A 4x4 array with entries from the set {0,1,2,3} is such that:

• The sum of each of the four rows is divisible by 4, and:
• The sum of each of the four columns is divisible by 4

Determine the total number of arrays having the above property.

Notes:

(i) Two or more cells of a given row or column may or may not contain the same number.
(ii) Disregard rotations or reflections.

To construct a row we can place any digit in the first 3 spots and that fixes one possibility for the last.  So there are 4^3=64 possible rows.
We can construct the first 3 rows this way.
64^3 = 262144.
These three rows will leave one possible fourth row because the columns have to add up.

The actual answer will be lower than 262144 because some arrays will have rotation or reflection symmetrical twins that part (ii) says not to count.

My plan is to remove these.

Posted by Jer on 2014-05-20 13:36:50