Maths Olympiad Prep

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, 2023

Combinatorics Difficulty 4.5 AIME Find the answer United States

Problem:

Compute the number of ways to tile a 3×53 \times 5 rectangle with one 1×11 \times 1 tile, one 1×21 \times 2 tile, one 1×31 \times 3 tile, one 1×41 \times 4 tile, and one 1×51 \times 5 tile. (The tiles can be rotated, and tilings that differ by rotation or reflection are considered distinct.)

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

Our strategy is to first place the 1×51 \times 5 and the 1×41 \times 4 tiles since their size restricts their location. We have three cases:

- Case 1: first row. There are 4 ways to place the 1×41 \times 4 tile. There is an empty cell next to the 1×41 \times 4 tile, which can either be occupied by the 1×11 \times 1 tile or the 1×21 \times 2 tile (see diagram). In both cases, there are 2 ways to place the remaining two tiles, so this gives 4⋅2⋅2=164 \cdot 2 \cdot 2=16 ways.

Figure 1

- Case 2: middle row. There are 4 ways to place the 1×41 \times 4 tile, and the 1×11 \times 1 tile must go next to it. There are 2 ways to place the remaining two tiles, so this gives 4⋅2=84 \cdot 2=8 ways.

- Case 3: bottom row. This is the same as Case 1 up to rotation, so there are also 16 ways to place the tiles here.

In total, we have 16+8+16=4016+8+16=40 ways to place the tiles.

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