An alphamagic square is a magic square in which the number of letters in the name of each number in the square generates another magic square. Thus alphamagic squares are language dependent. Alphamagic squares were invented by Lee Sallows in 1986. Here is an example in English. Start with the following magic square:
Now consider the array of corresponding number words:
Counting the letters in each number word generates the following square which turns out to also be magic:
Since the generated array is also a magic square, the original square is said to be alphamagic. Could the generated square also be alphamagic? It is not known if any such squares exist.
The above example enjoys another special property: the 9 numbers in the lower square are consecutive. This prompted Martin Gardner to describe it as "Surely the most fantastic magic square ever discovered." Of course, most alphamagic squares do not share this latter property.
However, Sallows then went on to produce a still more magical version in the form of the geometric magic square on the right. Any three shapes in a straight line tile the cross; the numbers printed on them sum to 45.
- A surprisingly large number of 3 × 3 alphamagic squares exist—in English and in other languages. French allows just one 3 × 3 alphamagic square involving numbers up to 200, but a further 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 there are 6 in Dutch, 13 in Finnish, and an incredible 221 in German. Yet to be determined is whether a 3 × 3 square exists from which a magic square can be derived that, in turn, yields a third magic square—a magic triplet. Also unknown is the number of 4 × 4 and 5 × 5 language-dependent alphamagic squares.
- Wolfram MathWorld: Alphamagic Squares
- ACM Digital Library, Volume 4 Issue 1, Fall 1986
- Mathematical Association of America, Ivars Peterson's MathTrek: Alphamagic Squares
- Gardner, Martin (1968), A Gardner's Workout: Training the Mind and Entertaining the Spirit, p. 161, A K Peters/CRC Press, Natick, Mass., July 2001, ISBN 1568811209
- The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes, by David Darling, p. 12, Hoboken, NJ: Wiley, 2004, ISBN 0471270474
- Encyclopedia of Science, Games & Puzzles: Alphamagic Squares