KA-0046Counting by position (sliding window)
4 points, difficulty 2 of 3Level 3–4about 75sHow many 2 by 2 squares fit on a 4 by 4 board, if they must line up with the grid?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Count where the top-left corner can sit.
2The strategy
A 2 by 2 square needs one more column to its right and one more row below it, so its top-left corner cannot use the last column or the last row.
3The full solution
The corner can sit in 3 of the 4 columns and 3 of the 4 rows, so 3 x 3 = 9 positions.
Solution
The reliable way
Fix each 2 by 2 square by its top-left corner. That corner needs room for one more column to the right and one more row below, so it can use 3 of the 4 columns and 3 of the 4 rows. That gives 3 x 3 = 9. The transferable idea: count the positions of one corner, sliding one step at a time.
The elegant way
On an n by n board there are (n-1) squared aligned 2 by 2 squares, so a 4 by 4 board gives 3 squared = 9.
Why this is on the test: It is the same sliding-window count one size up, which is where students who memorised the answer four rather than the method come unstuck.