## Task 4 - 010524

From a wooden board of length $a=60 \mathrm{~cm}$ and width $b=15 \mathrm{~cm}$, 12 small boards of size $5 \mathrm{~cm}$ by $15 \mathrm{~cm}$ are to be cut out. Lutz is trying to get by with as few cuts as possible.

How many cuts does he have to make at a minimum? (Sawing "in bulk" is not allowed.) How many centimeters is the saw path?
a)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0017.jpg?height=129&width=399&top_left_y=227&top_left_x=383)
b)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0017.jpg?height=126&width=411&top_left_y=228&top_left_x=845)
c)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0017.jpg?height=128&width=394&top_left_y=233&top_left_x=1325)

Lutz can make the cuts in three different ways.

In image a) he saws 11 times vertically, resulting in 12 small planks. He needs 11 cuts.

In image b) he cuts twice horizontally and then 9 times vertically, which again results in 11 cuts. He gets the same result if he cuts three times vertically and then 8 times horizontally.

In the third image c) he cuts the board into 4 smaller boards of size \(15 \mathrm{~cm} \times 15 \mathrm{~cm}\) with 3 cuts, and then each smaller board is cut twice, either horizontally or vertically as needed.

In all three cases, he needs 11 cuts. The total sawing distance is \(11 \cdot 15 \mathrm{~cm} = 165 \mathrm{~cm}\) in case a). In the other two cases, he also has to cut \(165 \mathrm{~cm}\).