Alphamagic square

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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]


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[edit]

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.


The geometric magic square is a variation of the alphamagic square.


External links[edit]