## 45. Math Puzzle $2 / 69$

Hans is supposed to nail a $90 \mathrm{~cm}$ wide gap in a wooden fence. He has eight $10 \mathrm{~cm}$ wide and sixteen $8 \mathrm{~cm}$ wide planks of suitable length. A saw, with which he could cut the planks lengthwise, is not available.

How many and which possibilities does Hans have to close the opening without any gaps?
Number of required boards: $m$ (10 cm wide) and $n$ (8 cm wide). Then, $10 m + 8 n = 90$. The equation is simplified by dividing by 2 and rearranged: $4 n = 5(9 - m)$.

From this, it follows that $n$ must be divisible by 5, so $n = 0, 5, 10, \ldots$

1st Case: $(n=0, m=9)$ Excluded, as only 8 cm boards are considered

2nd Case: $(n=5, m=5)$, i.e., 5 boards 10 cm wide, 5 boards 8 cm wide

3rd Case: $(n=10, m=1)$, i.e., 1 board 10 cm wide, 10 boards 8 cm wide

4th Case: $(n=15, m=-3)$, Excluded, as $m$ must be greater than 0

Only Case 2 and 3 are valid solutions.