If we made a five by five grid the max would be fifteen counters because if we did a two by two grid you can only get three counters ...
Gina's comments follow :
The students did not have time to write up their solutions in full. The text above is one student's attempt to begin a solution.
Some of the students placed counters on the grid and moved them around to "fit them in" without making a square. Others filled up the grid and then strategically removed counters until there were no squares.
The students found that they could fit a maximum of 6 counters on a 3x3 grid, and a maximum of 10 counters on a 4x4 grid. They initially thought that, since the maximum changed by 4, it would mean that going to a 5x5 grid would increase the maximum by 4 again, from 10 to 14. But then someone pointed out that on a 2x2 grid you could fit a maximum of 3 counters, which doesn't fit the +4 pattern. They thought then that maybe the difference between the maximum increases by 1 with every increase in grid size. They speculate that the maximum number of counters on a 5x5 grid would thus be 10+5=15, but they have not had time to check.