There is a 3x3 magic square (each row, each column and each main diagonal add to the same value), presented in a number base higher than ten but no higher than 36, so that A is a digit representing ten, B representing eleven, etc.

In this magic square, one of the cells contains what looks like the word DO (that's a letter O), which, by the rules, represents thirteen times whatever the base is, plus twenty-four.

Not only does the word DO appear as one of the numbers in the grid, but also the number representations in the bottom row, if concatenated together, form the word MAGIC.

What is this magic square?

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Click here to see the original New Scientist Enigma No. 1680 version of this puzzle.

With a bit of help from excel.

The cells of the magic square can be written as

a          b     -a-b+3c
-2a-b+4c   c     2a+b-2c
a+b-c    -b+2c    -a+2c

With magic constant 3c.

Let the base be x so we are looking for DO = 13x+24

MAGIC can be parsed in 6 ways, but only 3 have only two letters per number (I figured a 3 digit number would be too big):

M|AG|IC has sum 28x+50

MA|G|IC has sum 40x+38

MA|GI|C has sum 38x+40

Each of the above can make a magic square based on the value of x.  Also each is a multiple of 3 only if x=3n+1 which will make searching easier.

I then used excel to create a magic square for each parsing of MAGIC.  I really only had to use value of n from 8 to 11 since the base has to be at least 24 for O to be valid.

When I changed n to 11 (x=34), I noticed the magic square for second parsing of MAGIC had the center number 466, the same as 13x+24 = DO so i got it:

758 16  624
332 466 600
308 916 174

Which is in base 10.  In base 34 the top numbers are MA, G, IC
and the 466 is DO

(The others are 9Q, HM, 92, QW, 54)

[it turns out you can cut and past a spreadsheet into this window but afterward it doesn't display right.]

Edited on March 27, 2012, 12:00 pm

Posted by Jer on 2012-03-27 11:57:46