Maths Olympiad Prep

Library / /15 of 42

Combinatorics Difficulty 5.6 AIME, harder Prove it 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?

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.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic and difficulty added by this site.