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. Since different languages will have a different number of letters for the spelling of the same number, alphamagic squares are language dependent. Alphamagic squares were invented by Lee Sallows in 1986.
To verify that a magic square is also alphamagic square, the magic square is converted into and array of corresponding number words. For example
converts to ...
Counting the letters in each number word generates the following square which turns out to also be magic:
If the generated array is also a magic square, the original square is verified as alphamagic. It is not known if any verification squares exist which are also alphamagic.
The above example has been described as "the most fantastic magic square ever discovered" due to its unique property of being consecutive (three to eleven).
- 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.
The geometric magic square is a variation of the alphamagic square.
- 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