10 October 2010

Fiendishly tricky geometry puzzle

Start with a piece of paper 8" x 8", as shown on the left.  Make the two cuts shown.

Then shift the big lower-left piece up and to the left and move the triangle from the upper left corner to the lower right one.

The area that used to be 8x8 = 64 square inches now appears to be 9x7 = 63 square inches.  Where is the fallacy?

I pondered this for way too long, and finally had to cut some physical pieces of paper to solve the paradox.

The puzzle was originally created by or published by the famous Sam Loyd; I found it in the Futility Closet.


  1. Heh, I saw it within a couple of minutes. But I also have some background in computational geometry, so there's that.

  2. The original 8x8 square on the left is correctly labelled to spell out the puzzle. The rearranged rectangle on the right is actually incorrectly labelled based on a false assumption that is easy to make.

  3. expatQLD is correct; we'll see how many readers can spot (or figure out) the false assumption.

  4. It took me a while but I finally got it. It's funny how your brain works sometimes.

  5. "The area that used to be 8x8 = 64 square inches is now 9x7 = 63 square inches. How is that possible?"

    I don't get it. The area of the second configuration is not 9x7. It's 9 and 1/7 inches by 7 inches.

    How is this a puzzle? The question itself is false.

  6. Encountering this puzzle again two years later, I had to think it out again, so I'll put the answer here so I don't have to do it a third time.

    The fallacy lies in the little triangle that's cut off the top and moved. It looks like a 1x1 triangle (with a 1.41 hypotenuse)- but it's not (and it's not labeled as such - only the vertical is labeled 1).

    It's not a 1x1 triangle because the diagonal through the large 8x8 square doesn't go from one corner to the other. Because the bottom right termination of the diagonal is up on the side, then the base of that little triangle is not 1. It's actually 1.14, and the new quadrilateral is 7x9.14 = 64 square units, same as the original 8x8.

  7. we did this in grade 10 in 1968. we cut it in 4 pieces..but I forget how..4 larger pieces. math teacher has now passed on


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