Write 0 or 1 in each cell of the 5x5 table such that each 2x2 square of the 5x5 table contains exactly 3 equal numbers. What is the largest possible sum of all numbers in the table?

Question

Please give me an answer with explanation.

Answer 1

This problem offers much to learn about matrices.

For example, "each 2x2 square" means that you rally have a 4x4 matrix that has a 5x5 extended version. The restriction of 3 equal numbers means that each cell in the 4x4 matrix has a sum of 1 or 3. The cells of the 5x5 matrix have either 0 or 1.

Also, rotating the entire matrix (or flipping it) is really the same solution -- with all the same characteristics.

so, the lowest total of the 1's in the 5x5 matrix is 4 --:

0 0 0 0 0

0 1 0 1 0

0 0 0 0 0

0 1 0 1 0

0 0 0 0 0

and the largest total of the 1's is 21 --

1 1 1 1 1

1 0 1 0 1

1 1 1 1 1

1 0 1 0 1

1 1 1 1 1

Now, for fun, determine how many unique solutions exist (note: this is a valuable exercise).

Think about this: How many unique complete "pattern of moves" exist for tic-tac-toe? (if put into a "solution tree" ?) [note: rotation or "flipping" constitute the same pattern, so there are exactly three initial moves in the "solution tree": Corner, Edge, Center.]