Checkerboard Triangle

Geometry Level 1

Which area is larger?

Blue area Green area Equal area

3 solutions

Calvin Lin Staff
Sep 23, 2016

[Not a complete solution, fill in the missing details]

Can you generalize this result to a right triangle of sides $m$ and $n$ ?

If both halves are the same, then both halves have equal total areas of either green or blue. Hence, if there's more green squares than blue, then in each half, there's greater total area green than blue. This will be true in any case where both $m$ and $n$ are odd.

Contrariwise, if both $m$ and $n$ are even, then it becomes trivial to show that both colors have equal total area.

However in the case of mixed parity of $m$ and $n$ , the picture apparently gets more complicated.

Michael Mendrin - 4 years, 8 months ago

Great!

(restating) In the case when both sides have the same parity, then we can show that these 2 triangles (in the solution) are identical. As such, we can simply consider the squares in the rectangle, which is much easier to analyze

Calvin Lin Staff - 4 years, 8 months ago

Marta Reece
Apr 19, 2017

Green area = 9. Blue area = 8.5.

In green the six partial squares make 3 whole squares.

In blue the five partial squares make 2.5 whole squares.

A Former Brilliant Member
Apr 4, 2018

By observation, we can see that the square in the right angle is a green square and the polygons on the other vertices are colored green. Therefore, we can conclude that the green area is larger.

Not quite sure what you mean. Can you elaborate?

Calvin Lin Staff - 3 years, 2 months ago

Checkerboard Triangle

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