KA-0054Counting shapes hidden inside a figure
3 points, difficulty 1 of 3Level 3–4about 60sA rectangle is divided by two vertical lines into three parts. How many rectangles of any size are in the picture?
Text description of the figure
A wide rectangle divided by two vertical lines into three smaller rectangles side by side.
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Count runs of neighbouring cells.
2The strategy
Every rectangle here is a block of cells that sit next to each other. Count the blocks of one cell, then of two, then of three.
3The full solution
Blocks of one: 3. Blocks of two: the left pair and the right pair, so 2. Blocks of three: 1. Total 3 + 2 + 1 = 6.
Solution
The reliable way
Number the cells 1, 2, 3 from the left. Every rectangle is a run of neighbouring cells. Runs of one: three of them. Runs of two: 1-2 and 2-3, so two. Runs of three: the whole rectangle, one. In total 3 + 2 + 1 = 6. The transferable idea: counting by size stops you missing the middle sizes.
The elegant way
A rectangle is fixed by choosing 2 of the 4 vertical lines, and there are 4 x 3 divided by 2 = 6 such choices.
Why this is on the test: It is the hidden-shape count in its simplest form, so the by-size habit is established before the triangle and grid versions.