![](https://cdn.mathpix.com/cropped/2024_05_06_a1c7cb3d3682c5b3cf5fg-17.jpg?height=108&width=1541&top_left_y=755&top_left_x=13)

Is it possible to arrange 12 identical coins along the walls of a large square box so that along each wall there are exactly
a) 2 coins;
b) 3 coins;
c) 4 coins;
d) 5 coins;
e) 6 coins;
f) 7 coins;

(You are allowed to stack the coins on top of each other.) In the cases where it is possible, draw how to do it. In the other cases, prove that it is impossible to arrange the coins in this way.
a) Since, by condition, all coins must be placed along the walls, and each wall is touched by exactly two coins, the total number of coins is no more than 8.

b)-d) Examples of the required arrangements are shown in the figure

![](https://cdn.mathpix.com/cropped/2024_05_06_a1c7cb3d3682c5b3cf5fg-17.jpg?height=536&width=788&top_left_y=1438&top_left_x=40)

e) Note that a coin cannot touch two opposite walls of the box. Therefore, the total number of coins touching two opposite walls is $7+7=14$ > 12.

## Answer

a), e) no; b), c), d) yes.