Pythagorean Paradox




Introduction: Pythagorean Paradox

I have always enjoyed a good puzzle. The composite photo is of the same puzzle assembled two different ways. Both are squares with the sides being the sums of a large triangle's and a small triangle's hypotenuses. (I have intentionally left gaps between the pieces so they would photograph better. They fit together really tight.) The problem is that on the right is solved with one less piece. Having exact perimeters and shape they should have the same area. Thus means that conservation of area was only a working theory and not valid. Since the puzzle has a finite thickness, conservation of mass must also not be valid. Since mass is energy then this just might be the solution to our energy problems ...... or it might be a trick

I was introduced to this puzzle some 35 years ago and what made it interesting is that the better someone is at solving puzzles, the more difficult they will find it. It originally came from a article by Martin Gardner in Scientific American May 1961.

I cut out this puzzle with a laser using a DXF file that I have included. I made mine from a piece of 1/8" acrylic sheet. You could also make one out of some nice hardwood or plywood. If you do not have access to a laser, you can also cut out one with a scroll saw using the attached PDF pattern. This size puzzle will easily fit in a plastic sandwich bag.

Step 1: Materials

You will be a piece of sheet material about 6" square. I used 1/8" acrylic sheet. You could also use thin hardwood or plywood. The choice is yours.

Step 2: Cut Out Puzzle

Import the pattern into your laser and cut out the puzzle. If you are using a scroll saw, copy the pattern, glue it to the material, and cut it out. Peel off the paper off the acrylic when done. (Sand if using wood)

Step 3: Solution Spoiler Alert

The next step gives the solution to this puzzle.

If you want to figure it out on your own, stop now.

Step 4: Solution

The small triangle is 2 units high and 5 units long for a slope of 0.4. The large triangle is 3 units high and 8 units long for a slope of 0.375. From the drawing above you can see that the small triangle does have a slightly steeper slope. So this puzzle is not actually a square. When arranged with each side being a small triangle large triangle in a clockwise order, the puzzle has a slight budge. When the triangle is reversed the order is large triangle small triangle and the puzzle has a slight pucker. The reason it is harder for a puzzle solver is that puzzle solving is a left-brain operation. Ideas in the left-brain are processed as symbols. So the better you are at logic puzzles the faster you go to the left brain were it automatically defines the shape as a square and does not notice the discrepancies. The difference in area is only 2% which is at the limit of detection by the human eye. Only when you know the solution can you plainly see the bulge or pucker.

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    4 years ago

    A nice, classic puzzle; great introduction and explanation, by the way. :)