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Ireland 2023 counting and probability
Problem
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: 
The diagram shown below is known as the Aztec Diamond of order . 
Which number is larger, the maximum number of straight triominoes or the maximum number of corner triominoes that fit simultaneously into the Aztec Diamond?
Solution
There are many ways to fit corner triominoes into the Aztec Diamond. Here is one possibility.
To show that at most straight triomino fit into the Aztec Diamond, we colour the squares as shown below with three colours in a diagonal fashion.
There are green, blue and red cells. As any straight triomino covers exactly one cell of each colour, no more than straight triominoes fit in. Hence more corner triominoes can be fitted to the Aztec Diamond.
To show that at most straight triomino fit into the Aztec Diamond, we colour the squares as shown below with three colours in a diagonal fashion.
There are green, blue and red cells. As any straight triomino covers exactly one cell of each colour, no more than straight triominoes fit in. Hence more corner triominoes can be fitted to the Aztec Diamond.
Final answer
corner triominoes
Techniques
Coloring schemes, extremal argumentsPigeonhole principle