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Destroying a checkerboard

Question

You have just lost your 143rd straight game of checkers and have vowed never to play another game. To confirm your vow you decide to saw your wooden checkerboard into pieces that contain no more than a single (red or black) square. With each use of the saw you may pick up a piece of the board and make one straight cut, along boundaries of individual squares. You wish to inflict as much damage as possible with each cut, so you first calculate the minimum number of saw cuts needed to finish the job. And that number is ... (spoilers appreciated.)

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21:

7 horizontal cuts of length 7, which allows the entire board to be picked up at once.

7 vertical cuts of length 8 (each one is straight, and separates 8 squares)

this leaves a strip of 8 squares.

7 horizontal cuts, each removing one square from the remaining strip, last one separates 2.

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Spoiler

I can do six, I think. cut board in half, stack halves, cut in half again (2x8 chunks now), stack, cut again, stack (1x8), cut in half other way, stack (1x4), cut in half, stack (1x2), cut.

 

Edited by flamebirde
I can do even better!

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5 hours ago, flamebirde said:
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I can do six, I think. cut board in half, stack halves, cut in half again (2x8 chunks now), stack, cut again, stack (1x8), cut in half other way, stack (1x4), cut in half, stack (1x2), cut.

 

That would not be permitted. The OP says with each use of the saw you may pick up one piece of the board and make a straight cut.

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@CaptainEd Wow. Not the answer I had in mind -- a much better one! Nice.

Bonus (slightly modified) version.
Same question, but this time no cut may end short of the opposite edge. It must go entirely through the piece being cut.

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