Alphamagic square
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.[1] Alphamagic squares were invented by Lee Sallows in 1986.[2]
Verification
To verify that a magic square is also alphamagic square, the magic square is converted into and array of corresponding number words. For example
| 5 | 22 | 18 |
| 28 | 15 | 2 |
| 12 | 8 | 25 |
converts to ...
| five | twenty-two | eighteen |
| twenty-eight | fifteen | two |
| twelve | eight | twenty-five |
Counting the letters in each number word generates the following square which turns out to also be magic:
| 4 | 9 | 8 |
| 11 | 7 | 3 |
| 6 | 5 | 10 |
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.[3]
The above example has been described as "the most fantastic magic square ever discovered"[4] due to its unique property of being consecutive (three to eleven).
Other languages
The Universal Book of Mathematics provides the following information about Alphamagic Squares:[5][6]
- 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.
Variations
The geometric magic square is a variation of the alphamagic square.
References
- ↑ 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