# Cutting apart a chessboard

talzamir
Posers and Puzzles 27 Oct '11 20:25
1. talzamir
Art, not a Toil
27 Oct '11 20:25
Not that any of us would do such a thing, but if we did, and wanted to cut it to 64 separate squares, how many cuts would it take? A single cut goes between any two rows or columns to the other end, cutting the board or part thereof in two. For example the first cut could be between the columns e and f, making two rectangles, A1-E8 and F1-H8. The second would cut one of the pieces in two, so you might end up with, say, A1-E8, F1-H6 and F7-H8. That procedure would be continued until all 64 squares are separate.
2. forkedknight
Defend the Universe
27 Oct '11 22:111 edit
I get the same answer for each attempt I have tried.

63

It makes sense, since no cut can cross another cut.
3. talzamir
Art, not a Toil
27 Oct '11 22:58
Perfect sense indeed, as each cut, regardless of whether it splits off one square or many, increases the number of pieces by one. ðŸ™‚
4. 28 Oct '11 06:361 edit
And if the cuts didn't have to be straight (but still followed the edges of the squares) then the number would be considerably less.
5. forkedknight
Defend the Universe
28 Oct '11 15:01
I don't think so.

You can still increase the total number of pieces by only one.
6. talzamir
Art, not a Toil
28 Oct '11 15:481 edit
Agreed. If a "cut" ends when a piece of the board is falls into two pieces, n - 1 is how many it takes where n is the number of pieces. Of course, if a cut is defined differently, a whole new problem is found. If a cut is a line that follows goes between the squares, ending at an intersection from which there are no routes out.. it could be doable with as few as
fourteen
?
7. 28 Oct '11 22:30
Agreed it comes down to the definition of a cut. When I made my earlier statement I didn't define the separation of the board into 2 pieces as a complete cut - I envisaged cutting from one edge of the original board to another edge following a diagonal / stepped path but not turning back on a previous direction. Visualise cutting steps into the board starting at one corner. I also came up with 14.
8. 30 Oct '11 10:42
Interesting 14 is also the answer if the straight cuts can cross previous cuts (without rearranging)
9. talzamir
Art, not a Toil
30 Oct '11 16:31
Not a coincidence. Every intersection on the board has four lines leading into it, so no cut can end there if none starts there. That leaves 7 end points on each side of the board, 4 x 7 = 28; each cut removes 2 of them; 28 : 2 = 14.