# Puzzles

## 7 December

There are 15 dominos that can be made using the numbers 0 to 4 (inclusive):

The sum of all the numbers on all these dominos is 60.

Today's number is the sum of all the numbers on all the dominos that can be made using the numbers 5 to 10 (inclusive).

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Each number will appear 7 times: one time paired with the numbers six numbers 5 to 10, plus an extra appearance on the tile containing the same number twice.
The total of all the numbers is therefore 7×(5+6+...+10)=7×45=**315**.

## 4 December

There are 5 ways to tile a 3×2 rectangle with 2×2 squares and 2×1 dominos.

Today's number is the number of ways to tile a 9×2 rectangle with 2×2 squares and 2×1 dominos.

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Let \(a_n\) be the number of ways to tile an \(n\times2\) rectangle.

There is one way to tile a 1×2 rectangle; and there are three ways to tile a 2×2 rectangle. Therefore \(a_1=1\) and \(a_2=3\).

In a \(n\times2\) rectangle, the rightmost column will either contain a vertical 2×1 domino, two horizontal 2×1 dominoes, or a 2×2 square.
Therefore \(a_n=a_{n-1}+2a_{n-2}\).

Using this, we find that there are **341** ways to tile a 9×2 rectangle.