5. By arranging sticks in a hexagonal grid, students create shapes called hexes. In the image, a hex is shown that has five hexagons in the center (highlighted in the image). Sticks are sold in boxes containing 45 sticks each, and the price of one box is 1 euro and 25 cents.

What is the minimum amount that students must spend to buy enough sticks to form a hex that has 13 hexagons in the center?

![](https://cdn.mathpix.com/cropped/2024_05_30_e5ca9b3e8971fb159cf9g-5.jpg?height=483&width=448&top_left_y=110&top_left_x=1455)
## First Solution.

We can observe that all the matches are arranged in three directions. Let's first count the horizontal matches.

Notice that the hexagons are arranged in columns, with the number of columns equal to the number of hexagons in the middle column. In the middle column, there are 13 hexagons, which use 14 horizontal matches. In the columns to the left and right of it, there are 12 hexagons, which use 13 horizontal matches each. Since there are a total of 13 columns, the number of horizontal matches is

$$
14 + 2 \cdot 13 + 2 \cdot 12 + 2 \cdot 11 + 2 \cdot 10 + 2 \cdot 9 + 2 \cdot 8 = 140
$$

In each of the three directions, the number of matches is the same, which means the total number of matches is $3 \cdot 140 = 420$.

Each box contains 45 matches, $420 \div 45 = 9$ with a remainder of 15, so at least 10 boxes of matches are needed.

The price of one box is 1 euro and 25 cents, so they will spend 12 euros and 50 cents for 10 boxes.