KA-0003Counting shapes hidden inside a figure
4 points, difficulty 2 of 3Level 3–4about 90sA large triangle is cut by two lines from its top corner into three small triangles side by side. How many triangles of any size appear in the picture?
Text description of the figure
A large triangle with two straight lines drawn from its top vertex down to the base, dividing it into three small triangles side by side.
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Count by size: one cell, two cells, three cells.
2The strategy
Every triangle here is a run of neighbouring small triangles. Count how many runs of length one there are, then length two, then length three.
3The full solution
Runs of one: 3. Runs of two: the left pair and the right pair, so 2. Runs of three: the whole triangle, so 1. Total 3 + 2 + 1 = 6.
Solution
The reliable way
Number the small triangles 1, 2, 3 from the left. Every triangle in the picture is a block of neighbouring small ones. Blocks of size one: 1, 2, 3 - that is three. Blocks of size two: 1-2 and 2-3 - that is two. Blocks of size three: 1-2-3 - that is one. In total 3 + 2 + 1 = 6. The transferable idea: counting by size stops you missing the middle sizes.
The elegant way
Any triangle is fixed by choosing two of the four lines from the top corner, and there are 4 times 3 divided by 2 = 6 such choices.
Why this is on the test: Hidden-shape counting is a Kangaroo staple, and the standard error is stopping at the shapes that are drawn as separate cells.