Can 12 identical coins be arranged 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 coins on top of each other.)
Coins cannot be too many or too few. Understand how many is too many and how many is too few.

## Solution

a) Since all the 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_e20a6bbfe3399b1d06acg-06.jpg?height=348&width=349&top_left_y=955&top_left_x=200)

b)

![](https://cdn.mathpix.com/cropped/2024_05_06_e20a6bbfe3399b1d06acg-06.jpg?height=348&width=366&top_left_y=958&top_left_x=617)
B)

![](https://cdn.mathpix.com/cropped/2024_05_06_e20a6bbfe3399b1d06acg-06.jpg?height=352&width=355&top_left_y=953&top_left_x=1066)

c)

![](https://cdn.mathpix.com/cropped/2024_05_06_e20a6bbfe3399b1d06acg-06.jpg?height=357&width=355&top_left_y=959&top_left_x=1526)

d)

In figure c), there are two coins in each corner, and in figure d) there are three.

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) Not possible; b)-d) possible.