# Magic constant

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.

• 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.

## 11l

Translation of: Python
```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
```

## ALGOL 68

Translation of: FreeBasic

... with the inverse magic constant routine as in the Julia/Wren samples.

Works with: ALGOL 68G version Any - tested with release 2.8.3.win32

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
end```

### PureBasic

```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

Works with: QBasic version 1.1
Works with: QuickBasic version 4.5
```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#

Translation of: Java
```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

Works with: Delphi version 6.0

```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;
N:=10;
for I:=1 to 20 do
begin
J:=GetNumberLess(N);
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

Works with: Factor version 0.99 2021-06-02
```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

Translation of: Wren
Library: Go-rcu
```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)`

```   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

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;```

```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
```
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:

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)
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).

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]».&comma;
say "1000th magic constant: ", @magic-constants[999].&comma;
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

Library: Wren-seq
Library: Wren-fmt

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
```