order of digits in a number does not affect the final result so caches the digits of the number in sorted order for more hits.
>>> from functools import lru_cache
>>> @lru_cache(maxsize=1024)
def _ids(nt):
if nt in {('1',), ('8', '9')}: return nt
else: return _ids(tuple(sorted(str(sum(int(d) ** 2 for d in nt)))))
>>> def ids(n):
return int(''.join(_ids(tuple(sorted(str(n))))))
>>> ids(1), ids(15)
(1, 89)
>>> [ids(x) for x in range(1, 21)]
[1, 89, 89, 89, 89, 89, 1, 89, 89, 1, 89, 89, 1, 89, 89, 89, 89, 89, 1, 89]
>>> sum(ids(x) == 89 for x in range(1, 100000000))
85744333
>>> _ids.cache_info()
CacheInfo(hits=99991418, misses=5867462, maxsize=1024, currsize=1024)
>>>
This took tens of minutes to run.
Count digit sums
If we always count up to powers of 10, it's faster to just record how many different numbers have the same digit square sum. The check89() function is pretty simple-minded, because it doesn't need to be fancy here.
from __future__ import print_function
from itertools import count
def check89(n):
while True:
n, t = 0, n
while t: n, t = n + (t%10)**2, t//10
if n <= 1: return False
if n ==89: return True
a, sq, is89 = [1], [x**2 for x in range(1, 10)], [False]
for n in range(1, 500):
b, a = a, a + [0]*81
is89 += map(check89, range(len(b), len(a)))
for i,v in enumerate(b):
for s in sq: a[i + s] += v
x = sum(a[i] for i in range(len(a)) if is89[i])
print("10^%d" % n, x)
QBasic
REM Version 1: Brute force
T = TIMER
n1 = 0
FOR i = 1 TO 10000000
j = i
DO
k = 0
DO
k = INT(k + (j MOD 10) ^ 2)
j = INT(j / 10)
LOOP WHILE j <> 0
j = k
LOOP UNTIL j = 89 OR j = 1
IF j > 1 THEN n1 = n1 + 1
NEXT i
PRINT USING "Version 1: ####### in ##.### seconds."; n1; (TIMER - T)
END
Quackery
[ abs 0 swap
[ base share /mod
dup *
rot + swap
dup 0 = until ]
drop ] is digitsquaresum ( n --> n )
[ dup 1 != while
dup 89 != while
digitsquaresum
again ] is chainends ( n --> n )
0 1000000 times
[ i^ 1+ chainends
89 = + ]
echo- Output:
856929
Racket
This contains two versions (in one go). The naive version which can (and should, probably) be used for investigating a single number. The second version can count the IDSes leading to 89 for powers of 10.
#lang racket
;; Tim-brown 2014-09-11
;; The basic definition.
;; It is possible to memoise this or use fixnum (native) arithmetic, but frankly iterating over a
;; hundred million, billion, trillion numbers will be slow. No matter how you do it.
(define (digit^2-sum n)
(let loop ((n n) (s 0))
(if (= 0 n) s (let-values ([(q r) (quotient/remainder n 10)]) (loop q (+ s (sqr r)))))))
(define (iterated-digit^2-sum n)
(match (digit^2-sum n) [0 0] [1 1] [89 89] [(app iterated-digit^2-sum rv) rv]))
;; Note that: ids(345) = ids(354) = ids(435) = ids(453) = ids(534) = ids(543) = 50 --> 89
;; One calculation does for 6 candidates.
;; The plan:
;; - get all the ordered combinations of digits including 0's which can be used both as digits and
;; "padding" digits in the most significant digits. (n.b. all-zeros is not in the range to be
;; tested and should be dropped)
;; - find the digit sets that have an IDS of 89
;; - find out how many combinations there are of these digits
;; output: a list of n-digits long lists containing all of the digit combinations.
;; a smart bunny would figure out the sums of the digits as they're generated but I'll plod
;; along step-by-step. a truly smart bunny would also count the combinations. that said, I
;; don't think I do much unnecessary computation here.
(define (all-digit-lists n-digits)
(define (inner remain acc least-digit)
(cond
[(zero? remain) (list (list))]
[(= least-digit 10) null]
[else
(for*/list
((ld+ (in-range least-digit 10))
(rgt (in-list (inner (sub1 remain) empty ld+))))
(append acc (cons ld+ rgt)))]))
(inner n-digits '() 0))
;; We calculate IDS differently since we're presented with a list of digits rather than a number
(define (digit-list-IDS c)
(define (digit-combo-IDS c)
(apply + (map sqr c)))
(iterated-digit^2-sum (digit-combo-IDS c)))
;; ! (factiorial) -- everyone's favourite combinatorial function! (that's just an exclamation mark)
;; there's one in (require math/number-theory) for any heavy lifting, but we're not or I could import
;; it from math/number-theory -- but this is about all I need. A lookup table is going to be faster
;; than a more general function.
(define (! n)
(case n [(0 1) 1] [(2) 2] [(3) 6] [(4) 24] [(5) 120] [(6) 720] [(7) 5040] [(8) 40320] [(9) 362880]
[else (* n (! (sub1 n)))] ; I expect this clause'll never be called
))
;; We need to count the permutations -- digits are in order so we can use the tail (cdr) function for
;; determining my various k's. See: https://en.wikipedia.org/wiki/Combination
(define (count-digit-list-permutations c #:length (l (length c)) #:length! (l! (! l)))
(let loop ((c c) (i 0) (prev -1 #;"never a digit") (p l!))
(match c
[(list) (/ p (! i))]
[(cons (== prev) d) (loop d (+ i 1) prev p)]
[(cons a d) (loop d 1 a (/ p (! i)))])))
;; Wrap it all up in a neat function
(define (count-89s-in-100... n)
(define n-digits (order-of-magnitude n))
(define combos (drop (all-digit-lists n-digits) 1)) ; don't want first one which is "all-zeros"
(for/sum ((c (in-list combos)) #:when (= 89 (digit-list-IDS c)))
(count-digit-list-permutations c #:length n-digits)))
(displayln "Testing permutations:")
(time (printf "1000000:\t~a~%" (count-89s-in-100... 1000000)))
(time (printf "100000000:\t~a~%" (count-89s-in-100... 100000000)))
(time (printf "1000000000:\t~a~%" (count-89s-in-100... 1000000000)))
(time (printf "1000000000000:\t~a~%" (count-89s-in-100... 1000000000000)))
(newline)
;; Do these last, since the 10^8 takes longer than my ADHD can cope with
(displayln "Testing one number at a time (somewhat slower):")
(time (printf "1000000:\t~a~%" (for/sum ((n (in-range 1 1000000))
#:when (= 89 (iterated-digit^2-sum n))) 1)))
(time (printf "100000000:\t~a~%" (for/sum ((n (in-range 1 100000000))
#:when (= 89 (iterated-digit^2-sum n))) 1)))
{module+ test
(require tests/eli-tester)
[test
(iterated-digit^2-sum 15) => 89
(iterated-digit^2-sum 7) => 1
(digit-combo-perms '()) => 1
(digit-combo-perms '(1 2 3)) => 6
(digit-combo-perms '(1 1 3)) => 3
(for/sum ((n (in-range 1 1000000)) #:when (= 89 (iterated-digit^2-sum n))) 1) => 856929
(all-digit-lists 1) => '((0) (1) (2) (3) (4) (5) (6) (7) (8) (9))
(length (all-digit-lists 2)) => 55
(length (all-digit-lists 3)) => 220
(count-89s-in-100... 1000000) => 856929]
}
- Output:
Testing permutations: 1000000: 856929 cpu time: 8 real time: 8 gc time: 0 100000000: 85744333 cpu time: 44 real time: 43 gc time: 0 1000000000: 854325192 cpu time: 112 real time: 110 gc time: 20 1000000000000: 850878696414 cpu time: 1108 real time: 1110 gc time: 472 Testing one number at a time (somewhat slower): 1000000: 856929 cpu time: 1168 real time: 1171 gc time: 0 100000000: 85744333 cpu time: 130720 real time: 130951 gc time: 0
Ok, so maybe 131 seconds is not so flattering -- but I have not memoised or anything fancy like that, because even doing that isn't going to come anywhere near competing with 44ms.
Raku
(formerly Perl 6) This fairly abstract version does caching and filtering to reduce the number of values it needs to check and moves calculations out of the hot loop, but is still interminably slow... even for just up to 1,000,000.
constant @sq = ^10 X** 2;
my $cnt = 0;
sub Euler92($n) {
$n == any(1,89) ?? $n !!
(state %){$n} //= Euler92( [+] @sq[$n.comb] )
}
for 1 .. 1_000_000 -> $n {
my $i = +$n.comb.sort.join;
++$cnt if Euler92($i) == 89;
}
say $cnt;
- Output:
856929
All is not lost, however. Through the use of gradual typing, Raku scales down as well as up, so this jit-friendly version is performant enough to brute force the larger calculation:
my @cache;
@cache[1] = 1;
@cache[89] = 89;
sub Euler92(int $n) {
$n < 649 # 99,999,999 sums to 648, so no point remembering more
?? (@cache.AT-POS($n) //= ids($n))
!! ids($n)
}
sub ids(int $num --> int) {
my int $n = $num;
my int $ten = 10;
my int $sum = 0;
my int $t;
my int $c;
repeat until $n == 89 or $n == 1 {
$sum = 0;
repeat {
$t = $n div $ten;
$c = $n - $t * $ten;
$sum = $sum + $c * $c;
} while $n = $t;
$n = @cache.AT-POS($sum) // $sum;
}
$n;
}
my int $cnt = 0;
for 1 .. 100_000_000 -> int $n {
$cnt = $cnt + 1 if Euler92($n) == 89;
}
say $cnt;
- Output:
85744333
Which is better, but is still pretty slow.
The biggest gains are often from choosing the right algorithm.
sub endsWithOne($n --> Bool) {
my $digit;
my $sum = 0;
my $nn = $n;
loop {
while $nn > 0 {
$digit = $nn % 10;
$sum += $digit²;
$nn = $nn div 10;
}
return True if $sum == 1;
return False if $sum == 89;
$nn = $sum;
$sum = 0;
}
}
my $k = 8; # 10**8
my @sums is default(0) = 1,0;
my $s;
for 1 .. $k -> $n {
for $n*81 … 1 -> $i {
for 1 .. 9 -> $j {
$s = $j²;
last if $s > $i;
@sums[$i] += @sums[$i - $s];
}
}
}
my $ends-with-one = sum flat @sums[(1 .. $k*81).grep: { endsWithOne($_) }], +endsWithOne(10**$k);
say 10**$k - $ends-with-one;
- Output:
85744333
REXX
{Both REXX versions don't depend on a specific end-number.}
with memoization
/*REXX program performs the squaring of iterated digits (until the sum equals 1 or 89).*/
parse arg n . /*obtain optional arguments from the CL*/
if n=='' | n=="," then n=10 * 1000000 /*Not specified? Then use the default.*/
!.=0; do m=1 for 9; !.m=m**2; end /*m*/ /*build a short-cut for the squares. */
a.=.; #.=!. /*intermediate counts of some numbers. */
do j=1 for n; x=j /* [?] process the numbers in the range*/
do q=1 until s==89 | s==1; s=0 /*add sum of the squared decimal digits*/
do until x=='' /*process each of the dec. digits in X.*/
parse var x _ +1 x; s=s + !._ /*get a digit; sum the fast square, */
end /*until x ··· */ /* [?] S=is sum of the squared digits.*/
z.q=s /*assign sum to a temporary auxiliary. */
if a.s\==. then do; s=a.s; leave; end /*Found a previous sum? Then use that.*/
x=s /*substitute the sum for the "new" X. */
end /*q*/ /* [?] keep looping 'til S= 1 or 89.*/
do f=1 for q; _=a.f; a._=s /*use the auxiliary arrays (for lookup)*/
end /*f*/
#.s=#.s + 1 /*bump the counter for the 1's or 89's.*/
end /*j*/
do k=1 by 88 for 2; @k=right('"'k'"', 5) /*display two results; define a literal*/
say 'count of' @k " chains for all natural numbers up to " n ' is:' #.k
end /*k*/ /*stick a fork in it, we're all done. */
- output when using the default input:
(ten million)
count of "1" chains for all natural numbers up to 10000000 is: 1418854 count of "89" chains for all natural numbers up to 10000000 is: 8581146
process in chunks
This version is about 2½ times faster than the previous REXX version.
/*REXX program performs the squaring of iterated digits (until the sum equals 1 or 89).*/
parse arg n . /*obtain optional arguments from the CL*/
if n=='' | n=="," then n=10 * 1000000 /*Not specified? Then use the default.*/
!.=0; do m=1 for 9; !.m=m**2; end /*m*/ /*build a short-cut for the squares. */
$.=.; $.0=0; $.00=0; $.000=0; $.0000=0; @.=$. /*short-cuts for sub-group summations. */
#.=0 /*count of 1 and 89 results so far.*/
do j=1 for n; s=sumDs(j) /* [?] process each number in a range.*/
#.s=#.s + 1 /*bump the counter for 1's or 89's. */
end /*j*/
do k=1 by 88 for 2; @=right('"'k'"', 5) /*display two results; define a literal*/
say 'count of' @ " chains for all natural numbers up to " n ' is:' #.k
end /*k*/ /*stick a fork in it, we're all done. */
exit /*stick a fork in it, we're all done. */
/*--------------------------------------------------------------------------------------*/
sumDs: parse arg z; chunk=3 /*obtain the number (for adding digits)*/
p=0 /*set partial sum of the decimal digits*/
do m=1 by chunk to length(z) /*process the number, in chunks of four*/
y=substr(z, m, chunk) /*extract a 4-byte chunk of the number.*/
if @.y==. then do; oy=y; a=0 /*Not done before? Then sum the number*/
do until y=='' /*process each of the dec. digits in Y.*/
parse var y _ +1 y /*obtain a decimal digit; add it to A.*/
a=a + !._ /*obtain a decimal digit; add it to A.*/
end /*until y ···*/ /* [?] A = is the sum of squared digs*/
@.oy=a /*mark original Y as being summed. */
end
else a=@.y /*use the pre-summed digits of Y. */
p=p + a /*add all the parts of number together.*/
end /*m*/
if $.p\==. then return $.p /*Computed before? Then use the value.*/
y=p /*use a new copy of P. */
do until s==1 | s==89; s=0 /*add the squared decimal digits of P.*/
do until y=='' /*process each decimal digits in X.*/
parse var y _ +1 y; s=s + !._ /*get a dec. digit; sum the fast square*/
end /*until y ···*/ /* [?] S = is sum of the squared digs.*/
y=s /*substitute the sum for a "new" X. */
end /*until s ···*/ /* [?] keep looping 'til S=1 or 89.*/
$.p=s /*use this for memoization for the sum.*/
return s
- output when using the input of: 100000000
(one hundred million)
count of "1" chains for all natural numbers up to 100000000 is 14255667 count of "89" chains for all natural numbers up to 100000000 is 85744333
Ring
nr = 1000
num = 0
for n = 1 to nr
sum = 0
for m = 1 to len(string(n))
sum += pow(number(substr(string(n),m,1)),2)
if sum = 89 num += 1 see "" + n + " " + sum + nl ok
next
next
see "Total under 1000 is : " + num + nlOutput:
58 89 85 89 229 89 267 89 276 89 292 89 348 89 384 89 438 89 483 89 508 89 580 89 581 89 582 89 583 89 584 89 585 89 586 89 587 89 588 89 589 89 627 89 672 89 726 89 762 89 805 89 834 89 843 89 850 89 851 89 852 89 853 89 854 89 855 89 856 89 857 89 858 89 859 89 922 89 Total under 1000 is : 41
RPL
The IS89? subroutine handles unsigned integers rather than floating point numbers: it is a little bit longer, but rounding errors are then avoided.
| RPL code | Comment |
|---|---|
≪ { # 1d # 89d } → goal
≪ R→B DO
0 SWAP
WHILE DUP # 0d ≠ REPEAT
DUP 10 / SWAP OVER 10 * -
DUP * ROT + SWAP
END DROP
UNTIL goal OVER POS END
# 89d ==
≫ ≫ 'IS89?' STO
≪ → ndig
≪ { } ndig 81 * 1 + + 0 CON 1 1 PUT 'SUMS' STO
1 ndig FOR n
n 81 * 1 FOR ii
1 9 FOR j
j SQ
IF DUP ii > THEN DROP 9 'j' STO
ELSE
SUMS ii 1 + GET LAST 4 ROLL - GET +
'SUMS' ii 1 + ROT PUT
END
NEXT -1 STEP NEXT
0
2 SUMS SIZE 1 GET FOR j
IF j 1 - IS89? THEN SUMS j GET + END NEXT
≫ ≫ 'CNT89' STO
|
IS89? ( x → boolean ) convert to integer and loop s = 0 while x ≠ 0 get last digit square it and add it to s clean stack until x = 1 or x = 89 return boolean CNT89 ( nb_digits → count ) sums[32*81 + 1] = {1, 0}; for (int n = 1; ; n++) { for (int i = n*81; i; i--) { for (int j = 1; j < 10; j++) { int s = j*j; if (s > i) break; sums[i+1] += sums[i-s+1]; count89 = 0; for (int j = 1; j < n*81 + 1; j++) { if (!js89(i)) continue; count89 += sums[j+1]; } |
8 CNT89
- Output:
1: 85744333
Runs in 49 seconds on a iPhone Xr. A basic HP-28 needs 930 seconds for 3 CNT89, giving 85, whilst the emulator needs only 5 seconds, meaning that the 8 digits would probably need 2 h 30 on the HP-28.
Ruby
# Count how many number chains for Natural Numbers < 10**d end with a value of 1.
def iterated_square_digit(d)
f = Array.new(d+1){|n| (1..n).inject(1, :*)} #Some small factorials
g = -> (n) { res = n.digits.sum{|d| d*d}
res==89 ? 0 : res }
#An array: table[n]==0 means that n translates to 89 and 1 means that n translates to 1
table = Array.new(d*81+1){|n| n.zero? ? 1 : (i=g.call(n))==89 ? 0 : i}
table.collect!{|n| n = table[n] while n>1; n}
z = 0 #Running count of numbers translating to 1
[*0..9].repeated_combination(d) do |rc| #Iterate over unique digit combinations
next if table[rc.inject(0){|g,n| g+n*n}].zero? #Count only ones
nn = [0] * 10 #Determine how many numbers this digit combination corresponds to
rc.each{|n| nn[n] += 1}
z += nn.inject(f[d]){|gn,n| gn / f[n]} #Add to the count of numbers terminating in 1
end
puts "\nd=(#{d}) in the range 1 to #{10**d-1}",
"#{z} numbers produce 1 and #{10**d-1-z} numbers produce 89"
end
[8, 11, 14, 17].each do |d|
t0 = Time.now
iterated_square_digit(d)
puts " #{Time.now - t0} sec"
end
- Output:
d=(8) in the range 1 to 99999999 14255667 numbers produce 1 and 85744332 numbers produce 89 0.116007 sec d=(11) in the range 1 to 99999999999 15091199357 numbers produce 1 and 84908800642 numbers produce 89 0.921052 sec d=(14) in the range 1 to 99999999999999 13770853279685 numbers produce 1 and 86229146720314 numbers produce 89 5.503315 sec d=(17) in the range 1 to 99999999999999999 12024696404768025 numbers produce 1 and 87975303595231974 numbers produce 89 24.337392 sec
Rust
These are two naive solutions, one with lots of redundant calculations (memoizationless recursion) and one with a few precomputed values. All digit square sums are no greater than 648 for numbers < 100_000_000.
Both are slow algorithms, however, Rust is among faster languages, so this doesn't take minutes or hours.
Naive Recursion
fn digit_square_sum(mut num: usize) -> usize {
let mut sum = 0;
while num != 0 {
sum += (num % 10).pow(2);
num /= 10;
}
sum
}
fn last_in_chain(num: usize) -> usize {
match num {
1 | 89 => num,
_ => last_in_chain(digit_square_sum(num)),
}
}
fn main() {
let count = (1..100_000_000).filter(|&n| last_in_chain(n) == 89).count();
println!("{}", count);
}
- Output:
85744333
Runtime: 6s on a 2500k @ 4Ghz
With precomputation
fn dig_sq_sum(mut num : usize ) -> usize {
let mut sum = 0;
while num != 0 {
sum += (num % 10).pow(2);
num /= 10;
}
sum
}
fn last_in_chain(num: usize) -> usize {
match num {
0 => 0,
1 | 89 => num,
_ => last_in_chain(dig_sq_sum(num)),
}
}
fn main() {
let prec: Vec<_> = (0..649).map(|n| last_in_chain(n)).collect();
let count = (1..100_000_000).filter(|&n| prec[dig_sq_sum(n)] == 89).count();
println!("{}", count);
}
Runtime: 1.7s on a 2500k @ 4Ghz
- Output:
85744333
Scala
Naïve Version, conventional iteration and (tail) recursive in one
- Output:
Best seen running in your browser either by ScalaFiddle (ES aka JavaScript, non JVM) or Scastie (remote JVM) to compare the run times.
import scala.annotation.tailrec
object Euler92 extends App {
override val executionStart = compat.Platform.currentTime
@tailrec
private def calcRec(i: Int): Int = {
@tailrec
def iter0(n: Int, total: Int): Int =
if (n > 0) {
val rest = n % 10
iter0(n / 10, total + rest * rest)
}
else total
if (i == 89 || i == 1) i else calcRec(iter0(i, 0))
}
private def calcConv(i: Int) = {
var n: Int = i
while (n != 89 && n != 1) {
var total = 0
while (n > 0) {
val x = n % 10
total += (x * x)
n /= 10
}
n = total
}
n
}
println((1 until 100000000).par.count(calcConv(_) == 89))
println(s"Runtime conventional loop.[total ${compat.Platform.currentTime - executionStart} ms]")
val executionStart0 = compat.Platform.currentTime
println((1 until 100000000).par.count(calcRec(_) == 89))
println(s"Runtime recursive loop. [total ${compat.Platform.currentTime - executionStart0} ms]")
}
Sidef
func digit_square_sum_iter(n) is cached {
if ((n == 1) || (n == 89)) {
return n
}
__FUNC__(n.digits.sum { .sqr })
}
say (1..1e6 -> count_by { digit_square_sum_iter(_) == 89 })
- Output:
856929
Swift
With nth power support.
import BigInt
func is89(_ n: Int) -> Bool {
var n = n
while true {
var s = 0
repeat {
s += (n%10) * (n%10)
n /= 10
} while n > 0
if s == 89 {
return true
} else if s == 1 {
return false
}
n = s
}
}
func iterSquare(upToPower pow: Int) {
var sums = [BigInt](repeating: 0, count: pow * 81 + 1)
sums[0] = 1
for n in 1...pow {
var i = n * 81
while i > 0 {
for j in 1..<10 {
let s = j * j
guard s <= i else { break }
sums[i] += sums[i-s]
}
i -= 1
}
var count89 = BigInt(0)
for x in 1..<n*81 + 1 {
guard is89(x) else { continue }
count89 += sums[x]
}
print("1->10^\(n): \(count89)")
}
}
iterSquare(upToPower: 8)
- Output:
1->10^1: 7 1->10^2: 80 1->10^3: 857 1->10^4: 8558 1->10^5: 85623 1->10^6: 856929 1->10^7: 8581146 1->10^8: 85744333
Tcl
All three versions below produce identical output (85744333), but the third is fastest and the first is the slowest, both by substantial margins.
Very Naïve Version
proc ids n {
while {$n != 1 && $n != 89} {
set n [tcl::mathop::+ {*}[lmap x [split $n ""] {expr {$x**2}}]]
}
return $n
}
for {set i 1} {$i <= 100000000} {incr i} {
incr count [expr {[ids $i] == 89}]
}
puts $count
Intelligent Version
Conversion back and forth between numbers and strings is slow. Using math operations directly is much faster (around 4 times in informal testing).
proc ids n {
while {$n != 1 && $n != 89} {
for {set m 0} {$n} {set n [expr {$n / 10}]} {
incr m [expr {($n%10)**2}]
}
set n $m
}
return $n
}
for {set i 1} {$i <= 100000000} {incr i} {
incr count [expr {[ids $i] == 89}]
}
puts $count
Substantially More Intelligent Version
Using the observation that the maximum value after 1 step is obtained for 999999999, which is . Thus, running one step of the reduction and then using a lookup table (which we can construct quickly at the start of the run, and which has excellent performance) is much faster overall, approximately 3–4 times than the second version above (and over 12 times faster than the first version).
- Donald, you have 1 too many 9's the value after step 1 is 81*8 = 648. Not that that is the problem here, you can not afford to go around this loop 100 million times. Notice that IDS[21] == IDS[12], IDS[123] == IDS[132] == IDS[213} ... etc, etc. The Ruby version takes about a tenth of a second.--Nigel Galloway (talk) 12:47, 31 August 2014 (UTC)
# Basic implementation
proc ids n {
while {$n != 1 && $n != 89} {
for {set m 0} {$n} {set n [expr {$n / 10}]} {
incr m [expr {($n%10)**2}]
}
set n $m
}
return $n
}
# Build the optimised version
set body {
# Microoptimisation to avoid an unnecessary alloc in the loop
for {set m 0} {$n} {set n [expr {"$n[unset n]" / 10}]} {
incr m [expr {($n%10)**2}]
}
}
set map 0
for {set i 1} {$i <= 729} {incr i} {
lappend map [ids $i]
}
proc ids2 n [append body "return \[lindex [list $map] \$m\]"]
# Put this in a lambda context for a little extra speed.
apply {{} {
set count 0
for {set i 1} {$i <= 100000000} {incr i} {
incr count [expr {[ids2 $i] == 89}]
}
puts $count
}}
VBScript
start_time = Now
cnt = 0
For i = 1 To 100000000
n = i
sum = 0
Do Until n = 1 Or n = 89
For j = 1 To Len(n)
sum = sum + (CLng(Mid(n,j,1))^2)
Next
n = sum
sum = 0
Loop
If n = 89 Then
cnt = cnt + 1
End If
Next
end_time = Now
WScript.Echo "Elapse Time: " & DateDiff("s",start_time,end_time) &_
vbCrLf & "Count: " & cnt
- Output:
Elapse time is in seconds. Friends don't let friends do this in VBScript. :-)
Elapse Time: 2559 Count: 85744333
Wren
var endsWith89 = Fn.new { |n|
var digit = 0
var sum = 0
while (true) {
while (n > 0) {
digit = n % 10
sum = sum + digit*digit
n = (n/10).floor
}
if (sum == 89) return true
if (sum == 1) return false
n = sum
sum = 0
}
}
var start = System.clock
var sums = List.filled(8*81 + 1, 0)
sums[0] = 1
sums[1] = 0
var s = 0
for (n in 1..8) {
for (i in n*81..1) {
for (j in 1..9) {
s = j * j
if (s > i) break
sums[i] = sums[i] + sums[i-s]
}
}
if (n == 8) {
var count89 = 0
for (i in 1..n*81) {
if (endsWith89.call(i)) count89 = count89 + sums[i]
}
System.print("There are %(count89) numbers from 1 to 100 million ending with 89.")
}
}
System.print("Took %(((System.clock - start)*1000).round) milliseconds.")
- Output:
Timing for Intel Core i7-8565U machine running Ubuntu 18.04.
There are 85744333 numbers from 1 to 100 million ending with 89. Took 4 milliseconds.
X86 Assembly
section .data
count dd 0
section .text
global _main
_main:
mov ecx, 1
looping:
mov eax, ecx ;pass parameter in eax
push ecx
call doMath
pop ecx
add [count], eax ;doMath returns 0 or 1 in eax
inc ecx
cmp ecx, 100000001
jl looping
mov eax, count ;returns memory address of count
ret
addSquaredDigits:
push ebx
mov ebx, 0
mov esi, 10
looping2:
xor edx, edx ;clear edx for division
div esi ;div by 10 to get last digit in edx
mov ecx, eax ;save parameter
mov eax, edx ; get last digit
mul eax ;square last digit
add ebx, eax ;add the square to the result
jecxz aSDend ;if the parameter is 0 we have all digits
mov eax, ecx ;restore parameter before looping
jmp looping2
aSDend:
mov eax, ebx ;move result to return register
pop ebx
ret
doMath:
looping3:
call addSquaredDigits ;do until eax is 89 or 1
cmp eax, 89
je ret1
cmp eax, 1
je ret0
jmp looping3
ret1: ;if eax == 89 we return 1 -> inc count
mov eax, 1
ret
ret0: ;if eax == 1 we return 0 -> don't inc count
mov eax, 0
ret
XPL0
Simple minded brute force takes 30 seconds on Pi4.
int C, N, M, S;
[C:= 0;
for N:= 1 to 100_000_000-1 do
[M:= N;
loop [S:= 0;
repeat M:= M/10;
S:= S + rem(0)*rem(0);
until M = 0;
if S = 89 then
[C:= C+1; quit];
if S = 1 then quit;
M:= S;
];
];
IntOut(0, C);
]- Output:
85744333
zkl
Using brute force is a never ending process so need to be clever, which takes under a second.
fcn check(number){ // a list of digits: 13 is L(0,0,0,0,0,0,1,3)
candidate:=number.reduce(fcn(sum,n){ sum*10 + n },0); // digits to int
while(candidate != 89 and candidate != 1) // repeatedly sum squares of digits
{ candidate = candidate.split().reduce(fcn(sum,c){ sum + c*c },0); }
if(candidate == 89){ // count permutations of these digits, they all sum to 89
digitsCount:=List(0,0,0,0,0,0,0,0,0,0);
foreach d in (number){ digitsCount[d] += 1; }
return(digitsCount.reduce(fcn(r,c){ r/factorial(c) },cacheBang)); // cacheBang==number.len()!
}
0 // this number doesn't sum to 89 (ie sums to 1)
}
fcn factorial(n) { (1).reduce(n,fcn(N,n){ N*n },1) }
limit:=0d100_000_000; cacheSize:=limit.toFloat().log10().ceil().toInt();
number:=(0).pump(cacheSize,List().write,0); // list of zeros
result:=0; i:=cacheSize - 1;
var cacheBang=factorial(cacheSize); //== number.len()!
while(True){ // create numbers s.t. no set of digits is repeated
if(i == 0 and number[i] == 9) break;
if(i == cacheSize - 1 and number[i] < 9){ number[i] += 1; result += check(number); }
else if(number[i] == 9) i -= 1;
else{
number[i] += 1;
foreach j in ([i + 1 .. cacheSize - 1]){ number[j] = number[i]; }
i = cacheSize - 1;
result += check(number);
}
}
println(result);- Output:
85744333
ZX Spectrum Basic
Very, very slow. Use a ZX Spectrum emulator and run with maximum speed option enabled.
10 LET n=0
20 FOR i=1 TO 1000
30 LET j=i
40 LET k=0
50 LET k=INT (k+FN m(j,10)^2)
60 LET j=INT (j/10)
70 IF j<>0 THEN GO TO 50
80 LET j=k
90 IF j=89 OR j=1 THEN GO TO 100
95 GO TO 40
100 IF j>1 THEN LET n=n+1
110 NEXT i
120 PRINT "Version 1: ";n
200 DEF FN m(a,b)=a-INT (a/b)*b: REM modulo
- Output:
Version 1: 857
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