Geometry - Its got me Stumped.

This was discussed already...

http://cr4.globalspec.com/thread/13789/The-Puzzle-from-Hell

Oh Sorry. Thanks for the direction.

I have now read the posts and Thank you thank you thank you.

This was driving me nuts.

No problem...

Hello Johnoz

Much earlier than the above CR4 listing, this same puzzle was listed in Martin Gardner's column in Scientific American almost 40 years ago.

He had gotten the puzzle from the book written by the great puzzle expert: Henry Ernest Dudeney

I well remember the puzzle, which just about drove me nuts, until I carefully examined and located the interesting solution, after a couple of weeks.

Kind Regards....

Cheers Sparkstation.

I knew something was up when I worked out that the SA of the different parts added up to 32 cm² whilst the overall triangles size was 32.5 cm².

Its just nice to know.

This is the same principle our politicians use the balance our budget. They can't explain it either!

A while ago, I cut out the pieces of this puzzle for my kids to play around with. The difference in slope (that makes the long "side"* concave in one case and convex in the other) is subtle enough that even with the pieces in hand and a with a sheet of graph paper to lay them out upon, the illusion is pretty strong.

If you can rearrange the parts to fit inside the same shape, ...

This is the problem -- they are not the same shape: they are two different quadrilaterals.

*Of course this is not really one side in either case -- it is two legs of a quadrilateral.

These two overall shapes are not triangles, but are both actually four sided. On each shape the 'hypotenuse' is not actually a straight line. In the top figure it is slightly bent inward where the red and blue triangles meet, and in the bottom figure it is bent slightly outwards.

The slope of the hypotenuse on the red triangle (y/x = 3/8) is smaller than the slope of the hypotenuse on the blue triangle (y/x = 2/5), and as a result they are not actually similar triangles (2/5 > 3/8). They look the same to the eye, but they clearly are not. The larger triangle would need to be 7.5 units long for them to be similar.

If you hold a straight edge (such as a business card) up to the images on the screen, you can actually see that these are not straight lines. This are the kinds of rounding-off errors that can happen when you try to let a piece of graph paper do your thinking for you.

In the second drawing. The length of the line segment that is common to both the green triangle and the orange block seems to be 2 units. But this is absolutely wrong!

Note that the green triangle and the largest triangle are similar. Let x be the length of the line segment that is common to both the green triangle and the orange block.

5 / 13 = x / 5

x = 25 / 13 = 1.92

So the second drawing is impossible.

Clearly, the second drawing is not impossible -- it is there for all to see.

Note that the green triangle and the largest triangle are similar.

The largest triangle is the red triangle. The large figure made up of four smaller shapes is not a triangle in either the upper or lower figure. In both cases it is a quadrilateral.

The green triangle and the red triangle are not similar. One is 2 x 5 (hypotenuse slope = .4); one is 3 by 8 (hypotenuse slope .375).

In the the first figure: the intersection between the red and dark green triangle does not intersect the intersection of the graph. In the second figure: the intersection between the the red and dark green triangle intersect the intersection of the graph.