KA-0133Counting shapes hidden inside a figure
5 points, difficulty 3 of 3Level 3–4about 105sHow many squares of any size can be found on a 3 by 3 board?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Count by size: 1 by 1, then 2 by 2, then 3 by 3.
2The strategy
Take the sizes one at a time. For each size, slide the square across and down and count where its top-left corner can sit, rather than trying to see them all at once.
3The full solution
Size 1: 3 x 3 = 9. Size 2: 2 x 2 = 4. Size 3: 1. Total 9 + 4 + 1 = 14.
Solution
The reliable way
Take the sizes separately. A 1 by 1 square has 3 x 3 = 9 positions. A 2 by 2 square needs one spare column and one spare row, so its corner sits in 2 x 2 = 4 places. A 3 by 3 square fits in exactly 1 way. Adding: 9 + 4 + 1 = 14. The transferable idea: counting by size, largest or smallest first, is what stops the middle sizes being forgotten - and every wrong answer here is a missing middle size.
The elegant way
On an n by n board the total is the sum of the first n square numbers, so a 3-board gives 9 + 4 + 1 = 14.
Why this is on the test: Hidden-shape counting is a Kangaroo staple, and stopping at the sizes that are drawn is the error the question is built around.