How can this be true?

About: thanks,


  1. puzzle
  2. geometry
  3. question~answer


Seth says
This puzzle has been giving me a headache.   I think i finally solved it.   The problem is that we assume the draftsman is being honest with us.  But actually the triangles will not meet exactly two squares down and five squares from the left as shown in the diagram.   Were that to be the case, then the tangent of all of the angels would need to be exactly the same ... but they are not:  the overall tangent is 5/13 = .3846,  the large red triangel is 3/8 = .3750, and the small tangent is 2/5 = .4000.  So they are close enough that, within the line width of the draftsmamn's pen and a bit of fudging, they can be made to look square.  If you built the puzzle pieces to 4 place accuracy you would find that they  fit quite loosely inside the triangular space ... so loose in fact that they could absorb the extra square.


Mark de LA says
M 2007-10-21 05:41:52 8290
Or the sum of all the pieces has to equal all of it's parts:
5*13/2 is not equal 3*8/2 + 2*5/2 + 3*5
32.5 versus 32  i.e. the drawing is inacurate, the algebra disproves the geometry/drawing. If the first drawing is bad the second one can't be any better.

The coloring of the parts just supports an optical illusion.

Seth says
I don't see an illusion, i just see a small variation in drawing the hypotenuse ... too small to see unless you know what to look for.  But you are right ... the whole must equal the sum of it's parts.  In the top case, the 32 squares of the colored puzzel must equal the 32.5 squares you get from calculating the area of the overall trangle. In the bottom case it is 33 squares.  I think what they did in the top case was to draw the hypotenuse through the point 2 down and five from the right, and  in the bottom case through the point 3 down and 13 from the right ... but a straight line would not go through those points. Perhaps the way of coloring the pieces made that harder to see ... idk.

Mark de LA says
The illusion is that the drawing is correct & that the top & bottom triangles are equal along with the grid. When I first stared at the thing I kept wondering whether the lines that formed the hypotenuses were straight because to me they were not.  The coloring though seemed to support it.

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