Nine heart cards from an ordinary deck can easily be arranged to form a magic square so that each row, column and main diagonal has the largest possible constant sum, 27.
(Jacks count 11, queens 12, kings 13.)
Drop the requirement that each value must be different.
Allowing duplicate values, what is the largest constant sum for an order-3 magic square that can be formed with nine cards taken from a deck?
Attributed to the great Martin Gardner.
Consider the general 3x3 magic square with constant sum zero beginning with A,B,C as shown:
A B -A-B
C 2A+B+C -2A-B-2C
-A-C -2A-2B-C X
By the diagonal X=-3A-B-C but by the row X=3A+2B+2C.
So 2A+B+C=0 To make them integers close to zero: A=1, B=C=-1 and the square becomes
1 -1 0
-1 0 1
0 1 -1
Add 12 to each of them makes
13 11 12
11 12 13
12 13 11
With a magic sum 36
Edited to fix typo. Nice catch Charlie.
Edited on December 12, 2015, 8:27 pm
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Posted by Jer
on 2015-12-11 12:57:28 |
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