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. Thus alphamagic squares are language dependent.[1] Alphamagic squares were invented by Lee Sallows in 1986.[2] Here is an example in English. Start with the following magic square:

5 22 18
28 15 2
12 8 25

Now consider the array of corresponding number words:

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

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.[3]

Figure 1:   A geomagic square that is also alphamagic

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."[4] 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 of Figure 1. Any three shapes in a straight line tile the cross; the numbers printed on them sum to 45.

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.

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