[ Tiling with domino bones and tiles]

Giuseppe has a sheet of plywood, measuring $22 \times 15$. Giuseppe wants to cut out as many rectangular blanks of size $3 \times 5$ as possible from it. How can he do this?

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Note that more than 22 blanks cannot be obtained. Why?

## Solution

First of all, note that Giuseppe will not be able to get more blanks than \((22 \times 15) / (3 \times 5) = 22\) pieces. Now let's proceed to cutting.

Cut the sheet into three strips across the side of 22: \(5 \times 15\), \(5 \times 15\), and \(12 \times 15\). Now cut the third piece along the side of 12 into four equal pieces \(3 \times 15\). In total, we get 6 pieces - two \(5 \times 15\) and four \(3 \times 15\). From the first two pieces, we will get 5 blanks \(5 \times 3\) each, and from the remaining four - 3 blanks \(3 \times 5\) each. In total, we get 22 pieces (see figure).

![](https://cdn.mathpix.com/cropped/2024_05_06_1b5b6121a03c921b88efg-15.jpg?height=417&width=548&top_left_y=1378&top_left_x=755)

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