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

 (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:

• S2 = (p² + q² + r² + s²)².
 (+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:

• S2 = (a² + b² + c² + d²)(p² + q² + r² + s²).

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

• pr + qs = 0,
• a / c = [ d(pq + rs)  b(ps + qr)] / [b(pq + rs) + d(ps + qr)].

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.

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.

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:

• Squares generated by (1, 7, 10, 12) and (2, 4, 7, 15) can be permuted to get the square generated by (A)
• Squares generated by (2, 13, 20, 21) and (5, 6, 13, 28) can be permuted to get the square generated by (B)
• Squares generated by (2, 14, 20, 24) and (4, 8, 14, 30) can be permuted to get the square generated by (C)

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:

• The smallest prime number is the same in the two squares, 29, and it is located in the same place, the first cell.
• The biggest prime number is also the same in the two squares, 673.
• The same pair of twin prime numbers is used, 137 and 139.
• Two other identical numbers are used in the two squares, 101 and 277.
• There is a big gap in the numbers used in the two squares:
• 304 in the (CB16) square, nothing is used between 337 and 641
• 324 in the (CB17) square, nothing is used between 277 and 601.

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)