Arranging 10 counters in 5×5 grid

Question

Draw a 5×5 grid. Place 10 counters in it. You can place only one counter in each space. Get an even number of them in each row and column and two main diagonals. Unfortunately, I have not been able to solve this one :(

Answer 1

A useful approach is to start with two counters in the top row. Fill each row below by going down one row and over two or three spaces from the row above, wrapping around. That will give you two in each row and column, so you only need to check the diagonals. My first try worked.

Answer 2

\begin{array} {|r|r|}\hline X & X & & & \\ \hline X & X & & & \\ \hline X & X & & & \\ \hline X & & X & \phantom{X} &\phantom{X} \\ \hline & X & X & & \\ \hline \end{array}

Thought process: I initially wanted to prove that this was not possible. I started by considering partitions of $10$ in even numbers.

The first one to test was $4,4,2$. I decided to ignore the diagonals first so that the position of the markers do not matter, as long as they stay in the same column.

Now I have filled in the first column and three of the second that overlap with the first (by pigeonhole principle). Considering the 2 markers in the third column, I cannot place them in the first three rows. Then this table was generated and it satisfies all the conditions (if you allow one diagonal to be blank.)

Answer 3

I tried to get $2$ in each column, I realized I couldn't avoid both diagonals entirely so I tried to avoid one of them and get two in the other diagonal. Then it was just a matter of trying to get an even number in each row.

$\begin{pmatrix} 0 & 0 & 0 & 0 & 0 \\ x & 0 & x & x & x \\ x & 0 & 0 & 0 & x \\ 0 & x & x & 0 & 0 \\ 0 & x & 0 & x & 0 \\ \end{pmatrix}$