All the triangles are a big triangle

Geometry Level 4

A triangle can be cut into $N$ congruent smaller triangles by lines parallel to its sides.

What is the sum of all possible positive integers $N \leq 100?$

$The blue lines cut the larger triangle into $N=4$ smaller congruent triangles.$ The blue lines cut the larger triangle into $N=4$ smaller congruent triangles.

This arrangement is not allowed because it contains trapezoids.

Note : The lines must cut the larger triangle into smaller triangles—they cannot create any other kinds of polygons.

The answer is 384.

1 solution

L X
Mar 5, 2018

The first thing to notice is that you need an equal amount of lines in each direction to slice the triangle into congruent parts.

Then you can try out the different possibilities. One line in every direction slices into 4 equal parts (the original example). 2 lines slice into 9 parts, and 3 lines into 16, 4 lines into 25. A pattern emerges: $n$ lines per direction slice up the triangle into $(n - 1)^2$ parts. The most triangles for this given problem is exactly 100 using 9 lines in every direction.

To get to the solution you just have to add up all square numbers $\le100$ (leaving out 1, because that is not slicing the triangle at all). So:

$N = 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2$

$N = 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100$

$N = \boxed{384}$

All the triangles are a big triangle

The answer is 384.

1 solution

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