Supplement to the article
“Some Notes on the Magic Squares of Squares Problem”
published in The Mathematical Intelligencer, Vol. 27, N. 2, 2005, pages 52-64
by Christian Boyer

Summary

A) Lucas’s 3×3 semi-magic squares of squares

(p² + q² – r² – s²)²

[2(qr + ps)]²

[2(qs – pr)]²

[2(qr – ps)]²

(p² – q² + r² – s²)²

[2(rs + pq)]²

[2(qs + pr)]²

[2(rs – pq)]²

(p² – q² – r² + s²)²

EL1, from The Mathematical Intelligencer article

The 3 rows and 3 columns have the same magic sum:

B) Euler’s 4×4 magic squares of squares

(+ap+bq+cr+ds)²

(+ar–bs–cp+dq)²

(–as–br+cq+dp)²

(+aq–bp+cs–dr)²

(–aq+bp+cs–dr)²

(+as+br+cq+dp)²

(+ar–bs+cp–dq)²

(+ap+bq–cr–ds)²

(+ar+bs–cp–dq)²

(–ap+bq–cr+ds)²

(+aq+bp+cs+dr)²

(+as–br–cq+dp)²

(–as+br–cq+dp)²

(–aq–bp+cs+dr)²

(–ap+bq+cr–ds)²

(+ar+bs+cp+dq)²

LE3, from The Mathematical Intelligencer article

The 4 rows and 4 columns have the same magic sum:

Two supplemental conditions are given to get the two magic diagonals:

C) Using prime numbers

A) Lucas’s 3×3 semi-magic squares of squares

A1) The full list of examples of Lucas’s family, producing distinct numbers, six magic lines, and a magic sum ≤100²:

(p, q, r, s)

Magic sum

(1, 2, 4, 6)*

57² (the LE1cb square)

(1, 2, 3, 7)*

63²

(2, 3, 4, 6)

65² (the AB2 square)

(1, 3, 5, 6)*

71²

(1, 2, 5, 7)

79²

(2, 4, 5, 6)**

81² (the EL2 square)

(1, 2, 4, 8)

85²

(1, 4, 5, 7)

91²

(2, 3, 4, 8)

93²

(1, 3, 6, 7)

95²

(1, 3, 5, 8)*

99²

(3, 4, 5, 7)

99² (a different square with the same sum)

and of course all the possible permutations of positions and signs of p, q, r, s.

* : The four examples published by Euler in 1770. They were presented slightly differently, Euler using rational numbers: the nine numbers were signed, not squared, and they were divided by p²+q²+r²+s².

** : The example published by Lucas in 1876.

A2) The full list of examples of Lucas’s family, producing distinct numbers, seven magic lines, and a magic sum ≤ 2000² are:

(p, q, r, s)

Magic sum

(1, 3, 4, 11)

147² (the MS1 square)

(3, 5, 8, 14) (A)

294² (the MS2 square)

(4, 9, 11, 17)

507²

(2, 6, 8, 22)

588² (three identical permuted squares)

(3, 11, 13, 17)

588² (three identical permuted squares)

(5, 9, 11, 19)

588² (three identical permuted squares)

(7, 8, 15, 26) (B)

1014²

(8, 11, 13, 27)

1083²

(6, 10, 16, 28) (C)

1176²

(3, 9, 12, 33)

1323²

and of course all the possible permutations of positions and signs of p, q, r, s.

None of these examples were published by Lucas or Euler.

Some other (p, q, r, s) produce only 6 magic lines, but their cells can easily be permuted to get some of the above squares with 7 magic lines:

B) Euler’s 4×4 magic squares of squares

The full list of examples of Euler’s family, producing distinct numbers, ten magic lines (by definition of Euler’s family), and a magic sum ≤ 10000 are:

(a, b, c, d, p, q, r, s)

Magic sum

(2, 3, 5, 0, 1, 2, 8, –4) (D)

3230 (the CB1 magic square)

(1, 2, 3, 4, 2, 5, 10, –4)

4350

(1, 4, 6, 1, 1, 2, 8, –4)

4590

(2, 5, 5, 0, 1, 2, 8, –4) (D)

4590 (a different square with the same sum)

(5, 2, 9, 0, 2, 3, 6, –4) (E)

7150 (the Benneton square)

(5, 3, 9, 0, 2, 3, 6, –4) (E)

7475

(2, 8, 5, 0, 1, 2, 8, –4) (D)

7905

(5, 5, 9, 0, 2, 3, 6, –4) (E)*

8515 (a permutation of the LE2 magic square)

(5, 6, 9, 0, 2, 3, 6, –4) (E)

9230

(4, 1, 10, 0, 1, 2, 8, –4)

9945

and of course all the possible permutations of positions and signs of a, b, c, d, p, q, r, s, like the only example * given by Euler, the magic square LE2 sent to Lagrange.

(2, k, 5, 0, 1, 2, 8, –4) (D) and (5, k, 9, 0, 2, 3, 6, –4) (E) give these two very nice sub-families (CB2) and (CB15) of magic squares of squares. The only limitation: the 16 generated numbers are not always distinct for every k.

(2k + 42)²

(4k + 11)²

(8k – 18)²

(k + 16)²

(k – 24)²

(8k + 2)²

(4k + 21)²

(2k – 38)²

(4k –11)²

(2k – 42)²

(k – 16)²

(8k + 18)²

(8k – 2)²

(k + 24)²

(2k + 38)²

(4k – 21)²

CB2, from The Mathematical Intelligencer article.
A sub-family of Euler’s magic square of squares, S2 = 85(k² + 29).
Its 16 numbers are distinct for k = 3, 5, 8,…

(3k + 64)²

(4k + 12)²

(6k – 47)²

(2k + 21)²

(2k – 51)²

(6k + 7)²

(4k + 48)²

(3k – 44)²

(4k – 12)²

(3k – 64)²

(2k – 21)²

(6k + 47)²

(6k – 7)²

(2k + 51)²

(3k + 44)²

(4k – 48)²

CB15. Another sub-family of Euler’s magic square of squares, S2 = 65(k² + 106).
Its 16 numbers are distinct for k = 2, 3, 5,…

 There are only 6 other magic squares of squares, with a magic sum ≤10000, that are not part of the Euler’s family:

Magic sum

2823 (the AB3 magic square)

4875

6462

7150 (a square different from Benneton’s square)

7735

9775

C) Using prime numbers

What about the magic squares of squares problem, if only squares of distinct prime numbers are allowed? The research is more difficult, but it gives the following (CB16) and (CB17) 4×4 results.

29

293

641

227

277

659

73

181

643

101

337

109

241

137

139

673

CB16. The smallest 4×4 semi-bimagic square of prime numbers. S1 = 1190, S2 = 549100.

 

29²

191²

673²

137²

71²

647²

139²

257²

277²

211²

163²

601²

653²

97²

101²

251²

CB17. The smallest 4×4 magic square of squares of prime numbers, S2 = 509020.

 There are some interesting similarities if you analyze and compare the two squares:

See prime puzzles 287 and 288 on Carlos Rivera’s Web site at http://www.primepuzzles.net

11²

23²

53²

139²

107²

13²

103²

149²

31²

17²

71²

137²

47²

67²

61²

113²

59²

41²

97²

83²

127²

29²

73²

7²

109²

CB18. The smallest 5×5 magic square of squares of prime numbers, S2 = 34229.

 Two open problems from the ten published in The Mathematical Intelligencer article:
- Open problem 4. Construct a bimagic square of prime numbers (problem solved in November 2006, see here)
- Open problem 6. Construct a magic square of cubes of prime numbers (problem solved in July 2007, see here)

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