Est. 1998 — 27 years of woodworking knowledge
Tuesday Puzzle
Still filling in for Dan, this time with an oldie but goodie. If you have seen it, just say so, but hold off until those who are not familiar with it have had a chance.
Below is a picture with two triangles. In the first, the triangle has been separated into four areas, which are color-coded. In the second, those four areas have been rearranged, leaving a hole in the triangle. Where did that extra square come from?
Re: Tuesday Puzzle
It seems to me that the red and the green triangles are not exactly "friends". I came to this conclusion once I started measuring lengths by counting squares instead of just looking at the picture (like a baffled person).
You may be on to something
Okay, someone want to explain it?
Re: Okay, someone want to explain it?
The two "triangles" are not triangles, but quadrilaterals!
The one on the right is convex (there is an outward kink at the point where the hypotenuses of the green and the red triangle meet).
The one on the left is concave (there is an inward kink at the point where the hypotenuses of the green and the red triangle meet).
Both these irregularities are small, so there is an optical illusion that makes the figures look like triangles.
The red and the green triangles are not similar (i.e. not "friends" as I wrote in my previous posting).
One can easily see that the ratios of their sides are not equal. Hence, if they are positioned adjacent to each other as in the given figures,
their hypotenuses cannot form a straight line. They have different slopes.
The bottom line is that the two figures are not equal at all, and therefore there is a discrepancy when their parts are rearranged.
The left figure, if assumed a triangle, should have an area of 20 squares, but in reality it is 19.5 squares. (calculate area by dividing in parts)
The right figure, if assumed a triangle, should have an area of 20 squares, but in reality it is 20.5 squares. (calculate area by dividing in parts).
This difference of one square is exactly the white square in the right figure.
Very good!
Niko's explanation is right on target.
The key to this illusion is that the three triangles (total and red and green have nearly, but not quite identical slopes. the 5x8 outer triangle is .625, the small 2x3 triangle is .667 and the larger 3x5 triangle is .600.
To make the illusion better, I made the outer triangle true, and fudged the gridlines. To see that, look at (3,3) (which is not really on the hypotenuse of the larger triangle. In the first drawing it looks like it is, but in the second drawing it is well off the hypotenuse. The red and green triangles are NOT the size they appear, and the red one is larger in the bottom The "blocks making up the yellow and blue pieces are slightly smaller in the lower triangles.
Here is another picture showing the original triangles with my faked gridlines on the left and the same areas with true gridlines (and thinner boundary lines; the bold ones help cover the inaccuracy in the illusion version) on the right.