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Problem 1340

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Combinatorics Difficulty 5.6 Prove it Irish Mathematical Olympiad · Ireland

A triomino is formed by joining three equal squares edge to edge. Up to rotation there are only two types, namely straight and corner triominoes:
Figure 1
Figure 2
The diagram shown below is known as the Aztec Diamond of order 7/27/2.
Figure 3
Which number is larger, the maximum number of straight triominoes or the maximum number of corner triominoes that fit simultaneously into the Aztec Diamond?

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

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Official solution

There are many ways to fit 88 corner triominoes into the Aztec Diamond. Here is one possibility.
Figure 4

To show that at most 77 straight triomino fit into the Aztec Diamond, we colour the squares as shown below with three colours in a diagonal fashion.
Figure 5

There are 1111 green, 77 blue and 77 red cells. As any straight triomino covers exactly one cell of each colour, no more than 77 straight triominoes fit in. Hence more corner triominoes can be fitted to the Aztec Diamond.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.