Casting out nines
Write a procedure (say ) which implements Casting Out Nines as described by returning the checksum for . Demonstrate the procedure using the examples given there, or others you may consider lucky.
You are encouraged to solve this task according to the task description, using any language you may know.
- Task (in three parts)
- Part 1
Note that this function does nothing more than calculate the least positive residue, modulo 9. Many of the solutions omit Part 1 for this reason. Many languages have a modulo operator, of which this is a trivial application.
With that understanding, solutions to Part 1, if given, are encouraged to follow the naive pencil-and-paper or mental arithmetic of repeated digit addition understood to be "casting out nines", or some approach other than just reducing modulo 9 using a built-in operator. Solutions for part 2 and 3 are not required to make use of the function presented in part 1.
- Part 2
Notwithstanding past Intel microcode errors, checking computer calculations like this would not be sensible. To find a computer use for your procedure:
- Consider the statement "318682 is 101558 + 217124 and squared is 101558217124" (see: Kaprekar numbers#Casting Out Nines (fast)).
- note that has the same checksum as ();
- note that has the same checksum as () because for a Kaprekar they are made up of the same digits (sometimes with extra zeroes);
- note that this implies that for Kaprekar numbers the checksum of equals the checksum of .
Demonstrate that your procedure can be used to generate or filter a range of numbers with the property and show that this subset is a small proportion of the range and contains all the Kaprekar in the range.
- Part 3
Considering this MathWorld page, produce a efficient algorithm based on the more mathematical treatment of Casting Out Nines, and realizing:
- is the residual of mod ;
- the procedure can be extended to bases other than 9.
Demonstrate your algorithm by generating or filtering a range of numbers with the property and show that this subset is a small proportion of the range and contains all the Kaprekar in the range.
- related tasks
11l
F CastOut(Base, Start, End)
V ran = (0 .< Base - 1).filter(y -> y % (@Base - 1) == (y * y) % (@Base - 1))
V (x, y) = divmod(Start, Base - 1)
[Int] r
L
L(n) ran
V k = (Base - 1) * x + n
I k < Start
L.continue
I k > End
R r
r.append(k)
x++
L(v) CastOut(Base' 16, Start' 1, End' 255)
print(v, end' ‘ ’)
print()
L(v) CastOut(Base' 10, Start' 1, End' 99)
print(v, end' ‘ ’)
print()
L(v) CastOut(Base' 17, Start' 1, End' 288)
print(v, end' ‘ ’)
print()- Output:
1 6 10 15 16 21 25 30 31 36 40 45 46 51 55 60 61 66 70 75 76 81 85 90 91 96 100 105 106 111 115 120 121 126 130 135 136 141 145 150 151 156 160 165 166 171 175 180 181 186 190 195 196 201 205 210 211 216 220 225 226 231 235 240 241 246 250 255 1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 1 16 17 32 33 48 49 64 65 80 81 96 97 112 113 128 129 144 145 160 161 176 177 192 193 208 209 224 225 240 241 256 257 272 273 288
360 Assembly
The program uses two ASSIST macros (XDECO,XPRNT) to keep the code as short as possible.
* Casting out nines 08/02/2017
CASTOUT CSECT
USING CASTOUT,R13 base register
B 72(R15) skip savearea
DC 17F'0' savearea
STM R14,R12,12(R13) prolog
ST R13,4(R15) " <-
ST R15,8(R13) " ->
LR R13,R15 " addressability
L R1,LOW low
XDECO R1,XDEC edit low
MVC PGT+4(4),XDEC+8 output low
L R1,HIGH high
XDECO R1,XDEC edit high
MVC PGT+12(4),XDEC+8 output low
L R1,BASE base
XDECO R1,XDEC edit base
MVC PGT+24(4),XDEC+8 output base
XPRNT PGT,L'PGT print buffer
L R2,BASE base
BCTR R2,0 -1
ST R2,RM rm=base-1
LA R8,PG ipg=0
SR R7,R7 j=0
L R6,LOW i=low
DO WHILE=(C,R6,LE,HIGH) do i=low to high
LR R5,R6 i
SR R4,R4 clear for div
D R4,RM /rm
LR R2,R4 r2=i mod rm
LR R5,R6 i
MR R4,R6 i*i
SR R4,R4 clear for div
D R4,RM /rm
IF CR,R2,EQ,R4 THEN if (i//rm)=(i*i//rm) then
LA R7,1(R7) j=j+1
XDECO R6,XDEC edit i
MVC 0(4,R8),XDEC+8 output i
LA R8,4(R8) ipg=ipg+4
IF C,R7,EQ,=F'20' THEN if j=20 then
XPRNT PG,L'PG print buffer
LA R8,PG ipg=0
SR R7,R7 j=0
MVC PG,=CL80' ' clear buffer
ENDIF , end if
ENDIF , end if
LA R6,1(R6) i=i+1
ENDDO , end do i
IF LTR,R7,NE,R7 THEN if j<>0 then
XPRNT PG,L'PG print buffer
ENDIF , end if
L R13,4(0,R13) epilog
LM R14,R12,12(R13) " restore
XR R15,R15 " rc=0
BR R14 exit
LOW DC F'1' low
HIGH DC F'500' high
BASE DC F'10' base
RM DS F rm
PGT DC CL80'for ... to ... base ...' buffer
PG DC CL80' ' buffer
XDEC DS CL12 temp for xdeco
YREGS
END CASTOUT- Output:
for 1 to 500 base 10 1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 100 108 109 117 118 126 127 135 136 144 145 153 154 162 163 171 172 180 181 189 190 198 199 207 208 216 217 225 226 234 235 243 244 252 253 261 262 270 271 279 280 288 289 297 298 306 307 315 316 324 325 333 334 342 343 351 352 360 361 369 370 378 379 387 388 396 397 405 406 414 415 423 424 432 433 441 442 450 451 459 460 468 469 477 478 486 487 495 496
ABC
\ casting out nines - based on the Action! sample
HOW TO ADD v TO n: PUT n + v IN n
PUT 10, 2, 0, 0 IN base, n, count, total
FOR i IN { 1 .. base ** n }:
ADD 1 TO total
IF i mod ( base - 1 ) = ( i * i ) mod ( base - 1 ):
ADD 1 TO count
WRITE i
WRITE // "Trying", count, "numbers instead of", total, "numbers saves"
WRITE 100 - ( ( 100 * count ) / total ), "%" /- Output:
1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 100 Trying 23 numbers instead of 100 numbers saves 77 %
Action!
INT FUNC Power(INT a,b)
INT i,res
res=1
FOR i=1 TO b
DO
res==*a
OD
RETURN (res)
PROC Main()
DEFINE BASE="10"
DEFINE N="2"
INT i,max,count,total,perc
max=Power(BASE,N)
count=0 total=0
FOR i=1 TO max
DO
total==+1
IF i MOD (BASE-1)=(i*i) MOD (BASE-1) THEN
count==+1
PrintI(i) Put(32)
FI
OD
perc=100-100*count/total
PrintF("%E%ETrying %I numbers instead of %I numbers saves %I%%",count,total,perc)
RETURN- Output:
Screenshot from Atari 8-bit computer
1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 100 Trying 23 numbers instead of 100 numbers saves 77%
ALGOL 68
BEGIN # casting out nines - translated from the Action! sample #
INT base = 10;
INT n = 2;
INT count := 0;
INT total := 0;
FOR i TO base ^ n DO
total +:= 1;
IF i MOD ( base - 1 ) = ( i * i ) MOD ( base - 1 ) THEN
count +:= 1;
print( ( whole( i, 0 ), " " ) )
FI
OD;
print( ( newline, newline, "Trying ", whole( count, 0 )
, " numbers instead of ", whole( total, 0 )
, " numbers saves ", fixed( 100 - ( ( 100 * count ) / total ), -6, 2 )
, "%", newline
)
)
END- Output:
1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 100 Trying 23 numbers instead of 100 numbers saves 77.00%
Arturo
N: 2
base: 10
c1: 0
c2: 0
loop 1..(base^N)-1 'k [
c1: c1 + 1
if (k%base-1)= (k*k)%base-1 [
c2: c2 + 1
prints ~"|k| "
]
]
print ""
print ["Trying" c2 "numbers instead of" c1 "numbers saves" 100.0 - 100.0*c2//c1 "%"]
- Output:
1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 Trying 22 numbers instead of 99 numbers saves 77.77777777777777 %
AWK
# syntax: GAWK -f CASTING_OUT_NINES.AWK
# converted from C
BEGIN {
base = 10
for (k=1; k<=base^2; k++) {
c1++
if (k % (base-1) == (k*k) % (base-1)) {
c2++
printf("%d ",k)
}
}
printf("\nTrying %d numbers instead of %d numbers saves %.2f%%\n",c2,c1,100-(100*c2/c1))
exit(0)
}
- Output:
1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 100 Trying 23 numbers instead of 100 numbers saves 77.00%
BASIC
BASIC256
base = 10
c1 = 0
c2 = 0
for k = 1 to (base ^ 2) - 1
c1 += 1
if k % (base - 1) = (k * k) % (base - 1) then c2 += 1: print k; " ";
next k
print
print "Trying "; c2; " numbers instead of "; c1; " numbers saves "; 100 - (100 * c2 / c1); "%"
end
Chipmunk Basic
100 cls
110 bs = 10 : c1 = 0 : c2 = 0
120 for k = 1 to (bs^2)-1
130 c1 = c1+1
140 if k mod (bs-1) = (k*k) mod (bs-1) then c2 = c2+1 : print k;
150 next k
160 print
170 print "Trying ";c2;"numbers instead of ";c1;"numbers saves ";100-(100*c2/c1);"%"
180 end
Gambas
Public Sub Main()
Dim base10 As Integer = 10
Dim c1 As Integer = 0, c2 As Integer = 0, k As Integer
For k = 1 To base10 ^ 2
c1 += 1
If (k Mod (base10 - 1) = (k * k) Mod (base10 - 1)) Then
c2 += 1
Print k; " ";
End If
Next
Print "\nTrying "; c2; " numbers instead of "; c1; " numbers saves "; 100 - (100 * c2 / c1); "%"
End
GW-BASIC
The MSX-BASIC solution works without any changes.
MSX Basic
100 CLS
110 BS = 10 : C1 = 0 : C2 = 0
120 FOR K = 1 TO (BS^2)-1
130 C1 = C1+1
140 IF K MOD (BS-1) = (K*K) MOD (BS-1) THEN C2 = C2+1 : PRINT K;
150 NEXT K
160 PRINT
170 PRINT USING "Trying ## numbers instead of ### numbers saves ##.##%";C2;C1;100-(100*C2/C1)
180 END
PureBasic
OpenConsole()
Define.i base, c1, c2, k
base = 10
c1 = 0
c2 = 0
For k = 1 To Pow(base, 2) - 1
c1 + 1
If k % (base - 1) = (k * k) % (base - 1)
c2 + 1
Print(Str(k) + " ")
EndIf
Next k
PrintN(#CRLF$ + "Trying " + Str(c2) + " numbers instead of " + Str(c1) + " numbers saves " + Str(100 - (100 * c2 / c1)) + "%")
PrintN(#CRLF$ + "Press ENTER to exit"): Input()
CloseConsole()
QBasic
CLS
bs = 10: c1 = 0: c2 = 0
FOR k = 1 TO (bs ^ 2) - 1
c1 = c1 + 1
IF k MOD (bs - 1) = (k * k) MOD (bs - 1) THEN c2 = c2 + 1: PRINT k;
NEXT k
PRINT
PRINT USING "Trying ## numbers instead of ### numbers saves ##.##%"; c2; c1; 100 - (100 * c2 / c1)
Run BASIC
base = 10
c1 = 0
c2 = 0
for k = 1 to (base ^ 2) - 1
c1 = c1 + 1
if k mod (base - 1) = (k * k) mod (base - 1) then c2 = c2 + 1: print k; " ";
next k
print
print "Trying "; using("##", c2); " numbers instead of "; using("###", c1); " numbers saves "; using("##.##", (100 - (100 * c2 / c1))); "%"
end
True BASIC
LET bs = 10
LET c1 = 0
LET c2 = 0
FOR k = 1 TO (bs^2)-1
LET c1 = c1 + 1
IF REMAINDER(k,(bs-1)) = REMAINDER((k*k),(bs-1)) THEN
LET c2 = c2 + 1
PRINT k;
END IF
NEXT k
PRINT
PRINT USING "Trying ## numbers instead of ### numbers saves ##.##%": c2, c1, 100-(100*c2/c1)
END
XBasic
PROGRAM "Casting out nines"
VERSION "0.0000"
DECLARE FUNCTION Entry ()
FUNCTION Entry ()
bs = 10
c1 = 0
c2 = 0
FOR k = 1 TO (bs ** 2) - 1
INC c1
IF k MOD (bs - 1) = (k * k) MOD (bs - 1) THEN INC c2: PRINT k;
NEXT k
PRINT
PRINT "Trying "; c2; " numbers instead of "; c1; " numbers saves "; 100 - (100 * c2 / c1); "%"
END FUNCTION
END PROGRAM
Yabasic
base = 10
c1 = 0
c2 = 0
for k = 1 to (base ^ 2) - 1
c1 = c1 + 1
if mod(k, (base - 1)) = mod((k * k), (base - 1)) then c2 = c2 + 1: print k, " "; : fi
next k
print "\nTrying ", c2 using("##"), " numbers instead of ", c1 using("###"), " numbers saves ", (100 - (100 * c2 / c1)) using("##.##"), "%"
end
C
#include <stdio.h>
#include <math.h>
int main() {
const int N = 2;
int base = 10;
int c1 = 0;
int c2 = 0;
int k;
for (k = 1; k < pow(base, N); k++) {
c1++;
if (k % (base - 1) == (k * k) % (base - 1)) {
c2++;
printf("%d ", k);
}
}
printf("\nTring %d numbers instead of %d numbers saves %f%%\n", c2, c1, 100.0 - 100.0 * c2 / c1);
return 0;
}
- Output:
1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 Tring 22 numbers instead of 99 numbers saves 77.777778%
C#
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
namespace CastingOutNines {
public static class Helper {
public static string AsString<T>(this IEnumerable<T> e) {
var it = e.GetEnumerator();
StringBuilder builder = new StringBuilder();
builder.Append("[");
if (it.MoveNext()) {
builder.Append(it.Current);
}
while (it.MoveNext()) {
builder.Append(", ");
builder.Append(it.Current);
}
builder.Append("]");
return builder.ToString();
}
}
class Program {
static List<int> CastOut(int @base, int start, int end) {
int[] ran = Enumerable
.Range(0, @base - 1)
.Where(a => a % (@base - 1) == (a * a) % (@base - 1))
.ToArray();
int x = start / (@base - 1);
List<int> result = new List<int>();
while (true) {
foreach (int n in ran) {
int k = (@base - 1) * x + n;
if (k < start) {
continue;
}
if (k > end) {
return result;
}
result.Add(k);
}
x++;
}
}
static void Main() {
Console.WriteLine(CastOut(16, 1, 255).AsString());
Console.WriteLine(CastOut(10, 1, 99).AsString());
Console.WriteLine(CastOut(17, 1, 288).AsString());
}
}
}
- Output:
[1, 6, 10, 15, 16, 21, 25, 30, 31, 36, 40, 45, 46, 51, 55, 60, 61, 66, 70, 75, 76, 81, 85, 90, 91, 96, 100, 105, 106, 111, 115, 120, 121, 126, 130, 135, 136, 141, 145, 150, 151, 156, 160, 165, 166, 171, 175, 180, 181, 186, 190, 195, 196, 201, 205, 210, 211, 216, 220, 225, 226, 231, 235, 240, 241, 246, 250, 255] [1, 9, 10, 18, 19, 27, 28, 36, 37, 45, 46, 54, 55, 63, 64, 72, 73, 81, 82, 90, 91, 99] [1, 16, 17, 32, 33, 48, 49, 64, 65, 80, 81, 96, 97, 112, 113, 128, 129, 144, 145, 160, 161, 176, 177, 192, 193, 208, 209, 224, 225, 240, 241, 256, 257, 272, 273, 288]
C++
Filter
// Casting Out Nines
//
// Nigel Galloway. June 24th., 2012
//
#include <iostream>
int main() {
int Base = 10;
const int N = 2;
int c1 = 0;
int c2 = 0;
for (int k=1; k<pow((double)Base,N); k++){
c1++;
if (k%(Base-1) == (k*k)%(Base-1)){
c2++;
std::cout << k << " ";
}
}
std::cout << "\nTrying " << c2 << " numbers instead of " << c1 << " numbers saves " << 100 - ((double)c2/c1)*100 << "%" <<std::endl;
return 0;
}
- Produces:
1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 Trying 22 numbers instead of 99 numbers saves 77.7778%
The kaprekar numbers in this range 1 9 45 55 and 99 are included.
Changing: "int Base = 16;" Produces:
1 6 10 15 16 21 25 30 31 36 40 45 46 51 55 60 61 66 70 75 76 81 85 90 91 96 100 105 106 111 115 120 121 126 130 135 136 141 145 150 151 156 160 165 166 171 175 180 181 186 190 195 196 201 205 210 211 216 220 225 226 231 235 240 241 246 250 255 Trying 68 numbers instead of 255 numbers saves 73.3333%
The kaprekar numbers:
- 1 is 1
- 6 is 6
- a is 10
- f is 15
- 33 is 51
- 55 is 85
- 5b is 91
- 78 is 120
- 88 is 136
- ab is 171
- cd is 205
- ff is 255
in this range are included.
Changing: "int Base = 17;" Produces:
1 16 17 32 33 48 49 64 65 80 81 96 97 112 113 128 129 144 145 160 161 176 177 19 2 193 208 209 224 225 240 241 256 257 272 273 288 Trying 36 numbers instead of 288 numbers saves 87.5%
The kaprekar numbers:
- 1 is 1
- g is 16
- 3d is 64
- d4 is 225
- gg is 288
in this range are included.
C++11 For Each Generator
// Casting Out Nines Generator - Compiles with gcc4.6, MSVC 11, and CLang3
//
// Nigel Galloway. June 24th., 2012
//
#include <iostream>
#include <vector>
struct ran {
const int base;
std::vector<int> rs;
ran(const int base) : base(base) { for (int nz=0; nz<base-1; nz++) if(nz*(nz-1)%(base-1) == 0) rs.push_back(nz); }
};
class co9 {
private:
const ran* _ran;
const int _end;
int _r,_x,_next;
public:
bool operator!=(const co9& other) const {return operator*() <= _end;}
co9 begin() const {return *this;}
co9 end() const {return *this;}
int operator*() const {return _next;}
co9(const int start, const int end, const ran* r)
:_ran(r)
,_end(end)
,_r(1)
,_x(start/_ran->base)
,_next((_ran->base-1)*_x + _ran->rs[_r])
{
while (operator*() < start) operator++();
}
const co9& operator++() {
const int oldr = _r;
_r = ++_r%_ran->rs.size();
if (_r<oldr) _x++;
_next = (_ran->base-1)*_x + _ran->rs[_r];
return *this;
}
};
int main() {
ran r(10);
for (int i : co9(1,99,&r)) { std::cout << i << ' '; }
return 0;
}
- Produces:
1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99
An alternative implementation for struct ran using http://rosettacode.org/wiki/Sum_digits_of_an_integer#C.2B.2B which produces the same result is:
struct ran {
const int base;
std::vector<int> rs;
ran(const int base) : base(base) { for (int nz=0; nz<base-1; nz++) if(SumDigits(nz) == SumDigits(nz*nz)) rs.push_back(nz); }
};
Changing main:
int main() {
ran r(16);
for (int i : co9(1,255,&r)) { std::cout << i << ' '; }
return 0;
}
- Produces:
1 6 10 15 16 21 25 30 31 36 40 45 46 51 55 60 61 66 70 75 76 81 85 90 91 96 100 105 106 111 115 120 121 126 130 135 136 141 145 150 151 156 160 165 166 171 175 180 181 186 190 195 196 201 205 210 211 216 220 225 226 231 235 240 241 246 250 255
Changing main:
int main() {
ran r(17);
for (int i : co9(1,288,&r)) { std::cout << i << ' '; }
return 0;
}
- Produces:
1 16 17 32 33 48 49 64 65 80 81 96 97 112 113 128 129 144 145 160 161 176 177 192 193 208 209 224 225 240 241 256 257 272 273 288
Common Lisp
;;A macro was used to ensure that the filter is inlined.
;;Larry Hignight. Last updated on 7/3/2012.
(defmacro kaprekar-number-filter (n &optional (base 10))
`(= (mod ,n (1- ,base)) (mod (* ,n ,n) (1- ,base))))
(defun test (&key (start 1) (stop 10000) (base 10) (collect t))
(let ((count 0)
(nums))
(loop for i from start to stop do
(when (kaprekar-number-filter i base)
(if collect (push i nums))
(incf count)))
(format t "~d potential Kaprekar numbers remain (~~~$% filtered out).~%"
count (* (/ (- stop count) stop) 100))
(if collect (reverse nums))))
- Output:
CL-USER> (test :stop 99) 22 potential Kaprekar numbers remain (~77.78% filtered out). (1 9 10 18 19 27 28 36 37 45 ...) CL-USER> (test :stop 10000 :collect nil) 2223 potential Kaprekar numbers remain (~77.77% filtered out). NIL CL-USER> (test :stop 1000000 :collect nil) 222223 potential Kaprekar numbers remain (~77.78% filtered out). NIL CL-USER> (test :stop 255 :base 16) 68 potential Kaprekar numbers remain (~73.33% filtered out). (1 6 10 15 16 21 25 30 31 36 ...) CL-USER> (test :stop 288 :base 17) 36 potential Kaprekar numbers remain (~87.50% filtered out). (1 16 17 32 33 48 49 64 65 80 ...)
Craft Basic
precision 4
define base = 10, c1 = 0, c2 = 0
for k = 1 to (base ^ 2) - 1
let c1 = c1 + 1
if k % (base - 1) = (k * k) % (base - 1) then
let c2 = c2 + 1
print k
endif
next k
print "trying ", c2, " numbers instead of ", c1, " numbers saves ", 100 - (100 * c2 / c1), "%"
- Output:
1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99trying 22 numbers instead of 99 numbers saves 77.7778%
D
import std.stdio, std.algorithm, std.range;
uint[] castOut(in uint base=10, in uint start=1, in uint end=999999) {
auto ran = iota(base - 1)
.filter!(x => x % (base - 1) == (x * x) % (base - 1));
auto x = start / (base - 1);
immutable y = start % (base - 1);
typeof(return) result;
while (true) {
foreach (immutable n; ran) {
immutable k = (base - 1) * x + n;
if (k < start)
continue;
if (k > end)
return result;
result ~= k;
}
x++;
}
}
void main() {
castOut(16, 1, 255).writeln;
castOut(10, 1, 99).writeln;
castOut(17, 1, 288).writeln;
}
- Output (some newlines added):
[1, 6, 10, 15, 16, 21, 25, 30, 31, 36, 40, 45, 46, 51, 55, 60, 61, 66, 70, 75, 76, 81, 85, 90, 91, 96, 100, 105, 106, 111, 115, 120, 121, 126, 130, 135, 136, 141, 145, 150, 151, 156, 160, 165, 166, 171, 175, 180, 181, 186, 190, 195, 196, 201, 205, 210, 211, 216, 220, 225, 226, 231, 235, 240, 241, 246, 250, 255] [1, 9, 10, 18, 19, 27, 28, 36, 37, 45, 46, 54, 55, 63, 64, 72, 73, 81, 82, 90, 91, 99] [1, 16, 17, 32, 33, 48, 49, 64, 65, 80, 81, 96, 97, 112, 113, 128, 129, 144, 145, 160, 161, 176, 177, 192, 193, 208, 209, 224, 225, 240, 241, 256, 257, 272, 273, 288]
EasyLang
base = 10
for k = 1 to base * base - 1
c1 += 1
if k mod (base - 1) = (k * k) mod (base - 1)
c2 += 1
write k & " "
.
.
print ""
print "Trying " & c2 & " numbers instead of " & c1 & " numbers saves " & 100 - 100 * c2 / c1
Free Pascal
program castout9;
{$ifdef fpc}{$mode delphi}{$endif}
uses generics.collections;
type
TIntegerList = TSortedList<integer>;
procedure co9(const start,base,lim:integer;kaprekars:array of integer);
var
C1:integer = 0;
C2:integer = 0;
S:TIntegerlist;
k,i:integer;
begin
S:=TIntegerlist.Create;
for k := start to lim do
begin
inc(C1);
if k mod (base-1) = (k*k) mod (base-1) then
begin
inc(C2);
S.Add(k);
end;
end;
writeln('Valid subset: ');
for i in Kaprekars do
if not s.contains(i) then
writeln('invalid ',i);
for i in s do write(i:4);
writeln;
write('The Kaprekars in this range [');
for i in kaprekars do write(i:4);
writeln('] are included');
writeln('Trying ',C2, ' numbers instead of ', C1,' saves ',100-(C2 * 100 /C1):3:2,',%.');
writeln;
S.Free;
end;
begin
co9(1, 10, 99, [1,9,45,55,99]);
co9(1, 10, 1000, [1,9,45,55,99,297,703,999]);
end.
Output: Valid subset: 1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 The Kaprekars in this range [ 1 9 45 55 99] are included Trying 22 numbers instead of 99 saves 77.78,%. Valid subset: 1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 100 108 109 117 118 126 127 135 136 144 145 153 154 162 163 171 172 180 181 189 190 198 199 207 208 216 217 225 226 234 235 243 244 252 253 261 262 270 271 279 280 288 289 297 298 306 307 315 316 324 325 333 334 342 343 351 352 360 361 369 370 378 379 387 388 396 397 405 406 414 415 423 424 432 433 441 442 450 451 459 460 468 469 477 478 486 487 495 496 504 505 513 514 522 523 531 532 540 541 549 550 558 559 567 568 576 577 585 586 594 595 603 604 612 613 621 622 630 631 639 640 648 649 657 658 666 667 675 676 684 685 693 694 702 703 711 712 720 721 729 730 738 739 747 748 756 757 765 766 774 775 783 784 792 793 801 802 810 811 819 820 828 829 837 838 846 847 855 856 864 865 873 874 882 883 891 892 900 901 909 910 918 919 927 928 936 937 945 946 954 955 963 964 972 973 981 982 990 991 9991000 The Kaprekars in this range [ 1 9 45 55 99 297 703 999] are included Trying 223 numbers instead of 1000 saves 77.70,%.
FreeBASIC
Const base10 = 10
Dim As Integer c1 = 0, c2 = 0, k = 1
For k = 1 To base10^2
c1 += 1
If (k Mod (base10-1) = (k*k) Mod (base10-1)) Then c2 += 1: Print k;" ";
Next k
Print
Print Using "Intentar ## numeros en lugar de ### numeros ahorra un ##.##%"; c2; c1; 100-(100*c2/c1)
Sleep
- Output:
1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 100 Intentar 23 numeros en lugar de 100 numeros ahorra un 77.00%
FutureBasic
_base10 = 10
void local fn CastingOutNines
NSUInteger i, c1 = 0, c2 = 0
float percent
for i = 1 to _base10^2
c1++
if ( i mod ( _base10 -1 ) == ( i * i ) mod ( _base10 - 1 ) ) then c2++ : printf @"%d \b", i
next
print
percent = 100 -( 100 * c2 / c1 )
printf @"Trying %d numbers instead of %d numbers saves %.2f%%", c2, c1, percent
end fn
fn CastingOutNines
HandleEvents- Output:
1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 100 Trying 23 numbers instead of 100 numbers saves 77.00%
Go
package main
import (
"fmt"
"log"
"strconv"
)
// A casting out nines algorithm.
// Quoting from: http://mathforum.org/library/drmath/view/55926.html
/*
First, for any number we can get a single digit, which I will call the
"check digit," by repeatedly adding the digits. That is, we add the
digits of the number, then if there is more than one digit in the
result we add its digits, and so on until there is only one digit
left.
...
You may notice that when you add the digits of 6395, if you just
ignore the 9, and the 6+3 = 9, you still end up with 5 as your check
digit. This is because any 9's make no difference in the result.
That's why the process is called "casting out" nines. Also, at any
step in the process, you can add digits, not just at the end: to do
8051647, I can say 8 + 5 = 13, which gives 4; plus 1 is 5, plus 6 is
11, which gives 2, plus 4 is 6, plus 7 is 13 which gives 4. I never
have to work with numbers bigger than 18.
*/
// The twist is that co9Peterson returns a function to do casting out nines
// in any specified base from 2 to 36.
func co9Peterson(base int) (cob func(string) (byte, error), err error) {
if base < 2 || base > 36 {
return nil, fmt.Errorf("co9Peterson: %d invalid base", base)
}
// addDigits adds two digits in the specified base.
// People perfoming casting out nines by hand would usually have their
// addition facts memorized. In a program, a lookup table might be
// analogous, but we expediently use features of the programming language
// to add digits in the specified base.
addDigits := func(a, b byte) (string, error) {
ai, err := strconv.ParseInt(string(a), base, 64)
if err != nil {
return "", err
}
bi, err := strconv.ParseInt(string(b), base, 64)
if err != nil {
return "", err
}
return strconv.FormatInt(ai+bi, base), nil
}
// a '9' in the specified base. that is, the greatest digit.
s9 := strconv.FormatInt(int64(base-1), base)
b9 := s9[0]
// define result function. The result function may return an error
// if n is not a valid number in the specified base.
cob = func(n string) (r byte, err error) {
r = '0'
for i := 0; i < len(n); i++ { // for each digit of the number
d := n[i]
switch {
case d == b9: // if the digit is '9' of the base, cast it out
continue
// if the result so far is 0, the digit becomes the result
case r == '0':
r = d
continue
}
// otherwise, add the new digit to the result digit
s, err := addDigits(r, d)
if err != nil {
return 0, err
}
switch {
case s == s9: // if the sum is "9" of the base, cast it out
r = '0'
continue
// if the sum is a single digit, it becomes the result
case len(s) == 1:
r = s[0]
continue
}
// otherwise, reduce this two digit intermediate result before
// continuing.
r, err = cob(s)
if err != nil {
return 0, err
}
}
return
}
return
}
// Subset code required by task. Given a base and a range specified with
// beginning and ending number in that base, return candidate Kaprekar numbers
// based on the observation that k%(base-1) must equal (k*k)%(base-1).
// For the % operation, rather than the language built-in operator, use
// the method of casting out nines, which in fact implements %(base-1).
func subset(base int, begin, end string) (s []string, err error) {
// convert begin, end to native integer types for easier iteration
begin64, err := strconv.ParseInt(begin, base, 64)
if err != nil {
return nil, fmt.Errorf("subset begin: %v", err)
}
end64, err := strconv.ParseInt(end, base, 64)
if err != nil {
return nil, fmt.Errorf("subset end: %v", err)
}
// generate casting out nines function for specified base
cob, err := co9Peterson(base)
if err != nil {
return
}
for k := begin64; k <= end64; k++ {
ks := strconv.FormatInt(k, base)
rk, err := cob(ks)
if err != nil { // assertion
panic(err) // this would indicate a bug in subset
}
rk2, err := cob(strconv.FormatInt(k*k, base))
if err != nil { // assertion
panic(err) // this would indicate a bug in subset
}
// test for candidate Kaprekar number
if rk == rk2 {
s = append(s, ks)
}
}
return
}
var testCases = []struct {
base int
begin, end string
kaprekar []string
}{
{10, "1", "100", []string{"1", "9", "45", "55", "99"}},
{17, "10", "gg", []string{"3d", "d4", "gg"}},
}
func main() {
for _, tc := range testCases {
fmt.Printf("\nTest case base = %d, begin = %s, end = %s:\n",
tc.base, tc.begin, tc.end)
s, err := subset(tc.base, tc.begin, tc.end)
if err != nil {
log.Fatal(err)
}
fmt.Println("Subset: ", s)
fmt.Println("Kaprekar:", tc.kaprekar)
sx := 0
for _, k := range tc.kaprekar {
for {
if sx == len(s) {
fmt.Printf("Fail:", k, "not in subset")
return
}
if s[sx] == k {
sx++
break
}
sx++
}
}
fmt.Println("Valid subset.")
}
}
- Output:
Test case base = 10, begin = 1, end = 100: Subset: [1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 100] Kaprekar: [1 9 45 55 99] Valid subset. Test case base = 17, begin = 10, end = gg: Subset: [10 1f 1g 2e 2f 3d 3e 4c 4d 5b 5c 6a 6b 79 7a 88 89 97 98 a6 a7 b5 b6 c4 c5 d3 d4 e2 e3 f1 f2 g0 g1 gg] Kaprekar: [3d d4 gg] Valid subset.
Haskell
co9 n
| n <= 8 = n
| otherwise = co9 $ sum $ filter (/= 9) $ digits 10 n
task2 = filter (\n -> co9 n == co9 (n ^ 2)) [1 .. 100]
task3 k = filter (\n -> n `mod` k == n ^ 2 `mod` k) [1 .. 100]
Auxillary function, returning digits of a number for given base
digits base = map (`mod` base) . takeWhile (> 0) . iterate (`div` base)
or using unfolding:
digits base = Data.List.unfoldr modDiv
where modDiv 0 = Nothing
modDiv n = let (q, r) = (n `divMod` base) in Just (r, q)
Output
λ> co9 232345 1 λ> co9 34234234 7 λ> co9 (232345 + 34234234) == co9 232345 + co9 34234234 True λ> co9 (232345 * 34234234) == co9 232345 * co9 34234234 True λ> task2 [1,9,10,18,19,27,28,36,37,45,46,54,55,63,64,72,73,81,82,90,91,99,100] λ> task2 == (task3 9) True λ> task3 16 [1,16,17,32,33,48,49,64,65,80,81,96,97]
Finally it is possible to test usefull properties of co9 with QuickCheck:
λ> :m Test.QuickCheck λ> quickCheck (\a -> a > 0 ==> co9 a == a `mod` 9) +++ OK, passed 100 tests. λ> quickCheck (\a b -> a > 0 && b > 0 ==> co9 (co9 a + co9 b) == co9 (a+b)) +++ OK, passed 100 tests. λ> quickCheck (\a b -> a > 0 && b > 0 ==> co9 (co9 a * co9 b) == co9 (a*b)) +++ OK, passed 100 tests.
J
This is an implementation of: "given two numbers which mark the beginning and end of a range of integers, and another number which denotes an integer base, return numbers from within the range where the number is equal (modulo the base minus 1) to its square". At the time of this writing, this task is a draft task and this description does not precisely match the task description on this page. Eventually, either the task description will change to match this implementation (which means this paragraph should be removed) or the task description will change to conflict with this implementation (so this entire section should be re-written).
castout=: 1 :0
[: (#~ ] =&((m-1)&|) *:) <. + [: i. (+*)@-~
)
Example use:
0 (10 castout) 100
0 1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 100
Alternate implementation:
castout=: 1 :0
[: (#~ 0 = (m-1) | 0 _1 1&p.) <. + [: i. (+*)@-~
)
Note that about half of the code here is the code that implements "range of numbers". If we factor that out, and represent the desired values directly the code becomes much simpler:
(#~ 0=9|0 _1 1&p.) i.101
0 1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 100
(#~ ] =&(9&|) *:) i. 101
0 1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 100
And, of course, we can name parts of these expressions. For example:
(#~ ] =&(co9=: 9&|) *:) i. 101
0 1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 100
Or, if you prefer:
co9=: 9&|
(#~ ] =&co9 *:) i. 101
0 1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 100
Java
import java.util.*;
import java.util.stream.IntStream;
public class CastingOutNines {
public static void main(String[] args) {
System.out.println(castOut(16, 1, 255));
System.out.println(castOut(10, 1, 99));
System.out.println(castOut(17, 1, 288));
}
static List<Integer> castOut(int base, int start, int end) {
int[] ran = IntStream
.range(0, base - 1)
.filter(x -> x % (base - 1) == (x * x) % (base - 1))
.toArray();
int x = start / (base - 1);
List<Integer> result = new ArrayList<>();
while (true) {
for (int n : ran) {
int k = (base - 1) * x + n;
if (k < start)
continue;
if (k > end)
return result;
result.add(k);
}
x++;
}
}
}
[1, 6, 10, 15, 16, 21, 25, 30, 31, 36, 40, 45, 46, 51, 55, 60, 61, 66, 70, 75, 76, 81, 85, 90, 91, 96, 100, 105, 106, 111, 115, 120, 121, 126, 130, 135, 136, 141, 145, 150, 151, 156, 160, 165, 166, 171, 175, 180, 181, 186, 190, 195, 196, 201, 205, 210, 211, 216, 220, 225, 226, 231, 235, 240, 241, 246, 250, 255] [1, 9, 10, 18, 19, 27, 28, 36, 37, 45, 46, 54, 55, 63, 64, 72, 73, 81, 82, 90, 91, 99] [1, 16, 17, 32, 33, 48, 49, 64, 65, 80, 81, 96, 97, 112, 113, 128, 129, 144, 145, 160, 161, 176, 177, 192, 193, 208, 209, 224, 225, 240, 241, 256, 257, 272, 273, 288]
JavaScript
ES5
Assuming the context of a web page:
function main(s, e, bs, pbs) {
bs = bs || 10;
pbs = pbs || 10
document.write('start:', toString(s), ' end:', toString(e),
' base:', bs, ' printBase:', pbs)
document.write('<br>castOutNine: ');
castOutNine()
document.write('<br>kaprekar: ');
kaprekar()
document.write('<br><br>')
function castOutNine() {
for (var n = s, k = 0, bsm1 = bs - 1; n <= e; n += 1)
if (n % bsm1 == (n * n) % bsm1) k += 1,
document.write(toString(n), ' ')
document.write('<br>trying ', k, ' numbers instead of ', n = e - s + 1,
' numbers saves ', (100 - k / n * 100)
.toFixed(3), '%')
}
function kaprekar() {
for (var n = s; n <= e; n += 1)
if (isKaprekar(n)) document.write(toString(n), ' ')
function isKaprekar(n) {
if (n < 1) return false
if (n == 1) return true
var s = (n * n)
.toString(bs)
for (var i = 1, e = s.length; i < e; i += 1) {
var a = parseInt(s.substr(0, i), bs)
var b = parseInt(s.substr(i), bs)
if (b && a + b == n) return true
}
return false
}
}
function toString(n) {
return n.toString(pbs)
.toUpperCase()
}
}
main(1, 10 * 10 - 1)
main(1, 16 * 16 - 1, 16)
main(1, 17 * 17 - 1, 17)
main(parseInt('10', 17), parseInt('gg', 17), 17, 17)
- Output:
start:1 end:99 base:10 printBase:10 castOutNine: 1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 trying 22 numbers instead of 99 numbers saves 77.778% kaprekar: 1 9 45 55 99 start:1 end:255 base:16 printBase:10 castOutNine: 1 6 10 15 16 21 25 30 31 36 40 45 46 51 55 60 61 66 70 75 76 81 85 90 91 96 100 105 106 111 115 120 121 126 130 135 136 141 145 150 151 156 160 165 166 171 175 180 181 186 190 195 196 201 205 210 211 216 220 225 226 231 235 240 241 246 250 255 trying 68 numbers instead of 255 numbers saves 73.333% kaprekar: 1 6 10 15 51 85 91 120 136 171 205 255 start:1 end:288 base:17 printBase:10 castOutNine: 1 16 17 32 33 48 49 64 65 80 81 96 97 112 113 128 129 144 145 160 161 176 177 192 193 208 209 224 225 240 241 256 257 272 273 288 trying 36 numbers instead of 288 numbers saves 87.500% kaprekar: 1 16 64 225 288 start:10 end:GG base:17 printBase:17 castOutNine: 10 1F 1G 2E 2F 3D 3E 4C 4D 5B 5C 6A 6B 79 7A 88 89 97 98 A6 A7 B5 B6 C4 C5 D3 D4 E2 E3 F1 F2 G0 G1 GG trying 34 numbers instead of 272 numbers saves 87.500% kaprekar: 3D D4 GG
ES6
(() => {
'use strict';
// co9 :: Int -> Int
const co9 = n =>
n <= 8 ? n : co9(
digits(10, n)
.reduce((a, x) => x !== 9 ? a + x : a, 0)
);
// GENERIC FUNCTIONS
// digits :: Int -> Int -> [Int]
const digits = (base, n) => {
if (n < base) return [n];
const [q, r] = quotRem(n, base);
return [r].concat(digits(base, q));
};
// quotRem :: Integral a => a -> a -> (a, a)
const quotRem = (m, n) => [Math.floor(m / n), m % n];
// range :: Int -> Int -> [Int]
const range = (m, n) =>
Array.from({
length: Math.floor(n - m) + 1
}, (_, i) => m + i);
// squared :: Num a => a -> a
const squared = n => Math.pow(n, 2);
// show :: a -> String
const show = x => JSON.stringify(x, null, 2);
// TESTS
return show({
test1: co9(232345), //-> 1
test2: co9(34234234), //-> 7
test3: co9(232345 + 34234234) === co9(232345) + co9(34234234), //-> true
test4: co9(232345 * 34234234) === co9(232345) * co9(34234234), //-> true,
task2: range(1, 100)
.filter(n => co9(n) === co9(squared(n))),
task3: (k => range(1, 100)
.filter(n => (n % k) === (squared(n) % k)))(16)
});
})();
- Output:
{
"test1": 1,
"test2": 7,
"test3": true,
"test4": true,
"task2": [
1,
9,
10,
18,
19,
27,
28,
36,
37,
45,
46,
54,
55,
63,
64,
72,
73,
81,
82,
90,
91,
99,
100
],
"task3": [
1,
16,
17,
32,
33,
48,
49,
64,
65,
80,
81,
96,
97
]
}
jq
In the following, the filter is_kaprekar as defined at Kaprekar_numbers#jq is used. Since it is only defined for decimals, this section is correspondingly restricted.
Definition of co9:
def co9:
def digits: tostring | explode | map(. - 48); # "0" is 48
if . == 9 then 0
elif 0 <= . and . <= 8 then .
else digits | add | co9
end;For convenience, we also define a function to check whether co9(i) equals co9(i*i) for a given integer, i:
def co9_equals_co9_squared: co9 == ((.*.)|co9);Example:
Integers in 1 .. 100 satisfying co9(i) == co9(i*i):
[range (1;101) | select( co9_equals_co9_squared )produces:
[1,9,10,18,19,27,28,36,37,45,46,54,55,63,64,72,73,81,82,90,91,99,100]
Verification:
One way to verify that the Kaprekar numbers satisfy the co9_equals_co9_squared condition is by inspection. For the range 1..100 considered above, we have:
[ range(1;101) | select(is_kaprekar) ][1,9,45,55,99]
To check the condition programmatically for a given range of integers, we can define a function which will emit any exceptions, e.g.
def verify:
range(1; .)
| select(is_kaprekar and (co9_equals_co9_squared | not));For example, running (1000 | verify) produces an empty stream.
Proportion of integers in 1 .. n satisfying the mod (b-1) condition:
For a given base, "b", the following function computes the proportion of integers, i, in 1 .. n such that i % (b-1) == (i*i) % (b-1):
def proportion(base):
def count(stream): reduce stream as $i (0; . + 1);
. as $n
| (base - 1) as $b
| count( range(1; 1+$n) | select( . % $b == (.*.) % $b) ) / $n ;For example:
(10, 100, 1000, 10000, 100000) | proportion(16)
produces:
0.3
0.27
0.267
0.2667
0.26667
Julia
co9(x) = x == 9 ? 0 :
1<=x<=8 ? x :
co9(sum(digits(x)))
iskaprekar is defined in the task Kaprekar_numbers#Julia.
- Output:
julia> show(filter(x->co9(x)==co9(x^2), 1:100)) [1,9,10,18,19,27,28,36,37,45,46,54,55,63,64,72,73,81,82,90,91,99,100] julia> show(filter(iskaprekar, 1:100)) [1,9,45,55,99] julia> show(filter(x->x%15 == (x^2)%15, 1:100)) # base 16 [1,6,10,15,16,21,25,30,31,36,40,45,46,51,55,60,61,66,70,75,76,81,85,90,91,96,100]
Kotlin
// version 1.1.3
fun castOut(base: Int, start: Int, end: Int): List<Int> {
val b = base - 1
val ran = (0 until b).filter { it % b == (it * it) % b }
var x = start / b
val result = mutableListOf<Int>()
while (true) {
for (n in ran) {
val k = b * x + n
if (k < start) continue
if (k > end) return result
result.add(k)
}
x++
}
}
fun main(args: Array<String>) {
println(castOut(16, 1, 255))
println()
println(castOut(10, 1, 99))
println()
println(castOut(17, 1, 288))
}
- Output:
[1, 6, 10, 15, 16, 21, 25, 30, 31, 36, 40, 45, 46, 51, 55, 60, 61, 66, 70, 75, 76, 81, 85, 90, 91, 96, 100, 105, 106, 111, 115, 120, 121, 126, 130, 135, 136, 141, 145, 150, 151, 156, 160, 165, 166, 171, 175, 180, 181, 186, 190, 195, 196, 201, 205, 210, 211, 216, 220, 225, 226, 231, 235, 240, 241, 246, 250, 255] [1, 9, 10, 18, 19, 27, 28, 36, 37, 45, 46, 54, 55, 63, 64, 72, 73, 81, 82, 90, 91, 99] [1, 16, 17, 32, 33, 48, 49, 64, 65, 80, 81, 96, 97, 112, 113, 128, 129, 144, 145, 160, 161, 176, 177, 192, 193, 208, 209, 224, 225, 240, 241, 256, 257, 272, 273, 288]
Lua
local N = 2
local base = 10
local c1 = 0
local c2 = 0
for k = 1, math.pow(base, N) - 1 do
c1 = c1 + 1
if k % (base - 1) == (k * k) % (base - 1) then
c2 = c2 + 1
io.write(k .. ' ')
end
end
print()
print(string.format("Trying %d numbers instead of %d numbers saves %f%%", c2, c1, 100.0 - 100.0 * c2 / c1))
- Output:
1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 Trying 22 numbers instead of 99 numbers saves 77.777778%
Mathematica /Wolfram Language
Task 1: Simple referenced implementation that handles any base:
Co9[n_, b_: 10] :=
With[{ans = FixedPoint[Total@IntegerDigits[#, b] &, n]},
If[ans == b - 1, 0, ans]];
- Task 1 output:
Co9 /@ (vals = {1235, 2345, 4753})
{2, 5, 1}
Total[Co9 /@ vals] == Co9[Total[vals]]
True
Task 2:
task2 = Select[Range@100, Co9[#] == Co9[#^2] &]
- Task 2 output:
{1, 9, 10, 18, 19, 27, 28, 36, 37, 45, 46, 54, 55, 63, 64, 72, 73, 81, 82, 90, 91, 99, 100}
Task 3: Defines the efficient co9 using Mod.
Co9eff[n_, b_: 10] := Mod[n, b - 1];
- Task 3 output:
Testing bases 10 and 17
task2 == Select[Range@100, Co9eff[#] == Co9eff[#^2] &]
True
Select[Range@100, Co9eff[#, 17] == Co9eff[#^2, 17] &]
{1, 16, 17, 32, 33, 48, 49, 64, 65, 80, 81, 96, 97}
Nim
import sequtils
iterator castOut(base = 10, start = 1, ending = 999_999): int =
var ran: seq[int] = @[]
for y in 0 ..< base-1:
if y mod (base - 1) == (y*y) mod (base - 1):
ran.add(y)
var x = start div (base - 1)
var y = start mod (base - 1)
block outer:
while true:
for n in ran:
let k = (base - 1) * x + n
if k < start:
continue
if k > ending:
break outer
yield k
inc x
echo toSeq(castOut(base=16, start=1, ending=255))
- Output:
@[1, 6, 10, 15, 16, 21, 25, 30, 31, 36, 40, 45, 46, 51, 55, 60, 61, 66, 70, 75, 76, 81, 85, 90, 91, 96, 100, 105, 106, 111, 115, 120, 121, 126, 130, 135, 136, 141, 145, 150, 151, 156, 160, 165, 166, 171, 175, 180, 181, 186, 190, 195, 196, 201, 205, 210, 211, 216, 220, 225, 226, 231, 235, 240, 241, 246, 250, 255]
Objeck
class CastingNines {
function : Main(args : String[]) ~ Nil {
base := 10;
N := 2;
c1 := 0;
c2 := 0;
for (k:=1; k<base->As(Float)->Power(N->As(Float)); k+=1;){
c1+=1;
if (k%(base-1) = (k*k)%(base-1)){
c2+=1;
IO.Console->Print(k)->Print(" ");
};
};
IO.Console->Print("\nTrying ")->Print(c2)->Print(" numbers instead of ")
->Print(c1)->Print(" numbers saves ")->Print(100 - (c2->As(Float)/c1
->As(Float)*100))->PrintLine("%");
}
}- Output:
1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 Trying 22 numbers instead of 99 numbers saves 77.7777778%
PARI/GP
{base=10;
N=2;
c1=c2=0;
for(k=1,base^N-1,
c1++;
if (k%(base-1) == k^2%(base-1),
c2++;
print1(k" ")
);
);
print("\nTrying "c2" numbers instead of "c1" numbers saves " 100.-(c2/c1)*100 "%")}- Produces:
1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 Trying 22 numbers instead of 99 numbers saves 77.77777777777777777777777778%
Changing to: "base = 16;" produces:
1 6 10 15 16 21 25 30 31 36 40 45 46 51 55 60 61 66 70 75 76 81 85 90 91 96 100 105 106 111 115 120 121 126 130 135 136 141 145 150 151 156 160 165 166 171 175 180 181 186 190 195 196 201 205 210 211 216 220 225 226 231 235 240 241 246 250 255 Trying 68 numbers instead of 255 numbers saves 73.33333333333333333333333333%
PascalABC.NET
##
function co9(x: integer): integer;
begin
if x = 9 then result := 0
else if x < 9 then result := x
else result := co9(x.ToString.select(x -> StrToInt(x)).sum);
end;
var kaprekars := [1, 9, 45, 55, 99, 297, 703, 999];
Print('Checksums: '); (1..20).Select(x -> co9(x)).Println;
Print('co9(k) = co9(k*k): ');
Println((1..1000).Where(x -> co9(x) = co9(x * x)));
var part2 := (1..1000).Where(x -> co9(x) = co9(x * x)).ToSet;
Println('Kaprekars: ', kaprekars);
if kaprekars <= part2 then Println('Kaprekars are included.', part2.Count, 'numbers in range 1000');
- Output:
Checksums: 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 co9(k) = co9(k*k): [1,9,10,18,19,27,28,36,37,45,46,54,55,63,64,72,73,81,82,90,91,99,100,108,109,117,118,126,127,135,136,144,145,153,154,162,163,171,172,180,181,189,190,198,199,207,208,216,217,225,226,234,235,243,244,252,253,261,262,270,271,279,280,288,289,297,298,306,307,315,316,324,325,333,334,342,343,351,352,360,361,369,370,378,379,387,388,396,397,405,406,414,415,423,424,432,433,441,442,450,...] Kaprekars: [1,9,45,55,99,297,703,999] Kaprekars are included. 223 numbers in range 1000
Perl
sub co9 { # Follows the simple procedure asked for in Part 1
my $n = shift;
return $n if $n < 10;
my $sum = 0; $sum += $_ for split(//,$n);
co9($sum);
}
sub showadd {
my($n,$m) = @_;
print "( $n [",co9($n),"] + $m [",co9($m),"] ) [",co9(co9($n)+co9($m)),"]",
" = ", $n+$m," [",co9($n+$m),"]\n";
}
sub co9filter {
my $base = shift;
die unless $base >= 2;
my($beg, $end, $basem1) = (1, $base*$base-1, $base-1);
my @list = grep { $_ % $basem1 == $_*$_ % $basem1 } $beg .. $end;
($end, scalar(@list), @list);
}
print "Part 1: Create a simple filter and demonstrate using simple example.\n";
showadd(6395, 1259);
print "\nPart 2: Use this to filter a range with co9(k) == co9(k^2).\n";
print join(" ", grep { co9($_) == co9($_*$_) } 1..99), "\n";
print "\nPart 3: Use efficient method on range.\n";
for my $base (10, 17) {
my($N, $n, @l) = co9filter($base);
printf "[@l]\nIn base %d, trying %d numbers instead of %d saves %.4f%%\n\n",
$base, $n, $N, 100-($n/$N)*100;
}
- Output:
Part 1: Create a simple filter and demonstrate using simple example. ( 6395 [5] + 1259 [8] ) [4] = 7654 [4] Part 2: Use this to filter a range with co9(k) == co9(k^2). 1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 Part 3: Use efficient method on range. [1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99] In base 10, trying 22 numbers instead of 99 saves 77.7778% [1 16 17 32 33 48 49 64 65 80 81 96 97 112 113 128 129 144 145 160 161 176 177 192 193 208 209 224 225 240 241 256 257 272 273 288] In base 17, trying 36 numbers instead of 288 saves 87.5000%
Phix
with javascript_semantics procedure co9(integer start, integer base, integer lim, sequence kaprekars) integer c1=0, c2=0 sequence s = {} for k=start to lim do c1 += 1 if mod(k,base-1)=mod(k*k,base-1) then c2 += 1 s &= k end if end for string msg = "valid subset" for i=1 to length(kaprekars) do if not find(kaprekars[i],s) then msg = "***INVALID***" exit end if end for if length(s)>25 then s[10..-10] = {"..."} end if printf(1,"%V\nKaprekar numbers: %V - %s\n",{s,kaprekars,msg}) printf(1,"Trying %d numbers instead of %d saves %3.2f%%\n\n",{c2,c1,100-(c2/c1)*100}) end procedure co9(1, 10, 99, {1,9,45,55,99}) co9(0(17)10, 17, 17*17, {0(17)3d,0(17)d4,0(17)gg}) co9(1, 10, 1000, {1,9,45,55,99,297,703,999})
- Output:
{1,9,10,18,19,27,28,36,37,45,46,54,55,63,64,72,73,81,82,90,91,99}
Kaprekar numbers: {1,9,45,55,99} - valid subset
Trying 22 numbers instead of 99 saves 77.78%
{17,32,33,48,49,64,65,80,81,"...",225,240,241,256,257,272,273,288,289}
Kaprekar numbers: {64,225,288} - valid subset
Trying 35 numbers instead of 273 saves 87.18%
{1,9,10,18,19,27,28,36,37,"...",964,972,973,981,982,990,991,999,1000}
Kaprekar numbers: {1,9,45,55,99,297,703,999} - valid subset
Trying 223 numbers instead of 1000 saves 77.70%
Picat
go =>
Base10 = 10,
foreach(N in [2,6])
casting_out_nines(Base10,N)
end,
nl,
Base16 = 16,
foreach(N in [2,6])
casting_out_nines(Base16,N)
end,
nl.
casting_out_nines(Base,N) =>
println([base=Base,n=N]),
C1 = 0,
C2 = 0,
Ks = [],
LimitN = 3,
foreach(K in 1..Base**N-1)
C1 := C1 + 1,
if K mod (Base-1) == (K*K) mod (Base-1) then
C2 := C2+1,
if N <= LimitN then
Ks := Ks ++ [K]
end
end
end,
if C2 <= 100 then
println(ks=Ks)
end,
printf("Trying %d numbers instead of %d numbers saves %2.3f%%\n", C2, C1, 100 - ((C2/C1)*100)),
nl.- Output:
[base = 10,n = 2] ks = [1,9,10,18,19,27,28,36,37,45,46,54,55,63,64,72,73,81,82,90,91,99] Trying 22 numbers instead of 99 numbers saves 77.778% [base = 10,n = 6] Trying 222222 numbers instead of 999999 numbers saves 77.778% [base = 16,n = 2] ks = [1,6,10,15,16,21,25,30,31,36,40,45,46,51,55,60,61,66,70,75,76,81,85,90,91,96,100,105,106,111,115,120,121,126,130,135,136,141,145,150,151,156,160,165,166,171,175,180,181,186,190,195,196,201,205,210,211,216,220,225,226,231,235,240,241,246,250,255] Trying 68 numbers instead of 255 numbers saves 73.333% [base = 16,n = 6] Trying 4473924 numbers instead of 16777215 numbers saves 73.333%
PicoLisp
(de kaprekar (N)
(let L (cons 0 (chop (* N N)))
(for ((I . R) (cdr L) R (cdr R))
(NIL (gt0 (format R)))
(T (= N (+ @ (format (head I L)))) N) ) ) )
(de co9 (N)
(until
(> 9
(setq N
(sum
'((N) (unless (= "9" N) (format N)))
(chop N) ) ) ) )
N )
(println 'Part1:)
(println
(=
(co9 (+ 6395 1259))
(co9 (+ (co9 6395) (co9 1259))) ) )
(println 'Part2:)
(println
(filter
'((N) (= (co9 N) (co9 (* N N))))
(range 1 100) ) )
(println
(filter kaprekar (range 1 100)) )
(println 'Part3 '- 'base17:)
(println
(filter
'((N) (= (% N 16) (% (* N N) 16)))
(range 1 100) ) )
(bye)- Output:
Part1: T Part2: (1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 100) (1 9 45 55 99) Part3 - base17: (1 16 17 32 33 48 49 64 65 80 81 96 97)
Python
This works slightly differently, generating the "wierd" (as defined by Counting Out Nines) numbers which may be Kaprekar, rather than filtering all numbers in a range.
# Casting out Nines
#
# Nigel Galloway: June 27th., 2012,
#
def CastOut(Base=10, Start=1, End=999999):
ran = [y for y in range(Base-1) if y%(Base-1) == (y*y)%(Base-1)]
x,y = divmod(Start, Base-1)
while True:
for n in ran:
k = (Base-1)*x + n
if k < Start:
continue
if k > End:
return
yield k
x += 1
for V in CastOut(Base=16,Start=1,End=255):
print(V, end=' ')
Produces:
1 6 10 15 16 21 25 30 31 36 40 45 46 51 55 60 61 66 70 75 76 81 85 90 91 96 100 105 106 111 115 120 121 126 130 135 136 141 145 150 151 156 160 165 166 171 175 180 181 186 190 195 196 201 205 210 211 216 220 225 226 231 235 240 241 246 250 255
CastOut(Base=10,Start=1,End=99) produces:
1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99
CastOut(Base=17,Start=1,End=288) produces:
1 16 17 32 33 48 49 64 65 80 81 96 97 112 113 128 129 144 145 160 161 176 177 192 193 208 209 224 225 240 241 256 257 272 273 288
Quackery
kaprekar is defined at Kaprekar numbers#Quackery.
[ true unrot swap
witheach
[ over find
over found not if
[ dip not
conclude ] ]
drop ] is subset ( [ [ --> [ )
[ abs 0 swap
[ 10 /mod rot +
dup 8 > if [ 9 - ]
swap dup 0 = until ]
drop ] is co9 ( n --> n )
say "Part 1: Examples from Dr Math page." cr cr
say "6395 1259 + = " 6395 1259 + echo cr
say "6395 co9 = " 6395 co9 echo cr
say "1259 co9 = " 1259 co9 echo cr
say "5 8 + co9 = " 5 8 + co9 echo cr
say "7654 co9 = " 7654 co9 echo cr cr
say "6395 1259 * = " 6395 1259 * echo cr
say "6395 co9 = " 6395 co9 echo cr
say "1259 co9 = " 1259 co9 echo cr
say "5 8 * co9 = " 5 8 * co9 echo cr
say "8051305 co9 = " 7654 co9 echo cr cr
say "Part 2: Kaprekar numbers." cr cr
say "Kaprekar numbers less than one hundred: "
[]
100 times
[ i^ kaprekar if
[ i^ join ] ]
dup echo cr
say '0...99 with property "n co9 n 2 ** co9 =": '
[]
100 times
[ i^ co9
i^ 2 ** co9 = if
[ i^ join ] ]
dup echo cr
say "Is the former a subset of the latter? "
subset iff [ say "Yes." ] else [ say "No." ] cr cr
say "Part 3: Same as Part 2, but base 17." cr cr
say "Kaprekar (base 17) numbers less than one hundred: "
17 base put
[]
100 times
[ i^ kaprekar if
[ i^ join ] ]
base release
dup echo cr
say '0...99 with property "n 16 mod n 2 ** 16 mod =": '
[]
100 times
[ i^ 16 mod
i^ 2 ** 16 mod = if
[ i^ join ] ]
dup echo cr
say "Is the former a subset of the latter? "
subset iff [ say "Yes." ] else [ say "No." ]- Output:
Part 1: Examples from Dr Math page. 6395 1259 + = 7654 6395 co9 = 5 1259 co9 = 8 5 8 + co9 = 4 7654 co9 = 4 6395 1259 * = 8051305 6395 co9 = 5 1259 co9 = 8 5 8 * co9 = 4 8051305 co9 = 4 Part 2: Kaprekar numbers. Kaprekar numbers less than one hundred: [ 1 9 45 55 99 ] 0...99 with property "n co9 n 2 ** co9 =": [ 0 1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 ] Is the former a subset of the latter? Yes. Part 3: Same as Part 2, but base 17. Kaprekar (base 17) numbers less than one hundred: [ 1 16 64 ] 0...99 with property "n 16 mod n 2 ** 16 mod =": [ 0 1 16 17 32 33 48 49 64 65 80 81 96 97 ] Is the former a subset of the latter? Yes.
R
co9 <- function(base) {
x <- 1:(base^2-1)
x[(x %% (base-1)) == (x^2 %% (base-1))]
}
Map(co9,c(10,16,17))
- Output:
[[1]] [1] 1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 [[2]] [1] 1 6 10 15 16 21 25 30 31 36 40 45 46 51 55 60 61 66 70 75 76 81 85 90 91 96 [27] 100 105 106 111 115 120 121 126 130 135 136 141 145 150 151 156 160 165 166 171 175 180 181 186 190 195 [53] 196 201 205 210 211 216 220 225 226 231 235 240 241 246 250 255 [[3]] [1] 1 16 17 32 33 48 49 64 65 80 81 96 97 112 113 128 129 144 145 160 161 176 177 192 193 208 [27] 209 224 225 240 241 256 257 272 273 288
Racket
#lang racket
(require math)
(define (digits n)
(map (compose1 string->number string)
(string->list (number->string n))))
(define (cast-out-nines n)
(with-modulus 9
(for/fold ([sum 0]) ([d (digits n)])
(mod+ sum d))))
Raku
(formerly Perl 6)
sub cast-out(\BASE = 10, \MIN = 1, \MAX = BASE**2 - 1) {
my \B9 = BASE - 1;
my @ran = ($_ if $_ % B9 == $_**2 % B9 for ^B9);
my $x = MIN div B9;
gather loop {
for @ran -> \n {
my \k = B9 * $x + n;
take k if k >= MIN;
}
$x++;
} ...^ * > MAX;
}
say cast-out;
say cast-out 16;
say cast-out 17;
- Output:
(1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99) (1 6 10 15 16 21 25 30 31 36 40 45 46 51 55 60 61 66 70 75 76 81 85 90 91 96 100 105 106 111 115 120 121 126 130 135 136 141 145 150 151 156 160 165 166 171 175 180 181 186 190 195 196 201 205 210 211 216 220 225 226 231 235 240 241 246 250 255) (1 16 17 32 33 48 49 64 65 80 81 96 97 112 113 128 129 144 145 160 161 176 177 192 193 208 209 224 225 240 241 256 257 272 273 288)
REXX
/*REXX program demonstrates the casting─out─nines algorithm (with Kaprekar numbers). */
parse arg LO HI base . /*obtain optional arguments from the CL*/
if LO=='' | LO=="," then do; LO=1; HI=1000; end /*Not specified? Then use the default*/
if HI=='' | HI=="," then HI= LO /* " " " " " " */
if base=='' | base=="," then base= 10 /* " " " " " " */
numeric digits max(9, 2*length(HI**2) ) /*insure enough decimal digits for HI².*/
numbers= castOut(LO, HI, base) /*generate a list of (cast out) numbers*/
@cast_out= 'cast-out-' || (base-1) "test" /*construct a shortcut text for output.*/
say 'For' LO "through" HI', the following passed the' @cast_out":"
say numbers; say /*display the list of cast out numbers.*/
q= HI - LO + 1 /*Q: is the range of numbers in list.*/
p= words(numbers) /*P" " " number " " " " */
pc= format(p/q * 100, , 2) / 1 || '%' /*calculate the percentage (%) cast out*/
say 'For' q "numbers," p 'passed the' @cast_out "("pc') for base' base"."
if base\==10 then exit /*if radix isn't ten, then exit program*/
Kaps= Kaprekar(LO, HI) /*generate a list of Kaprekar numbers. */
say; say 'The Kaprekar numbers in the same range are:' Kaps
say
do i=1 for words(Kaps); x= word(Kaps, i) /*verify 'em in list.*/
if wordpos(x, numbers)\==0 then iterate /*it's OK so far ··· */
say 'Kaprekar number' x "isn't in the numbers list." /*oops─ay! */
exit 13 /*go spank the coder.*/
end /*i*/
say 'All Kaprekar numbers are in the' @cast_out "numbers list." /*OK*/
exit 0 /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
castOut: procedure; parse arg low,high,radix; rm= word(radix 10, 1) - 1; $=
do j=low to word(high low, 1) /*test a range of numbers. */
if j//rm == j*j//rm then $= $ j /*did number pass the test?*/
end /*j*/ /* [↑] Then add # to list.*/
return strip($) /*strip and leading blanks from result.*/
/*──────────────────────────────────────────────────────────────────────────────────────*/
Kaprekar: procedure; parse arg L,H; $=; if L<=1 then $= 1 /*add unity if in range*/
do j=max(2, L) to H; s= j*j /*a slow way to find Kaprekar numbers. */
do m=1 for length(s)%2
if j==left(s, m) + substr(s, m+1) then do; $= $ j; leave; end
end /*m*/ /* [↑] found a Kaprekar number. */
end /*j*/
return strip($) /*return Kaprekar numbers to invoker. */
- output when using the default inputs:
For 1 through 1000, the following passed the cast-out-9 test: 1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 100 108 109 117 118 126 127 135 136 144 145 153 154 162 163 171 172 180 181 189 190 198 199 207 208 216 217 225 226 234 235 243 244 252 253 261 262 270 271 279 280 288 289 297 298 306 307 315 316 324 325 333 334 342 343 351 352 360 361 369 370 378 379 387 388 396 397 405 406 414 415 423 424 432 433 441 442 450 451 459 460 468 469 477 478 486 487 495 496 504 505 513 514 522 523 531 532 540 541 549 550 558 559 567 568 576 577 585 586 594 595 603 604 612 613 621 622 630 631 639 640 648 649 657 658 666 667 675 676 684 685 693 694 702 703 711 712 720 721 729 730 738 739 747 748 756 757 765 766 774 775 783 784 792 793 801 802 810 811 819 820 828 829 837 838 846 847 855 856 864 865 873 874 882 883 891 892 900 901 909 910 918 919 927 928 936 937 945 946 954 955 963 964 972 973 981 982 990 991 999 1000 For 1000 numbers, 223 passed the cast-out-9 test (22.3%) for base 10. The Kaprekar numbers in the same range are: 1 9 45 55 99 297 703 999 All Kaprekar numbers are in the cast-out-9 test numbers list.
- output when using the input of: 1 256 16
For 1 through 256, the following passed the cast-out-15 test: 1 6 10 15 16 21 25 30 31 36 40 45 46 51 55 60 61 66 70 75 76 81 85 90 91 96 100 105 106 111 115 120 121 126 130 135 136 141 145 150 151 156 160 165 166 171 175 180 181 186 190 195 196 201 205 210 211 216 220 225 226 231 235 240 241 246 250 255 256 For 256 numbers, 69 passed the cast-out-15 test (26.95%) for base 16.
Ring
# Project : Casting out nines
co9(1, 10, 99, [1,9,45,55,99])
co9(1, 10, 1000, [1,9,45,55,99,297,703,999])
func co9(start,base,lim,kaprekars)
c1=0
c2=0
s = []
for k = start to lim
c1 = c1 + 1
if k % (base-1) = (k*k) % (base-1)
c2 = c2 + 1
add(s,k)
ok
next
msg = "Valid subset" + nl
for i = 1 to len(kaprekars)
if not find(s,kaprekars[i])
msg = "***Invalid***" + nl
exit
ok
next
showarray(s)
see "Kaprekar numbers:" + nl
showarray(kaprekars)
see msg
see "Trying " + c2 + " numbers instead of " + c1 + " saves " + (100-(c2/c1)*100) + "%" + nl + nl
func showarray(vect)
see "{"
svect = ""
for n = 1 to len(vect)
svect = svect + vect[n] + ", "
next
svect = left(svect, len(svect) - 2)
see svect + "}" + nlOutput:
{1, 9, 10, 18, 19, 27, 28, 36, 37, 45, 46, 54, 55, 63, 64, 72, 73, 81, 82, 90, 91, 99}
Kaprekar numbers:
{1, 9, 45, 55, 99}
Valid subset
Trying 22 numbers instead of 99 saves 77.78%
{1, 9, 10, 18, 19, 27, 28, 36, 37, 45, 46, 54, 55, 63, 64, 72, 73, 81, 82, 90, 91, 99, 100, 108, 109, 117, 118, 126, 127, 135, 136, 144, 145, 153, 154, 162, 163, 171, 172, 180, 181, 189, 190, 198, 199, 207, 208, 216, 217, 225, 226, 234, 235, 243, 244, 252, 253, 261, 262, 270, 271, 279, 280, 288, 289, 297, 298, 306, 307, 315, 316, 324, 325, 333, 334, 342, 343, 351, 352, 360, 361, 369, 370, 378, 379, 387, 388, 396, 397, 405, 406, 414, 415, 423, 424, 432, 433, 441, 442, 450, 451, 459, 460, 468, 469, 477, 478, 486, 487, 495, 496, 504, 505, 513, 514, 522, 523, 531, 532, 540, 541, 549, 550, 558, 559, 567, 568, 576, 577, 585, 586, 594, 595, 603, 604, 612, 613, 621, 622, 630, 631, 639, 640, 648, 649, 657, 658, 666, 667, 675, 676, 684, 685, 693, 694, 702, 703, 711, 712, 720, 721, 729, 730, 738, 739, 747, 748, 756, 757, 765, 766, 774, 775, 783, 784, 792, 793, 801, 802, 810, 811, 819, 820, 828, 829, 837, 838, 846, 847, 855, 856, 864, 865, 873, 874, 882, 883, 891, 892, 900, 901, 909, 910, 918, 919, 927, 928, 936, 937, 945, 946, 954, 955, 963, 964, 972, 973, 981, 982, 990, 991, 999, 1000}
Kaprekar numbers:
{1, 9, 45, 55, 99, 297, 703, 999}
Valid subset
Trying 223 numbers instead of 1000 saves 77.70%
RPL
Task part 1: naive approach
« WHILE DUP 9 > REPEAT
→STR 0
1 3 PICK SIZE FOR j
OVER j DUP SUB STR→ +
NEXT
SWAP DROP
END
» 'CO9' STO @ ( n → remainder )
Kaprekar number checker (any base)
« OVER SQ → n b n2
« 1 CF
1 n2 LN b LN / IP 1 + FOR j
n2 b j ^ MOD LASTARG / IP
IF OVER THEN
IF + n == THEN 1 SF n 'j' STO END
ELSE DROP2 END
NEXT
1 FS?
» » 'ISBKAR?' STO @ ( n base → boolean )
Task parts 2 & 3
« { } → n base kaprekar
« 1 n FOR j
IF j base ISBKAR? THEN 'kaprekar' j STO+ END
NEXT
{ }
1 n FOR k
IF k base 1 - MOD LASTARG SWAP SQ SWAP MOD == THEN k + END
NEXT
0
1 kaprekar SIZE FOR j
IF OVER kaprekar j GET POS NOT THEN 1 + END
NEXT
"Missing K#" →TAG
1 3 PICK SIZE n / - "% saved" →TAG
» » 'CASTOUT' STO @ ( span base → results )
255 10 CASTOUT 255 17 CASTOUT
- Output:
6: { 1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 100 108 109 117 118 126 127 135 136 144 145 153 154 162 163 171 172 180 181 189 190 198 199 207 208 216 217 225 226 234 235 243 244 252 253 }
5: Missing K#:0
4: % saved: .776470588235
3: { 1 16 17 32 33 48 49 64 65 80 81 96 97 112 113 128 129 144 145 160 161 176 177 192 193 208 209 224 225 240 241 }
2: Missing K#:0
1: % saved: .878431372549
Ruby
N = 2
base = 10
c1 = 0
c2 = 0
for k in 1 .. (base ** N) - 1
c1 = c1 + 1
if k % (base - 1) == (k * k) % (base - 1) then
c2 = c2 + 1
print "%d " % [k]
end
end
puts
print "Trying %d numbers instead of %d numbers saves %f%%" % [c2, c1, 100.0 - 100.0 * c2 / c1]
- Output:
1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 Trying 22 numbers instead of 99 numbers saves 77.777778%
Rust
fn compare_co9_efficiency(base: u64, upto: u64) {
let naive_candidates: Vec<u64> = (1u64..upto).collect();
let co9_candidates: Vec<u64> = naive_candidates.iter().cloned()
.filter(|&x| x % (base - 1) == (x * x) % (base - 1))
.collect();
for candidate in &co9_candidates {
print!("{} ", candidate);
}
println!();
println!(
"Trying {} numbers instead of {} saves {:.2}%",
co9_candidates.len(),
naive_candidates.len(),
100.0 - 100.0 * (co9_candidates.len() as f64 / naive_candidates.len() as f64)
);
}
fn main() {
compare_co9_efficiency(10, 100);
compare_co9_efficiency(16, 256);
}
- Output:
1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 Trying 22 numbers instead of 99 saves 77.78% 1 6 10 15 16 21 25 30 31 36 40 45 46 51 55 60 61 66 70 75 76 81 85 90 91 96 100 105 106 111 115 120 121 126 130 135 136 141 145 150 151 156 160 165 166 171 175 180 181 186 190 195 196 201 205 210 211 216 220 225 226 231 235 240 241 246 250 255 Trying 68 numbers instead of 255 saves 73.33%
Scala
Code written in scala follows functional paradigm of programming, finds list of candidates for Kaprekar numbers within given range.
object kaprekar{
// PART 1
val co_base = ((x:Int,base:Int) => (x%(base-1) == (x*x)%(base-1)))
//PART 2
def get_cands(n:Int,base:Int):List[Int] = {
if(n==1) List[Int]()
else if (co_base(n,base)) n :: get_cands(n-1,base)
else get_cands(n-1,base)
}
def main(args:Array[String]) : Unit = {
//PART 3
val base = 31
println("Candidates for Kaprekar numbers found by casting out method with base %d:".format(base))
println(get_cands(1000,base))
}
}
- Output:
Output for base 10 within range of 100: Candidates for Kaprekar numbers found by casting out method with base 10: List(100, 99, 91, 90, 82, 81, 73, 72, 64, 63, 55, 54, 46, 45, 37, 36, 28, 27, 19, 18, 10, 9) Output for base 17 with range 1000: Candidates for Kaprekar numbers found by casting out method with base 17: List(993, 992, 977, 976, 961, 960, 945, 944, 929, 928, 913, 912, 897, 896, 881, 880, 865, 864, 849, 848, 833, 832, 817, 816, 801, 800, 785, 784, 769, 768, 753, 752, 737, 736, 721, 720, 705, 704, 689, 688, 673, 672, 657, 656, 641, 640, 625, 624, 609, 608, 593, 592, 577, 576, 561, 560, 545, 544, 529, 528, 513, 512, 497, 496, 481, 480, 465, 464, 449, 448, 433, 432, 417, 416, 401, 400, 385, 384, 369, 368, 353, 352, 337, 336, 321, 320, 305, 304, 289, 288, 273, 272, 257, 256, 241, 240, 225, 224, 209, 208, 193, 192, 177, 176, 161, 160, 145, 144, 129, 128, 113, 112, 97, 96, 81, 80, 65, 64, 49, 48, 33, 32, 17, 16)
Seed7
$ include "seed7_05.s7i";
const func bitset: castOut (in integer: base, in integer: start, in integer: ending) is func
result
var bitset: casted is {};
local
var bitset: ran is {};
var integer: x is 0;
var integer: n is 0;
var integer: k is 0;
var boolean: finished is FALSE;
begin
for x range 0 to base - 2 do
if x rem pred(base) = x ** 2 rem pred(base) then
incl(ran, x);
end if;
end for;
x := start div pred(base);
repeat
for n range ran until finished do
k := pred(base) * x + n;
if k >= start then
if k > ending then
finished := TRUE;
else
incl(casted, k);
end if;
end if;
end for;
incr(x);
until finished;
end func;
const proc: main is func
begin
writeln(castOut(16, 1, 255));
end func;- Output:
{1, 6, 10, 15, 16, 21, 25, 30, 31, 36, 40, 45, 46, 51, 55, 60, 61, 66, 70, 75, 76, 81, 85, 90, 91, 96, 100, 105, 106, 111, 115, 120, 121, 126, 130, 135, 136, 141, 145, 150, 151, 156, 160, 165, 166, 171, 175, 180, 181, 186, 190, 195, 196, 201, 205, 210, 211, 216, 220, 225, 226, 231, 235, 240, 241, 246, 250, 255}
Sidef
func cast_out(base = 10, min = 1, max = (base**2 - 1)) {
var b9 = base-1
var ran = b9.range.grep {|n| n%b9 == (n*n % b9) }
var x = min//b9
var r = []
loop {
ran.each {|n|
var k = (b9*x + n)
return r if (k > max)
r << k if (k >= min)
}
++x
}
return r
}
say cast_out().join(' ')
say cast_out(16).join(' ')
say cast_out(17).join(' ')
- Output:
1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 1 6 10 15 16 21 25 30 31 36 40 45 46 51 55 60 61 66 70 75 76 81 85 90 91 96 100 105 106 111 115 120 121 126 130 135 136 141 145 150 151 156 160 165 166 171 175 180 181 186 190 195 196 201 205 210 211 216 220 225 226 231 235 240 241 246 250 255 1 16 17 32 33 48 49 64 65 80 81 96 97 112 113 128 129 144 145 160 161 176 177 192 193 208 209 224 225 240 241 256 257 272 273 288
Tcl
proc co9 {x} {
while {[string length $x] > 1} {
set x [tcl::mathop::+ {*}[split $x ""]]
}
return $x
}
# Extended to the general case
proc coBase {x {base 10}} {
while {$x >= $base} {
for {set digits {}} {$x} {set x [expr {$x / $base}]} {
lappend digits [expr {$x % $base}]
}
set x [tcl::mathop::+ {*}$digits]
}
return $x
}
# Simple helper
proc percent {part whole} {format "%.2f%%" [expr {($whole - $part) * 100.0 / $whole}]}
puts "In base 10..."
set satisfying {}
for {set i 1} {$i < 100} {incr i} {
if {[co9 $i] == [co9 [expr {$i*$i}]]} {
lappend satisfying $i
}
}
puts $satisfying
puts "Trying [llength $satisfying] numbers instead of 99 numbers saves [percent [llength $satisfying] 99]"
puts "In base 16..."
set satisfying {}
for {set i 1} {$i < 256} {incr i} {
if {[coBase $i 16] == [coBase [expr {$i*$i}] 16]} {
lappend satisfying $i
}
}
puts $satisfying
puts "Trying [llength $satisfying] numbers instead of 255 numbers saves [percent [llength $satisfying] 255]"
- Output:
With some newlines inserted…
In base 10... 1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 Trying 22 numbers instead of 99 numbers saves 77.78% In base 16... 1 6 10 15 16 21 25 30 31 36 40 45 46 51 55 60 61 66 70 75 76 81 85 90 91 96 100 105 106 111 115 120 121 126 130 135 136 141 145 150 151 156 160 165 166 171 175 180 181 186 190 195 196 201 205 210 211 216 220 225 226 231 235 240 241 246 250 255 Trying 68 numbers instead of 255 numbers saves 73.33%
Visual Basic .NET
Module Module1
Sub Print(ls As List(Of Integer))
Dim iter = ls.GetEnumerator
Console.Write("[")
If iter.MoveNext Then
Console.Write(iter.Current)
End If
While iter.MoveNext
Console.Write(", ")
Console.Write(iter.Current)
End While
Console.WriteLine("]")
End Sub
Function CastOut(base As Integer, start As Integer, last As Integer) As List(Of Integer)
Dim ran = Enumerable.Range(0, base - 1).Where(Function(y) y Mod (base - 1) = (y * y) Mod (base - 1)).ToArray()
Dim x = start \ (base - 1)
Dim result As New List(Of Integer)
While True
For Each n In ran
Dim k = (base - 1) * x + n
If k < start Then
Continue For
End If
If k > last Then
Return result
End If
result.Add(k)
Next
x += 1
End While
Return result
End Function
Sub Main()
Print(CastOut(16, 1, 255))
Print(CastOut(10, 1, 99))
Print(CastOut(17, 1, 288))
End Sub
End Module
- Output:
[1, 6, 10, 15, 16, 21, 25, 30, 31, 36, 40, 45, 46, 51, 55, 60, 61, 66, 70, 75, 76, 81, 85, 90, 91, 96, 100, 105, 106, 111, 115, 120, 121, 126, 130, 135, 136, 141, 145, 150, 151, 156, 160, 165, 166, 171, 175, 180, 181, 186, 190, 195, 196, 201, 205, 210, 211, 216, 220, 225, 226, 231, 235, 240, 241, 246, 250, 255] [1, 9, 10, 18, 19, 27, 28, 36, 37, 45, 46, 54, 55, 63, 64, 72, 73, 81, 82, 90, 91, 99] [1, 16, 17, 32, 33, 48, 49, 64, 65, 80, 81, 96, 97, 112, 113, 128, 129, 144, 145, 160, 161, 176, 177, 192, 193, 208, 209, 224, 225, 240, 241, 256, 257, 272, 273, 288]
V (Vlang)
fn main() {
println(cast_out(16, 1, 255))
println("")
println(cast_out(10, 1, 99))
println("")
println(cast_out(17, 1, 288))
}
fn cast_out(base int, start int, end int) []int {
b := []int{len: base - 1, init: index}
ran := b.filter(it % b.len == (it * it) % b.len)
mut x, mut k := start / b.len, 0
mut result := []int{}
for {
for n in ran {
k = b.len * x + n
if k < start {continue}
if k > end {return result}
result << k
}
x++
}
return result
}
- Output:
[1, 6, 10, 15, 16, 21, 25, 30, 31, 36, 40, 45, 46, 51, 55, 60, 61, 66, 70, 75, 76, 81, 85, 90, 91, 96, 100, 105, 106, 111, 115, 120, 121, 126, 130, 135, 136, 141, 145, 150, 151, 156, 160, 165, 166, 171, 175, 180, 181, 186, 190, 195, 196, 201, 205, 210, 211, 216, 220, 225, 226, 231, 235, 240, 241, 246, 250, 255] [1, 9, 10, 18, 19, 27, 28, 36, 37, 45, 46, 54, 55, 63, 64, 72, 73, 81, 82, 90, 91, 99] [1, 16, 17, 32, 33, 48, 49, 64, 65, 80, 81, 96, 97, 112, 113, 128, 129, 144, 145, 160, 161, 176, 177, 192, 193, 208, 209, 224, 225, 240, 241, 256, 257, 272, 273, 288]
Wren
var castOut = Fn.new { |base, start, end|
var b = base - 1
var ran = (0...b).where { |n| n % b == (n * n) % b }
var x = (start/b).floor
var result = []
while (true) {
for (n in ran) {
var k = b*x + n
if (k >= start) {
if (k > end) return result
result.add(k)
}
}
x = x + 1
}
}
System.print(castOut.call(16, 1, 255))
System.print()
System.print(castOut.call(10, 1, 99))
System.print()
System.print(castOut.call(17, 1, 288))
- Output:
[1, 6, 10, 15, 16, 21, 25, 30, 31, 36, 40, 45, 46, 51, 55, 60, 61, 66, 70, 75, 76, 81, 85, 90, 91, 96, 100, 105, 106, 111, 115, 120, 121, 126, 130, 135, 136, 141, 145, 150, 151, 156, 160, 165, 166, 171, 175, 180, 181, 186, 190, 195, 196, 201, 205, 210, 211, 216, 220, 225, 226, 231, 235, 240, 241, 246, 250, 255] [1, 9, 10, 18, 19, 27, 28, 36, 37, 45, 46, 54, 55, 63, 64, 72, 73, 81, 82, 90, 91, 99] [1, 16, 17, 32, 33, 48, 49, 64, 65, 80, 81, 96, 97, 112, 113, 128, 129, 144, 145, 160, 161, 176, 177, 192, 193, 208, 209, 224, 225, 240, 241, 256, 257, 272, 273, 288]
XPL0
include xpllib;
def N = 2.;
int Base, C1, C2, K;
\int Main\ [
Base:= 10; C1:= 0; C2:= 0;
for K:= 1 to fix(Pow(float(Base), N)) - 1 do [
C1:= C1+1;
if rem(K/(Base-1)) = rem((K*K)/(Base-1)) then [
C2:= C2+1;
Print("%d ", K);
]
];
Print("\nTrying %d numbers instead of %d numbers saves %1f%%\n",
C2, C1, 100.0 - 100.0*float(C2)/float(C1));
return 0;
]- Output:
1 9 10 18 19 27 28 36 37 45 46 54 55 63 64 72 73 81 82 90 91 99 Trying 22 numbers instead of 99 numbers saves 77.77778%
zkl
fcn castOut(base=10, start=1, end=999999){
base-=1;
ran:=(0).filter(base,'wrap(n){ n%base == (n*n)%base });
result:=Sink(List);
foreach a,b in ([start/base ..],ran){ // foreach{ foreach {} }
k := base*a + b;
if (k < start) continue;
if (k > end) return(result.close());
result.write(k);
}
// doesn't get here
}castOut(16, 1, 255).toString(*).println("\n-----");
castOut(10, 1, 99).toString(*).println("\n-----");
castOut(17, 1, 288).toString(*).println();- Output:
L(1,6,10,15,16,21,25,30,31,36,40,45,46,51,55,60,61,66,70,75, 76,81,85,90,91,96,100,105,106,111,115,120,121,126,130,135,136, 141,145,150,151,156,160,165,166,171,175,180,181,186,190,195,196, 201,205,210,211,216,220,225,226,231,235,240,241,246,250,255) ----- L(1,9,10,18,19,27,28,36,37,45,46,54,55,63,64,72,73,81,82,90,91,99) ----- L(1,16,17,32,33,48,49,64,65,80,81,96,97,112,113,128, 129,144,145,160,161,176,177,192,193,208,209,224,225,240, 241,256,257,272,273,288)
ZX Spectrum Basic
10 LET Base=10
20 LET N=2
30 LET c1=0
40 LET c2=0
50 LET k=1
60 IF k>=(Base^N)-1 THEN GO TO 150
70 LET c1=c1+1
80 IF FN m(k,Base-1)=FN m(k*k,Base-1) THEN LET c2=c2+1: PRINT k;" ";
90 LET k=k+1
100 GO TO 60
150 PRINT '"Trying ";c2;" numbers instead of ";c1;" numbers saves ";100-(c2/c1)*100;"%"
160 STOP
170 DEF FN m(a,b)=a-INT (a/b)*b