# How Many Queens to Threat Themselves?

How many queens can be placed in a standard $8*8$ chessboard to threat each of them exactly once by other queens?

• Are other pieces allow on the board? – d'alar'cop Oct 11 '14 at 8:01
• I might come up with a proper proof about those limits btw – d'alar'cop Oct 11 '14 at 12:49
• that would be nice :) – Rafe Oct 11 '14 at 16:03

I found a 10 queen answer. I realized that d'alar'cop's reasoning was slightly off, because threatening is part of the puzzle, so 8 wasn't necessarily a limit. With the 8 inner queens in this position, you can put the final 2 queens in any 2 corners.

Update: I have a proof for a maximum of 10 queens.

Given that 2 queens in a row or a column use 1x2 and 2 queens diagonally use 2x2, it is advantageous to avoid the diagonals.

Put 2 queens in a row. You have now used up 1 row and 2 columns.
Do this again. 2 rows, 4 columns.
Again. 3 rows, 6 columns.
If you do it a 4th time, you will have used all 8 columns, so don't.
Put 2 queens in a column. 5 rows, 7 columns.
Do this again. 7 rows, 8 columns.

You can no longer add any queens, as your columns are used up.
Repeat swapping columns and rows and you end up with 8 rows and 7 columns, so you still cannot add any queens.

Any attempts at using diagonals simply make the problem end faster (exception: a single diagonal, like my displayed answer, gets you to 8 rows, 8 columns).

Therefore, 10 queens is the maximum.

Now as for allowing other pieces, I noticed that there might be extra space from d'alar'cop's 24 queens answer, and I was able to get 2 more queens in.

• Nicely done. As expected all the columns and rows are occupied – d'alar'cop Oct 12 '14 at 3:12
• awesome! I have been convinced that 8 is a limit :D thanks – Rafe Oct 12 '14 at 6:59

A queen cannot have a paired queen on the same column if it also has one on the same row. So, queens will pair up every other row and every other column. This will yield 8 queens. There are many configurations (100s probably). Here is one example:

The question didn't strictly state if this matters, but I believe this is a maximum.

If other pieces are allowed, then other pawns can be used to block queens from each other. 24 queens can be placed on the board in this case.

• Note that the original where this question comes from (how many queens can you place on a n*n board without any threatening another, without other pieces) also has n as solution. There's probably a paper worth of logic behind it, but it makes sense. – Mast Oct 11 '14 at 13:13
• @Mast I thought that would be easier to prove. Because you can use the pigeon-hole-pricinple. With more than 8 queens, you'd have one row or column with 2 or more queens - i.e. threatening. – d'alar'cop Oct 11 '14 at 13:15