A 3 by 3 board is drawn. How many 2 by 2 squares can be found on it, if they must line up with the grid?
Text description of the figure
A square board divided into a 3 by 3 arrangement of nine equal small squares.
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Slide the small square one step at a time.
2The strategy
A 2 by 2 square is fixed by where its top-left corner sits. Count the corner positions instead of trying to see every square at once.
3The full solution
The top-left corner of a 2 by 2 square can sit in the left or middle column, and in the top or middle row. That is 2 times 2, so 4 positions in total.
Solution
The reliable way
Fix the 2 by 2 square by its top-left corner. That corner must leave room for one more column to the right and one more row below, so it can sit in 2 of the 3 columns and 2 of the 3 rows. That gives 2 times 2 = 4 positions. The transferable idea: count the positions of one corner, not the shapes.
The elegant way
On an n by n board the number of 2 by 2 squares is (n-1) squared, because each of the two directions loses exactly one position at the edge.
Why this is on the test: Students who cut a board into blocks instead of sliding a window undercount every question of this shape.