Magic constant
You are encouraged to solve this task according to the task description, using any language you may know.
A magic square is a square grid containing consecutive integers from 1 to N², arranged so that every row, column and diagonal adds up to the same number. That number is a constant. There is no way to create a valid N x N magic square that does not sum to the associated constant.
- EG
A 3 x 3 magic square always sums to 15.
┌───┬───┬───┐
│ 2 │ 7 │ 6 │
├───┼───┼───┤
│ 9 │ 5 │ 1 │
├───┼───┼───┤
│ 4 │ 3 │ 8 │
└───┴───┴───┘
A 4 x 4 magic square always sums to 34.
Traditionally, the sequence leaves off terms for n = 0 and n = 1 as the magic squares of order 0 and 1 are trivial; and a term for n = 2 because it is impossible to form a magic square of order 2.
- Task
- Starting at order 3, show the first 20 magic constants.
- Show the 1000th magic constant. (Order 1003)
- Find and show the order of the smallest N x N magic square whose constant is greater than 10¹ through 10¹⁰.
- Stretch
- Find and show the order of the smallest N x N magic square whose constant is greater than 10¹¹ through 10²⁰.
- See also
- Wikipedia: Magic constant
- OEIS: A006003 (Similar sequence, though it includes terms for 0, 1 & 2.)
- Magic squares of odd order
- Magic squares of singly even order
- Magic squares of doubly even order
11l
F a(=n)
n += 2
R n * (n ^ 2 + 1) / 2
F inv_a(x)
V k = 0
L k * (k ^ 2.0 + 1) / 2 + 2 < x
k++
R k
print(‘The first 20 magic constants are:’)
L(n) 1..19
print(Int(a(n)), end' ‘ ’)
print("\nThe 1,000th magic constant is: "Int(a(1000)))
L(e) 1..19
print(‘10^’e‘: ’inv_a(10.0 ^ e))- Output:
The first 20 magic constants are: 15 34 65 111 175 260 369 505 671 870 1105 1379 1695 2056 2465 2925 3439 4010 4641 The 1,000th magic constant is: 503006505 10^1: 3 10^2: 6 10^3: 13 10^4: 28 10^5: 59 10^6: 126 10^7: 272 10^8: 585 10^9: 1260 10^10: 2715 10^11: 5849 10^12: 12600 10^13: 27145 10^14: 58481 10^15: 125993 10^16: 271442 10^17: 584804 10^18: 1259922 10^19: 2714418
Ada
-- Magic constants
-- J. Carter 2024 May
with Ada.Text_IO;
with System;
procedure Magic_Constant is
type Big_U is mod System.Max_Binary_Modulus;
function Magic (Order : in Big_U) return Big_U with
Pre => Order > 2;
-- Returns the constant for the magic square of order Order
function Order (Guess : in Big_U) return Big_U;
-- Returns the order of the smallest magic square with constant > Guess
function Image (N : in Big_U; Width : in Positive) return String;
-- Returns a blank-filled image of N of at least width characters
function Magic (Order : in Big_U) return Big_U is
(Order * (Order ** 2 + 1) / 2);
function Order (Guess : in Big_U) return Big_U is
Min : Big_U := 3;
Max : Big_U := 5_850_000;
Mid : Big_U;
Prev : Big_U := 0;
begin -- Order
Search : loop
Mid := Min + (Max - Min) / 2;
exit Search when Mid = Prev;
Prev := Mid;
if Magic (Mid) > Guess and (Mid = 3 or else Magic (Mid - 1) <= Guess) then
return Mid;
end if;
if Magic (Mid) > Guess then
Max := Mid;
else
Min := Mid;
end if;
end loop Search;
raise Program_Error with "Order: no solution for" & Guess'Image;
end Order;
function Image (N : in Big_U; Width : in Positive) return String is
Raw : String renames N'Image;
Img : String renames Raw (2 .. Raw'Last);
begin -- Image
return (1 .. Width - Img'Length => ' ') & Img;
end Image;
begin -- Magic_Constant
First_20 : for N in Big_U range 3 .. 22 loop
Ada.Text_IO.Put_Line (Item => Image (N, 2) & Image (Magic (N), 5) );
end loop First_20;
Ada.Text_IO.New_Line;
Ada.Text_IO.Put_Line (Item => "1000" & Magic (1000)'Image);
Ada.Text_IO.New_Line;
Ten_To : for P in 1 .. (if Big_U'Size < 64 then 9 elsif Big_U'Size < 128 then 18 else 20) loop
Ada.Text_IO.Put_Line (Item => "10 ** " & Image (Big_U (P), 2) & Image (Order (10 ** P), 8) );
end loop Ten_To;
end Magic_Constant;
- Output:
3 15 4 34 5 65 6 111 7 175 8 260 9 369 10 505 11 671 12 870 13 1105 14 1379 15 1695 16 2056 17 2465 18 2925 19 3439 20 4010 21 4641 22 5335 1000 500000500 10 ** 1 3 10 ** 2 6 10 ** 3 13 10 ** 4 28 10 ** 5 59 10 ** 6 126 10 ** 7 272 10 ** 8 585 10 ** 9 1260 10 ** 10 2715 10 ** 11 5849 10 ** 12 12600 10 ** 13 27145 10 ** 14 58481 10 ** 15 125993 10 ** 16 271442 10 ** 17 584804 10 ** 18 1259922 10 ** 19 2714418 10 ** 20 5848036
ALGOL 68
... with the inverse magic constant routine as in the Julia/Wren samples.
Uses ALGOL 68G's LONG LONG INT whose default precision is large enough to cope with 10^20.
BEGIN # find some magic constants - the row, column and diagonal sums of a magin square #
# translation of the Free Basic sample with the Julia/Wren inverse function #
# returns the magic constant of a magic square of order n + 2 #
PROC a = ( INT n )LONG LONG INT:
BEGIN
LONG LONG INT n2 = n + 2;
( n2 * ( ( n2 * n2 ) + 1 ) ) OVER 2
END # a # ;
# returns the order of the magic square whose magic constant is at least x #
PROC inv a = ( LONG LONG INT x )LONG LONG INT:
ENTIER long long exp( long long ln( x * 2 ) / 3 ) + 1;
print( ( "The first 20 magic constants are " ) );
FOR n TO 20 DO
print( ( whole( a( n ), 0 ), " " ) )
OD;
print( ( newline ) );
print( ( "The 1,000th magic constant is ", whole( a( 1000 ), 0 ), newline ) );
LONG LONG INT e := 1;
FOR n TO 20 DO
e *:= 10;
print( ( "10^", whole( n, -2 ), ": ", whole( inv a( e ), -9 ), newline ) )
OD
END- Output:
The first 20 magic constants are 15 34 65 111 175 260 369 505 671 870 1105 1379 1695 2056 2465 2925 3439 4010 4641 5335 The 1,000th magic constant is 503006505 10^ 1: 3 10^ 2: 6 10^ 3: 13 10^ 4: 28 10^ 5: 59 10^ 6: 126 10^ 7: 272 10^ 8: 585 10^ 9: 1260 10^10: 2715 10^11: 5849 10^12: 12600 10^13: 27145 10^14: 58481 10^15: 125993 10^16: 271442 10^17: 584804 10^18: 1259922 10^19: 2714418 10^20: 5848036
Arturo
a: function [n][
n: n+2
return (n*(1 + n^2))/2
]
aInv: function [x][
k: new 0
while [x > 2 + k*(1+k^2)/2]
-> inc 'k
return k
]
print "The first 20 magic constants are:"
print map 1..19 => a
print ""
print "The 1,000th magic constant is:"
print a 1000
print ""
loop 1..19 'z ->
print ["10 ^" z "=>" aInv 10^z]
- Output:
The first 20 magic constants are: 15 34 65 111 175 260 369 505 671 870 1105 1379 1695 2056 2465 2925 3439 4010 4641 The 1,000th magic constant is: 503006505 10 ^ 1 => 3 10 ^ 2 => 6 10 ^ 3 => 13 10 ^ 4 => 28 10 ^ 5 => 59 10 ^ 6 => 126 10 ^ 7 => 272 10 ^ 8 => 585 10 ^ 9 => 1260 10 ^ 10 => 2715 10 ^ 11 => 5849 10 ^ 12 => 12600 10 ^ 13 => 27145 10 ^ 14 => 58481 10 ^ 15 => 125993 10 ^ 16 => 271442 10 ^ 17 => 584804 10 ^ 18 => 1259922 10 ^ 19 => 2714418
AWK
# syntax: GAWK -f MAGIC_CONSTANT.AWK
# converted from FreeBASIC
BEGIN {
printf("The first 20 magic constants are:")
for (i=1; i<=20; i++) {
printf(" %d",a(i))
}
printf("\n")
printf("The 1,000th magic constant is: %d\n",a(1000))
for (i=1; i<=20; i++) {
printf("10^%02d: %8d\n",i,inv_a(10^i))
}
exit(0)
}
function a(n) {
n += 2
return(n*(n^2+1)/2)
}
function inv_a(x, k) {
while (k*(k^2+1)/2+2 < x) {
k++
}
return(k)
}
- Output:
The first 20 magic constants are: 15 34 65 111 175 260 369 505 671 870 1105 1379 1695 2056 2465 2925 3439 4010 4641 5335 The 1,000th magic constant is: 503006505 10^01: 3 10^02: 6 10^03: 13 10^04: 28 10^05: 59 10^06: 126 10^07: 272 10^08: 585 10^09: 1260 10^10: 2715 10^11: 5849 10^12: 12600 10^13: 27145 10^14: 58481 10^15: 125993 10^16: 271442 10^17: 584804 10^18: 1259922 10^19: 2714418 10^20: 5848036
Basic
FreeBASIC
function a(byval n as uinteger) as ulongint
n+=2
return n*(n^2 + 1)/2
end function
function inv_a(x as double) as ulongint
dim as ulongint k = 0
while k*(k^2+1)/2+2 < x
k+=1
wend
return k
end function
dim as ulongint n
print "The first 20 magic constants are ":
for n = 1 to 20
print a(n);" ";
next n
print
print "The 1,000th magic constant is ";a(1000)
for e as uinteger = 1 to 20
print using "10^##: #########";e;inv_a(10^cast(double,e))
next e
- Output:
The first 20 magic constants are 15 34 65 111 175 260 369 505 671 870 1105 1379 1695 2056 2465 2925 3439 4010 4641 5335 The 1,000th magic constant is 503006505 10^ 1: 3 10^ 2: 6 10^ 3: 13 10^ 4: 28 10^ 5: 59 10^ 6: 126 10^ 7: 272 10^ 8: 585 10^ 9: 1260 10^10: 2715 10^11: 5849 10^12: 12600 10^13: 27145 10^14: 58481 10^15: 125993 10^16: 271442 10^17: 584804 10^18: 1259922 10^19: 2714418 10^20: 5848036
QB64
$NOPREFIX
DIM order AS INTEGER
DIM target AS INTEGER64
PRINT "First 20 magic constants:"
FOR i = 3 TO 22
PRINT USING "####, "; MagicSum(i);
IF i MOD 5 = 2 THEN PRINT
NEXT i
PRINT
PRINT USING "1000th magic constant: ##########,"; MagicSum(1002)
PRINT
PRINT "Smallest order magic square with a constant greater than:"
FOR i = 1 TO 13 ' 64-bit integers can take us no further, unsigned or not
target = 10 ^ i
DO
order = order + 1
LOOP UNTIL MagicSum(order) > target
PRINT USING "10^**: #####,"; i; order
order = order * 2 - 1
NEXT i
FUNCTION MagicSum&& (n AS INTEGER)
MagicSum&& = (n * n + 1) / 2 * n
END FUNCTION
- Output:
First 20 magic constants: 15 34 65 111 175 260 369 505 671 870 1,105 1,379 1,695 2,056 2,465 2,925 3,439 4,010 4,641 5,335 1000th magic constant: 503,006,505 Smallest order magic square with a constant greater than: 10^*1: 3 10^*2: 6 10^*3: 13 10^*4: 28 10^*5: 59 10^*6: 126 10^*7: 272 10^*8: 585 10^*9: 1,260 10^10: 2,715 10^11: 5,849 10^12: 12,600 10^13: 27,145
BASIC256
function a(n)
n = n + 2
return n*(n^2 + 1)/2
end function
function inv_a(x)
k = 0
while k*(k^2+1)/2+2 < x
k += 1
end while
return k
end function
print "The first 20 magic constants are:"
for n = 1 to 20
print int(a(n));" ";
next n
print : print
print "The 1,000th magic constant is "; int(a(1000)); chr(10)
for e = 1 to 20
print "10^"; e; ": "; chr(9); inv_a(10^e)
next e
endPureBasic
Procedure.i a(n.i)
n + 2
ProcedureReturn n*(Pow(n,2) + 1)/2
EndProcedure
Procedure.i inv_a(x.i)
k.i = 0
While k*(Pow(k,2)+1)/2+2 < x
k + 1
Wend
ProcedureReturn k
EndProcedure
OpenConsole()
PrintN("The first 20 magic constants are:")
For n.i = 1 To 20
Print(Str(a(n)) + " ")
Next n
PrintN("") : PrintN("")
PrintN("The 1,000th magic constant is " + Str(a(1000)))
For e.i = 1 To 20
PrintN("10^" + Str(e) + ": " + #TAB$ + Str(inv_a(Pow(10,e))))
Next e
CloseConsole()
QBasic
FUNCTION a (n)
n = n + 2
a = n * (n ^ 2 + 1) / 2
END FUNCTION
FUNCTION inva (x)
k = 0
WHILE k * (k ^ 2 + 1) / 2 + 2 < x
k = k + 1
WEND
inva = k
END FUNCTION
PRINT "The first 20 magic constants are: ";
FOR n = 1 TO 20
PRINT a(n); " ";
NEXT n
PRINT
PRINT "The 1,000th magic constant is "; a(1000)
PRINT
FOR e = 1 TO 20
PRINT USING "10^##: #########"; e; inva(10 ^ e)
NEXT e
END
True BASIC
FUNCTION a(n)
LET n = n + 2
LET a = n*(n^2 + 1)/2
END FUNCTION
FUNCTION inv_a(x)
LET k = 0
DO WHILE k*(k^2+1)/2+2 < x
LET k = k + 1
LOOP
LET inv_a = k
END FUNCTION
PRINT "The first 20 magic constants are: ";
FOR n = 1 TO 20
PRINT a(n);" ";
NEXT n
PRINT
PRINT "The 1,000th magic constant is "; a(1000)
FOR e = 1 TO 20
PRINT USING "10^##": e;
PRINT USING": #########": inv_a(10^e)
NEXT e
END
Yabasic
sub a(n)
n = n + 2
return n*(n^2 + 1)/2
end sub
sub inv_a(x)
k = 0
while k*(k^2+1)/2+2 < x
k = k + 1
wend
return k
end sub
print "The first 20 magic constants are: "
for n = 1 to 20
print a(n), " ";
next n
print "\nThe 1,000th magic constant is ", a(1000), "\n"
for e = 1 to 20
print "10^", e using"##", ": ", inv_a(10^e) using "#########"
next e
end
C#
using System;
public class MagicConstant {
private const int OrderFirstMagicSquare = 3;
public static void Main(string[] args) {
Console.WriteLine("The first 20 magic constants:");
for (int i = 1; i <= 20; i++) {
Console.Write(" " + MagicConstantValue(Order(i)));
}
Console.WriteLine("\n");
Console.WriteLine("The 1,000th magic constant: " + MagicConstantValue(Order(1_000)) + "\n");
Console.WriteLine("Order of the smallest magic square whose constant is greater than:");
for (int i = 1; i <= 20; i++) {
string powerOf10 = "10^" + i + ":";
Console.WriteLine($"{powerOf10,6}{MinimumOrder(i),8}");
}
}
// Return the magic constant for a magic square of the given order
private static int MagicConstantValue(int n) {
return n * (n * n + 1) / 2;
}
// Return the smallest order of a magic square such that its magic constant is greater than 10 to the given power
private static int MinimumOrder(int n) {
return (int)Math.Exp((Math.Log(2.0)+ n * Math.Log(10.0)) / 3) + 1;
}
// Return the order of the magic square at the given index
private static int Order(int index) {
return OrderFirstMagicSquare + index - 1;
}
}
- Output:
The first 20 magic constants: 15 34 65 111 175 260 369 505 671 870 1105 1379 1695 2056 2465 2925 3439 4010 4641 5335 The 1,000th magic constant: 503006505 Order of the smallest magic square whose constant is greater than: 10^1: 3 10^2: 6 10^3: 13 10^4: 28 10^5: 59 10^6: 126 10^7: 272 10^8: 585 10^9: 1260 10^10: 2715 10^11: 5849 10^12: 12600 10^13: 27145 10^14: 58481 10^15: 125993 10^16: 271442 10^17: 584804 10^18: 1259922 10^19: 2714418 10^20: 5848036
C++
#include <cmath>
#include <cstdint>
#include <iomanip>
#include <iostream>
#include <string>
constexpr int32_t ORDER_FIRST_MAGIC_SQUARE = 3;
constexpr double LN2 = log(2.0);
constexpr double LN10 = log(10.0);
// Return the magic constant for a magic square of the given order
int32_t magicConstant(int32_t n) {
return n * ( n * n + 1 ) / 2;
}
// Return the smallest order of a magic square such that its magic constant is greater than 10 to the given power
int32_t minimumOrder(int32_t n) {
return (int) exp( ( LN2 + n * LN10 ) / 3 ) + 1;
}
// Return the order of the magic square at the given index
int32_t order(int32_t index) {
return ORDER_FIRST_MAGIC_SQUARE + index - 1;
}
int main() {
std::cout << "The first 20 magic constants:" << std::endl;
for ( int32_t i = 1; i <= 20; ++i ) {
std::cout << " " << magicConstant(order(i));
}
std::cout << std::endl << std::endl;
std::cout << "The 1,000th magic constant: " << magicConstant(order(1'000)) << std::endl << std::endl;
std::cout << "Order of the smallest magic square whose constant is greater than:" << std::endl;
for ( int32_t i = 1; i <= 20; ++i ) {
std::string power_of_10 = "10^" + std::to_string(i) + ":";
std::cout << std::setw(6) << power_of_10 << std::setw(8) << minimumOrder(i) << std::endl;
}
}
- Output:
The first 20 magic constants: 15 34 65 111 175 260 369 505 671 870 1105 1379 1695 2056 2465 2925 3439 4010 4641 5335 The 1,000th magic constant: 503006505 Order of the smallest magic square whose constant is greater than: 10^1: 3 10^2: 6 10^3: 13 10^4: 28 10^5: 59 10^6: 126 10^7: 272 10^8: 585 10^9: 1260 10^10: 2715 10^11: 5849 10^12: 12600 10^13: 27145 10^14: 58481 10^15: 125993 10^16: 271442 10^17: 584804 10^18: 1259922 10^19: 2714418 10^20: 5848036
Delphi
function GetMagicNumber(N: double): double;
begin
Result:=N * (((N * N) + 1) / 2);
end;
function GetNumberLess(N: double): integer;
var M: double;
begin
for Result:=1 to High(Integer) do
begin
M:=GetMagicNumber(Result);
if M>N then break;
end;
end;
procedure ShowMagicNumber(Memo: TMemo);
var I,J: integer;
var N,M: double;
var S: string;
begin
S:='';
for I:=3 to 23 do
begin
S:=S+Format('%8.0n',[GetMagicNumber(I)]);
if (I mod 5)=2 then S:=S+#$0D#$0A;
end;
Memo.Lines.Add(S);
Memo.Lines.Add('');
Memo.Lines.Add('1000th: '+Format('%8.0n',[GetMagicNumber(1002)]));
Memo.Lines.Add('');
N:=10;
for I:=1 to 20 do
begin
J:=GetNumberLess(N);
Memo.Lines.Add('M^'+Format('%d%8d',[I,J]));
N:=N * 10;
end;
end;
- Output:
15 34 65 111 175
260 369 505 671 870
1,105 1,379 1,695 2,056 2,465
2,925 3,439 4,010 4,641 5,335
6,095
1000th: 503,006,505
M^1 3
M^2 6
M^3 13
M^4 28
M^5 59
M^6 126
M^7 272
M^8 585
M^9 1260
M^10 2715
M^11 5849
M^12 12600
M^13 27145
M^14 58481
M^15 125993
M^16 271442
M^17 584804
M^18 1259922
M^19 2714418
M^20 5848036
EasyLang
func a n .
n += 2
return n * (n * n + 1) / 2
.
func inva x .
while k * (k * k + 1) / 2 + 2 < x
k += 1
.
return k
.
write "The first 20 magic constants: "
for n to 20
write a n & " "
.
print ""
print ""
print "The 1,000th magic constant: " & a 1000
print ""
print "Smallest magic square with constant greater than:"
for e to 10
print "10^" & e & ": " & inva pow 10 e
.
Factor
USING: formatting io kernel math math.functions.integer-logs
math.ranges prettyprint sequences ;
: magic ( m -- n ) dup sq 1 + 2 / * ;
"First 20 magic constants:" print
3 22 [a,b] [ bl ] [ magic pprint ] interleave nl
nl
"1000th magic constant: " write 1002 magic .
nl
"Smallest order magic square with a constant greater than:" print
1 0 20 [
[ 10 * ] dip
[ dup magic pick < ] [ 1 + ] while
over integer-log10 over "10^%02d: %d\n" printf
dup + 1 -
] times 2drop
- Output:
First 20 magic constants: 15 34 65 111 175 260 369 505 671 870 1105 1379 1695 2056 2465 2925 3439 4010 4641 5335 1000th magic constant: 503006505 Smallest order magic square with a constant greater than: 10^01: 3 10^02: 6 10^03: 13 10^04: 28 10^05: 59 10^06: 126 10^07: 272 10^08: 585 10^09: 1260 10^10: 2715 10^11: 5849 10^12: 12600 10^13: 27145 10^14: 58481 10^15: 125993 10^16: 271442 10^17: 584804 10^18: 1259922 10^19: 2714418 10^20: 5848036
Go
package main
import (
"fmt"
"math"
"rcu"
)
func magicConstant(n int) int {
return (n*n + 1) * n / 2
}
var ss = []string{
"\u2070", "\u00b9", "\u00b2", "\u00b3", "\u2074",
"\u2075", "\u2076", "\u2077", "\u2078", "\u2079",
}
func superscript(n int) string {
if n < 10 {
return ss[n]
}
if n < 20 {
return ss[1] + ss[n-10]
}
return ss[2] + ss[0]
}
func main() {
fmt.Println("First 20 magic constants:")
for n := 3; n <= 22; n++ {
fmt.Printf("%5d ", magicConstant(n))
if (n-2)%10 == 0 {
fmt.Println()
}
}
fmt.Println("\n1,000th magic constant:", rcu.Commatize(magicConstant(1002)))
fmt.Println("\nSmallest order magic square with a constant greater than:")
for i := 1; i <= 20; i++ {
goal := math.Pow(10, float64(i))
order := int(math.Cbrt(goal*2)) + 1
fmt.Printf("10%-2s : %9s\n", superscript(i), rcu.Commatize(order))
}
}
- Output:
Same as Wren example.
J
Implementation:
mgc=: 0 0.5 0 0.5&p.
In other words, the magic constant for a magic square of order x is the result of the polynomial (0.5*x)+(0.5*x^3)
Task examples:
mgc 3+i.20
15 34 65 111 175 260 369 505 671 870 1105 1379 1695 2056 2465 2925 3439 4010 4641 5335
mgc 1003x
504514015
(#\,.],.mgc) x:(mgc i.3000) I.10^1+i.10
1 3 15
2 6 111
3 13 1105
4 28 10990
5 59 102719
6 126 1000251
7 272 10061960
8 585 100101105
9 1260 1000188630
10 2715 10006439295
stretch example:
((10+#\),.],.mgc) x:(mgc i.6e6) I.10^11+i.10
11 5849 100049490449
12 12600 1000188006300
13 27145 10000910550385
14 58481 100003310078561
15 125993 1000021311323825
16 271442 10000026341777165
17 584804 100000232056567634
18 1259922 1000002262299152685
19 2714418 10000004237431278525
20 5848036 100000026858987459346
Java
public final class MagicConstant {
public static void main(String[] aArgs) {
System.out.println("The first 20 magic constants:");
for ( int i = 1; i <= 20; i++ ) {
System.out.print(" " + magicConstant(order(i)));
}
System.out.println(System.lineSeparator());
System.out.println("The 1,000th magic constant: " + magicConstant(order(1_000)) + System.lineSeparator());
System.out.println("Order of the smallest magic square whose constant is greater than:");
for ( int i = 1; i <= 20; i++ ) {
String powerOf10 = "10^" + i + ":";
System.out.println(String.format("%6s%8s", powerOf10, minimumOrder(i)));
}
}
// Return the magic constant for a magic square of the given order
private static int magicConstant(int aN) {
return aN * ( aN * aN + 1 ) / 2;
}
// Return the smallest order of a magic square such that its magic constant is greater than 10 to the given power
private static int minimumOrder(int aN) {
return (int) Math.exp( ( LN2 + aN * LN10 ) / 3 ) + 1;
}
// Return the order of the magic square at the given index
private static int order(int aIndex) {
return ORDER_FIRST_MAGIC_SQUARE + aIndex - 1;
}
private static final int ORDER_FIRST_MAGIC_SQUARE = 3;
private static final double LN2 = Math.log(2.0);
private static final double LN10 = Math.log(10.0);
}
- Output:
The first 20 magic constants: 15 34 65 111 175 260 369 505 671 870 1105 1379 1695 2056 2465 2925 3439 4010 4641 5335 The 1,000th magic constant: 503006505 Order of the smallest magic square whose constant is greater than: 10^1: 3 10^2: 6 10^3: 13 10^4: 28 10^5: 59 10^6: 126 10^7: 272 10^8: 585 10^9: 1260 10^10: 2715 10^11: 5849 10^12: 12600 10^13: 27145 10^14: 58481 10^15: 125993 10^16: 271442 10^17: 584804 10^18: 1259922 10^19: 2714418 10^20: 5848036
jq
Adapted from Wren
Works with jq (*)
Works with gojq, the Go implementation of jq
(*) The arithmetic precision of the C implementation of jq is insufficient for the extended task.
Preliminaries
# To take advantage of gojq's arbitrary-precision integer arithmetic:
def power($b): . as $in | reduce range(0;$b) as $i (1; . * $in);
# nth-root
def iroot($n):
. as $in
| if $n == 1 then .
else (. < 0) as $neg
| if $neg and (n % 2) == 0
then "Cannot take the \($n)th root of a negative number." | error
else ($n-1) as $n
| {t: (if $neg then -. else . end)}
| .s = .t + 1
| .u = .t
| until (.u >= .s;
.s = .u
| .u = ((.u * $n) + (.t / (.u|power($n)))) / ($n + 1) )
| if $neg then - .s else .s end
end
end;
# input: an array
# output: a stream of arrays of size size except possibly for the last array
def group(size):
recurse( .[size:]; length>0) | .[0:size];
def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;
def ss : ["\u2070", "\u00b9", "\u00b2", "\u00b3", "\u2074",
"\u2075", "\u2076", "\u2077", "\u2078", "\u2079"];
def superscript:
if . < 10 then ss[.]
elif . < 20 then ss[1] + ss[. - 10]
else ss[2] + ss[0]
end;The Task
def magicConstant: (.*. + 1) * . / 2;
"First 20 magic constants:",
([range(3;23)
| magicConstant]
| group(10) | map(lpad(5)) | join(" ")),
"",
"1,000th magic constant: \( 1002| magicConstant)",
"",
"Smallest order magic square with a constant greater than:",
(range(1; 21) as $i
| (10 | power($i)) as $goal
| ((($goal * 2)|iroot(3) + 1) | floor) as $order
| ("10\($i|superscript)" | lpad(5)) + ": \($order|lpad(9))" )
- Output:
As for Wren except for commatization.
Julia
Uses the inverse of the magic constant function for the last part of the task.
using Lazy
magic(x) = (1 + x^2) * x ÷ 2
magics = @>> Lazy.range() map(magic) filter(x -> x > 10) # first 2 values are filtered out
println("First 20 magic constants: ", Int.(take(20, magics)))
println("Thousandth magic constant is: ", collect(take(1000, magics))[end])
println("Smallest magic square with constant greater than:")
for expo in 1:20
goal = big"10"^expo
ordr = Int(floor((2 * goal)^(1/3))) + 1
println("10^", string(expo, pad=2), " ", ordr)
end
- Output:
First 20 magic constants: [15, 34, 65, 111, 175, 260, 369, 505, 671, 870, 1105, 1379, 1695, 2056, 2465, 2925, 3439, 4010, 4641, 5335] Thousandth magic constant is: 503006505 Smallest magic square with constant greater than: 10^01 3 10^02 6 10^03 13 10^04 28 10^05 59 10^06 126 10^07 272 10^08 585 10^09 1260 10^10 2715 10^11 5849 10^12 12600 10^13 27145 10^14 58481 10^15 125993 10^16 271442 10^17 584804 10^18 1259922 10^19 2714418 10^20 5848036
Lua
function magic (x)
return x * (1 + x^2) / 2
end
print("Magic constants of orders 3 to 22:")
for i = 3, 22 do
io.write(magic(i) .. " ")
end
print("\n\nMagic constant 1003: " .. magic(1003) .. "\n")
print("Orders of smallest magic constant greater than...")
print("-----\t-----\nValue\tOrder\n-----\t-----")
local order = 1
for i = 1, 20 do
repeat
order = order + 1
until magic(order) > 10 ^ i
print("10^" .. i, order)
end
- Output:
Magic constants of orders 3 to 22: 15 34 65 111 175 260 369 505 671 870 1105 1379 1695 2056 2465 2925 3439 4010 4641 5335 Magic constant 1003: 504514015 Orders of smallest magic constant greater than... ----- ----- Value Order ----- ----- 10^1 3 10^2 6 10^3 13 10^4 28 10^5 59 10^6 126 10^7 272 10^8 585 10^9 1260 10^10 2715 10^11 5849 10^12 12600 10^13 27145 10^14 58481 10^15 125993 10^16 271442 10^17 584804 10^18 1259922 10^19 2714418 10^20 5848036
Mathematica /Wolfram Language
ClearAll[i, n, MagicSumHelper, MagicSum, InverseMagicSum]
MagicSumHelper[n_] = Sum[i, {i, n^2}]/n;
MagicSum[n_] := MagicSumHelper[n + 2]
InverseMagicSum[lim_] := Ceiling[-(1/(3^(1/3) (9 lim + Sqrt[3] Sqrt[1 + 27 lim^2])^(1/3))) + (9 lim + Sqrt[3] Sqrt[1 + 27 lim^2])^(1/3)/3^(2/3)]
MagicSum /@ Range[20]
MagicSum[1000]
exps = Range[1, 50];
nums = 10^exps;
Transpose[{Superscript[10, #] & /@ exps, InverseMagicSum[nums]}] // Grid
- Output:
{15, 34, 65, 111, 175, 260, 369, 505, 671, 870, 1105, 1379, 1695, 2056, 2465, 2925, 3439, 4010, 4641, 5335}
503006505
10^1 3
10^2 6
10^3 13
10^4 28
10^5 59
10^6 126
10^7 272
10^8 585
10^9 1260
10^10 2715
10^11 5849
10^12 12600
10^13 27145
10^14 58481
10^15 125993
10^16 271442
10^17 584804
10^18 1259922
10^19 2714418
10^20 5848036
10^21 12599211
10^22 27144177
10^23 58480355
10^24 125992105
10^25 271441762
10^26 584803548
10^27 1259921050
10^28 2714417617
10^29 5848035477
10^30 12599210499
10^31 27144176166
10^32 58480354765
10^33 125992104990
10^34 271441761660
10^35 584803547643
10^36 1259921049895
10^37 2714417616595
10^38 5848035476426
10^39 12599210498949
10^40 27144176165950
10^41 58480354764258
10^42 125992104989488
10^43 271441761659491
10^44 584803547642574
10^45 1259921049894874
10^46 2714417616594907
10^47 5848035476425733
10^48 12599210498948732
10^49 27144176165949066
10^50 58480354764257322
Nim
import std/[math, unicode]
func magicConstant(n: int): int =
## Return the magic constant for a magic square of order "n".
n * (n * n + 1) div 2
func minOrder(n: int): int =
## Return the smallest order such as the magic constant is greater than "10^n".
const Ln2 = ln(2.0)
const Ln10 = ln(10.0)
result = int(exp((Ln2 + n.toFloat * Ln10) / 3)) + 1
const First = 3
const Superscripts: array['0'..'9', string] = ["⁰", "¹", "²", "³", "⁴", "⁵", "⁶", "⁷", "⁸", "⁹"]
template order(idx: Positive): int =
## Compute the order of the magic square at index "idx".
idx + (First - 1)
func superscript(n: Natural): string =
## Return the Unicode string to use to represent an exponent.
for d in $n:
result.add(Superscripts[d])
echo "First 20 magic constants:"
for idx in 1..20:
stdout.write ' ', order(idx).magicConstant
echo()
echo "\n1000th magic constant: ", order(1000).magicConstant
echo "\nOrder of the smallest magic square whose constant is greater than:"
for n in 1..20:
let left = "10" & n.superscript & ':'
echo left.alignLeft(6), ($minOrder(n)).align(7)
- Output:
First 20 magic constants: 15 34 65 111 175 260 369 505 671 870 1105 1379 1695 2056 2465 2925 3439 4010 4641 5335 1000th magic constant: 503006505 Order of the smallest magic square whose constant is greater than: 10¹: 3 10²: 6 10³: 13 10⁴: 28 10⁵: 59 10⁶: 126 10⁷: 272 10⁸: 585 10⁹: 1260 10¹⁰: 2715 10¹¹: 5849 10¹²: 12600 10¹³: 27145 10¹⁴: 58481 10¹⁵: 125993 10¹⁶: 271442 10¹⁷: 584804 10¹⁸: 1259922 10¹⁹: 2714418 10²⁰: 5848036
Pascal
Free Pascal
program MagicConst;
{$IFDEF FPC}{$MODE DELPHI}{$OPTIMIZATION ON,ALL}{$ENDIF}
{$IFDEF WINDOWS}{$APPTYPE CONSOLE}{$ENDIF}
function MagicSum(n :Uint32):Uint64; inline;
var
k : Uint64;
begin
k := n*Uint64(n);
result := (k*k+k) DIV 2;
end;
function MagSumPerRow(n:Uint32):Uint32;
begin
//result := MagicSum(n) DIV n;
//(n^3 + n) /2
result := ((Uint64(n)*n+1)*n) DIV 2;
end;
var
s : String[31];
i : Uint32;
lmt,rowcnt : extended;
Begin
writeln('First Magic constants 3..20');
For i := 3 to 20 do
write(MagSumPerRow(i),' ');
writeln;
writeln('1000.th ',MagSumPerRow(1002));
writeln('First Magic constants > 10^xx');
//lmt = (rowcnt^3 + rowcnt) /2 -> rowcnt > (lmt*2 )^(1/3)
lmt := 2.0 * 10.0;
For i := 1 to 50 do
begin
rowcnt := Int(exp(ln(lmt)/3))+1.0;//+1 suffices
str(trunc(rowcnt),s);
writeln('10^',i:2,#9,s:18);
f := 10.0*lmt;
end;
end.
- Output:
First Magic constants 3..20 15 34 65 111 175 260 369 505 671 870 1105 1379 1695 2056 2465 2925 3439 4010 1000.th 503006505 First Magic constants > 10^xx 10^ 1 3 10^ 2 6 10^ 3 13 10^ 4 28 10^ 5 59 10^ 6 126 10^ 7 272 10^ 8 585 10^ 9 1260 10^10 2715 10^11 5849 10^12 12600 10^13 27145 10^14 58481 10^15 125993 10^16 271442 10^17 584804 10^18 1259922 10^19 2714418 10^20 5848036 ... same as Mathematica 10^46 2714417616594907 10^47 5848035476425733 10^48 12599210498948732 10^49 27144176165949066 10^50 58480354764257322
Perl
#!/usr/bin/perl
use strict; # https://rosettacode.org/wiki/Magic_constant
use warnings;
my @twenty = map $_ * ( $_ ** 2 + 1 ) / 2, 3 .. 22;
print "first twenty: @twenty\n\n" =~ s/.{50}\K /\n/gr;
my $thousandth = 1002 * ( 1002 ** 2 + 1 ) / 2;
print "thousandth: $thousandth\n\n";
print "10**N order\n";
for my $i ( 1 .. 20 )
{
printf "%3d %9d\n", $i, (10 ** $i * 2) ** ( 1 / 3 ) + 1;
}
- Output:
first twenty: 15 34 65 111 175 260 369 505 671 870 1105 1379 1695 2056 2465 2925 3439 4010 4641 5335 thousandth: 503006505 10**N order 1 3 2 6 3 13 4 28 5 59 6 126 7 272 8 585 9 1260 10 2715 11 5849 12 12600 13 27145 14 58481 15 125993 16 271442 17 584804 18 1259922 19 2714418 20 5848036
Phix
with javascript_semantics function magic(integer nth) integer order = nth+2 return (order*order+1)/2 * order end function printf(1,"First 20 magic constants: %V\n",{apply(tagset(20),magic)}) printf(1,"1000th magic constant: %,d\n",{magic(1000)}) include mpfr.e mpz {goal, order} = mpz_inits(2) for i=1 to 20 do mpz_ui_pow_ui(goal,10,i) mpz_mul_si(order,goal,2) mpz_nthroot(order,order,3) mpz_add_si(order,order,1) printf(1,"1e%d: %s\n",{i,mpz_get_str(order,10,true)}) end for
- Output:
First 20 magic constants: {15,34,65,111,175,260,369,505,671,870,1105,1379,1695,2056,2465,2925,3439,4010,4641,5335}
1000th magic constant: 503,006,505
1e1: 3
1e2: 6
1e3: 13
1e4: 28
1e5: 59
1e6: 126
1e7: 272
1e8: 585
1e9: 1,260
1e10: 2,715
1e11: 5,849
1e12: 12,600
1e13: 27,145
1e14: 58,481
1e15: 125,993
1e16: 271,442
1e17: 584,804
1e18: 1,259,922
1e19: 2,714,418
1e20: 5,848,036
Prolog
Minimalistic, efficient approach
m(X,Y):- Y is X*(X*X+1)/2.
l(L,R,T,X):- L > R -> X is L; M is div(L+R,2), m(M,F),
(T < F -> R_ is M-1, l(L,R_,T,X); L_ is M+1, l(L_,R,T,X)).
l(B,X):- l(1,B,B,X).
task:-
write("First 20 magic constants are:"), forall(between(3,22,N), (m(N,X), format(" ~d",X))), nl,
write("The 1000th magic constant is:"), forall(m(1002,X), format(" ~d",X)), nl,
forall(between(1,20,N), (l(10**N,X), format("10^~d:\t~d\n",[N,X]))).
- Output:
?- task. First 20 magic constants are: 15 34 65 111 175 260 369 505 671 870 1105 1379 1695 2056 2465 2925 3439 4010 4641 5335 The 1000th magic constant is: 503006505 10^1: 3 10^2: 6 10^3: 13 10^4: 28 10^5: 59 10^6: 126 10^7: 272 10^8: 585 10^9: 1260 10^10: 2715 10^11: 5849 10^12: 12600 10^13: 27145 10^14: 58481 10^15: 125993 10^16: 271442 10^17: 584804 10^18: 1259922 10^19: 2714418 10^20: 5848036 true.
Python
#!/usr/bin/python
def a(n):
n += 2
return n*(n**2 + 1)/2
def inv_a(x):
k = 0
while k*(k**2+1)/2+2 < x:
k+=1
return k
if __name__ == '__main__':
print("The first 20 magic constants are:");
for n in range(1, 20):
print(int(a(n)), end = " ");
print("\nThe 1,000th magic constant is:",int(a(1000)));
for e in range(1, 20):
print(f'10^{e}: {inv_a(10**e)}');
- Output:
The first 20 magic constants are: 15 34 65 111 175 260 369 505 671 870 1105 1379 1695 2056 2465 2925 3439 4010 4641 The 1,000th magic constant is: 503006505 10^1: 3 10^2: 6 10^3: 13 10^4: 28 10^5: 59 10^6: 126 10^7: 272 10^8: 585 10^9: 1260 10^10: 2715 10^11: 5849 10^12: 12600 10^13: 27145 10^14: 58481 10^15: 125993 10^16: 271442 10^17: 584804 10^18: 1259922 10^19: 2714418
Quackery
[ 3 + dup 3 ** + 2 / ] is magicconstant ( n --> n )
20 times [ i^ magicconstant echo sp ] cr cr
1000 magicconstant echo cr cr
0 1
[ over magicconstant over > if
[ over 3 + echo cr
10 * ]
dip 1+
[ 10 21 ** ] constant
over = until ]
2drop- Output:
15 34 65 111 175 260 369 505 671 870 1105 1379 1695 2056 2465 2925 3439 4010 4641 5335 504514015 3 4 6 13 28 59 126 272 585 1260 2715 5849 12600 27145 58481 125993 271442 584804 1259922 2714418 5848036
Raku
use Lingua::EN::Numbers:ver<2.8+>;
my @magic-constants = lazy (3..∞).hyper.map: { (1 + .²) * $_ / 2 };
put "First 20 magic constants: ", @magic-constants[^20]».,
say "1000th magic constant: ", @magic-constants[999].,
say "\nSmallest order magic square with a constant greater than:";
(1..20).map: -> $p {printf "10%-2s: %s\n", $p.&super, comma 3 + @magic-constants.first( * > exp($p, 10), :k ) }
- Output:
First 20 magic constants: 15 34 65 111 175 260 369 505 671 870 1,105 1,379 1,695 2,056 2,465 2,925 3,439 4,010 4,641 5,335 1000th magic constant: 503,006,505 Smallest order magic square with a constant greater than: 10¹ : 3 10² : 6 10³ : 13 10⁴ : 28 10⁵ : 59 10⁶ : 126 10⁷ : 272 10⁸ : 585 10⁹ : 1,260 10¹⁰: 2,715 10¹¹: 5,849 10¹²: 12,600 10¹³: 27,145 10¹⁴: 58,481 10¹⁵: 125,993 10¹⁶: 271,442 10¹⁷: 584,804 10¹⁸: 1,259,922 10¹⁹: 2,714,418 10²⁰: 5,848,036
RPL
This task can be solved through a few one-liners, by using algebraic and equation-solving features.
First, let's store the equation of a(n):
'X*(SQ(X)+1)/2' 'A6003' STO
Then, evaluate it for n=3 to 22 to get the first 20 magic constants:
≪ {} 3 22 FOR j j 'X' STO A6003 EVAL + NEXT ≫ EVAL
We need now to define the inverse function of a(n):
≪ A6003 OVER - 'X' ROT ROOT CEIL ≫ 'A6003→' STO
And finally, look for a-1( 10 ^ j ), for j=1 to 10:
≪ {} 1 10 FOR j 10 j ^ A6003→ + NEXT ≫ EVAL
which is a one-minute job for a basic HP-28S.
- Output:
2: { 15 34 65 111 175 260 369 505 671 870 1105 1379 1695 2056 2465 2925 3439 4010 4641 5335 }
1: { 3 6 13 28 59 126 272 585 1260 2715 }
Sidef
func f(n) {
(n+2) * ((n+2)**2 + 1) / 2
}
func order(n) {
iroot(2*n, 3) + 1
}
say ("First 20 terms: ", f.map(1..20).join(' '))
say ("1000th term: ", f(1000), " with order ", order(f(1000)))
for n in (1 .. 20) {
printf("order(10^%-2s) = %s\n", n, order(10**n))
}
- Output:
First 20 terms: 15 34 65 111 175 260 369 505 671 870 1105 1379 1695 2056 2465 2925 3439 4010 4641 5335 1000th term: 503006505 with order 1003 order(10^1 ) = 3 order(10^2 ) = 6 order(10^3 ) = 13 order(10^4 ) = 28 order(10^5 ) = 59 order(10^6 ) = 126 order(10^7 ) = 272 order(10^8 ) = 585 order(10^9 ) = 1260 order(10^10) = 2715 order(10^11) = 5849 order(10^12) = 12600 order(10^13) = 27145 order(10^14) = 58481 order(10^15) = 125993 order(10^16) = 271442 order(10^17) = 584804 order(10^18) = 1259922 order(10^19) = 2714418 order(10^20) = 5848036
Wren
This uses Julia's approach for the final parts.
import "./seq" for Lst
import "./fmt" for Fmt
var magicConstant = Fn.new { |n| (n*n + 1) * n / 2 }
var ss = ["\u2070", "\u00b9", "\u00b2", "\u00b3", "\u2074",
"\u2075", "\u2076", "\u2077", "\u2078", "\u2079"]
var superscript = Fn.new { |n| (n < 10) ? ss[n] : (n < 20) ? ss[1] + ss[n - 10] : ss[2] + ss[0] }
System.print("First 20 magic constants:")
var mc20 = (3..22).map { |n| magicConstant.call(n) }.toList
for (chunk in Lst.chunks(mc20, 10)) Fmt.print("$5d", chunk)
Fmt.print("\n1,000th magic constant: $,d", magicConstant.call(1002))
System.print("\nSmallest order magic square with a constant greater than:")
for (i in 1..20) {
var goal = 10.pow(i)
var order = (goal * 2).cbrt.floor + 1
Fmt.print("10$-2s : $,9d", superscript.call(i), order)
}
- Output:
First 20 magic constants: 15 34 65 111 175 260 369 505 671 870 1105 1379 1695 2056 2465 2925 3439 4010 4641 5335 1,000th magic constant: 503,006,505 Smallest order magic square with a constant greater than: 10¹ : 3 10² : 6 10³ : 13 10⁴ : 28 10⁵ : 59 10⁶ : 126 10⁷ : 272 10⁸ : 585 10⁹ : 1,260 10¹⁰ : 2,715 10¹¹ : 5,849 10¹² : 12,600 10¹³ : 27,145 10¹⁴ : 58,481 10¹⁵ : 125,993 10¹⁶ : 271,442 10¹⁷ : 584,804 10¹⁸ : 1,259,922 10¹⁹ : 2,714,418 10²⁰ : 5,848,036
XPL0
A magic square of side N contains N^2 items. The sum of a sequence 1..N^2 is given by: Sum = (N^2+1) * N^2 / 2. A grid row adds to the magic constant, and N rows add to the Sum. Thus the magic constant = Sum/N = (N^3+N)/2.
int N, X;
real M, Thresh, MC;
[Text(0, "First 20 magic constants:^M^J");
for N:= 3 to 20+3-1 do
[IntOut(0, (N*N*N+N)/2); ChOut(0, ^ )];
CrLf(0);
Text(0, "1000th magic constant: ");
N:= 1000+3-1;
IntOut(0, (N*N*N+N)/2);
CrLf(0);
Text(0, "Smallest order magic square with a constant greater than:^M^J");
Thresh:= 10.;
M:= 3.;
Format(1, 0);
for X:= 1 to 10 do
[repeat MC:= (M*M*M+M)/2.;
M:= M+1.;
until MC > Thresh;
Text(0, "10^^");
if X < 10 then ChOut(0, ^0);
IntOut(0, X);
Text(0, ": ");
RlOut(0, M-1.);
CrLf(0);
Thresh:= Thresh*10.;
];
]- Output:
First 20 magic constants: 15 34 65 111 175 260 369 505 671 870 1105 1379 1695 2056 2465 2925 3439 4010 4641 5335 1000th magic constant: 503006505 Smallest order magic square with a constant greater than: 10^01: 3 10^02: 6 10^03: 13 10^04: 28 10^05: 59 10^06: 126 10^07: 272 10^08: 585 10^09: 1260 10^10: 2715
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