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:
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 |
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: |
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