Posted by Mark on .
The squares of a 3×3 grid are filled with nonnegative integers such that the sum of each row and the sum of each column is 7. How many different ways can the squares be filled? The numbers in each grid square does not need to be distinct. Rotations and reflections are distinct arrangements.

Math (please help me Steve) 
MathMate,
Here's a lower bound solution. You may want to investigate other possibilities.
The number 7 can be partitioned into 3 ascending nonnegative integers in 8 ways, namely:
007#
016
025
034
115#
124
133#
233#
There are 3 permutations of partitions (indicated #) with nondistinct digits and 6 permutations of those with distinct digits for a total of
4*3+4*6=36 permutations of nonnegative integers whose sum is 7.
Each of these permutations can make a 3x3 grid as follows by putting a permutation as the first line, then rotate the digits to the left:
124
241
412
We can also make another grid by rotating them to the right, as follows:
124
412
241
This doubles the number of grids for a total of
36*2=72 grids. 
Math (please help me Steve) 
math,
what about this case, i didn't enumerated it
1 2 4
0 5 2
6 0 1 
Math (please help me Steve) 
exactly,
Hint: you need these:
(1) a,b,c,d are nonnegative integers
(2) 7−a−b≥0⇔a+b≤7
(3) 7−c−d≥0⇔c+d≤7
(4) 7−a−c≥0⇔a+c≤7
(5) 7−b−d≥0⇔b+d≤7
(6) a+b+c+d−7≥0⇔a+b+c+d≥7
Try to count how many satisfies these conditions (: