This is not a chess problem!

In the following position you can see six pawns that have been arranged into lines of three. Each pawn stands at the intersection of exactly two lines and each line contains exactly three pawns.

For this problem 2 pawns don't count as a line.

enter image description here

Can you arrange ten pawns in such a way that each pawn is at the intersection of exactly three lines and each line contains exactly 3 pawns?

Each pawn must be positioned in the exact center of a square of a standard 8x8 chessboard.

  • 5
    $\begingroup$ The pawns should be in the center of a tile in a standard chessboard, right? $\endgroup$
    – Magma
    Jul 18, 2020 at 17:24
  • 1
    $\begingroup$ Oops! Indeed. Corrected. I wanted to check if you people pay attention. :-) $\endgroup$
    – Florian F
    Jul 18, 2020 at 21:36
  • $\begingroup$ a line could also be drawn through the f7 and h3 pawns, only touching those two, but I think you meant that lines with < 3 pawns can be ignored. $\endgroup$
    – ilkkachu
    Jul 19, 2020 at 8:01
  • $\begingroup$ Yes. I am interested in alignments and 2 pawns don't count as an alignment. I updated the question. $\endgroup$
    – Florian F
    Jul 19, 2020 at 11:03
  • $\begingroup$ never mind, something was wrong with my vision I guess. $\endgroup$ Aug 1, 2020 at 0:07

1 Answer 1


It's possible

even on a 6x6 chessboard:

enter image description here


If we skew the previous solution we can obtain 6x5:

enter image description here

  • $\begingroup$ OK, nice, you could have made it a 6x5, but still good. $\endgroup$
    – Florian F
    Jul 18, 2020 at 21:41
  • 1
    $\begingroup$ (after the update) OK, now I can admit I am impressed how fast you got it. $\endgroup$
    – Florian F
    Jul 18, 2020 at 22:00
  • $\begingroup$ How? Did you just guess and check or what? $\endgroup$ Jul 19, 2020 at 4:14
  • $\begingroup$ At first I ignored the chessboard and just thought about the incidence structure of pawns and lines. One possibility for this incidence structure is the Desargues configuration. So I placed various Desargues configurations onto the chessboard until they fit onto centers of squares. $\endgroup$
    – Magma
    Jul 19, 2020 at 10:27

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