alphamagic square

An alphamagic square is a form of magic square, introduced by Lee Sallows,1, 2, 3 in which the number of letters in the word for each number, in whatever language is being used, gives rise to another magic square. In English, for example, the alphamagic square:

5 (five) 22 (twenty-two) 18 (eighteen)
28 (twenty-eight) 15 (fifteen) 2 (two)
12 (twelve) 8 (eight) 25 (twenty-five)


generates the square:


4 9 8
11 7 3
6 5 10

It turns out that there is a surprisingly large number of alphamagic squares, not only in English but also in many other languages. In French, there is just one alphamagic square involving numbers up to 200, but an additional 255 squares if the size of the entries is increased to 300. For entries less than 100, none occurs in Danish or in Latin, but 6 occur in Dutch, 13 in Finnish, and an incredible 221 in German.


Other possibilities suggest themselves, such as a three-by-three square from which a magic square can be derived that, in turn, yields a third magic square – a magic triplet. How many four-by-four and five-by-five language-dependent alphamagic squares are there? Here, for example, is a four-by-four alphamagic square in English:

26 37 48 59
49 58 27 36
57 46 39 28
38 29 56 47



1, Sallows, L. C. F. "Alphamagic squares." Abacus 4 (No. 1): 28-45, 1986.
2. Sallows, L. C. F. "Alphamagic squares, part II". Abacus 4 (No. 2): 20-29, 43, 1987.
3. Sallows, L. C. F. "Alphamagic squares." In The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and Its History, R. K. Guy and R. E. Woodrow, eds. Washington, D.C.: Mathematical Association of America, 1994.