the hardest question known to mathematicians.?

Question

http://www.click2amuse.com/fun/optical/opt.php?ID=37 <<< thats the link, IF ANYBODY WORKS THIS OUT PLEASE TELL ME!!!!

i have tried this over and over again on paper, i have cut shapes to giv the correct measurements, there is always that cube missing, it disproves pythagoras

and for all the people giving bullsh*t about linear equations and stuff, why dont u get some paper a ruler and some scissors and trying it becoz it does not add up, so stop trying to act clever, the only people that are answering this properly is if they dont know or if they are pondering for a sensible answer.

Answer 1

The large shape isn't a triangle. Look at the "hypotenuse" - it consists of one segment with a slope of 2/5 and a second segment with a slope of 3/8 -- these do not form a straight line. Rather, in the top shape, they form a lightly concave quadrilateral, whereas in the bottom shape, they form a slightly convex quadrilateral - the slight difference between the concave and convex "hypotenuse" is hidden by the black lines, and contains the "missing" one square of area.

Answer 2

Pascal and In.Taco have got it right. The large triangle (By that I mean the whole one) isn't really a triangle. It's a 'nearly' right angled triangle! The tangent of the smallest angle of the green triangle is 2/5 which equals 21.80 deg. The tangent of the 'same' angle of the green triangle is 3/8 which equals 20.56 deg, whilst the equivalent angle for the whole triangle is 5/13 which equals 21.04deg.

This itself leaves us with a small scalene triangle which will be approximately equal to the area of the missing square. Any discrepancies will be taken up by the clever use of thick lines.

Sorry mate but it looks like this Pythagoras geezer new what he was talking about! Seriously though, I think it's an excellent puzzle.

Hope that helps.

Answer 3

The two overall traingles do not have the same area because the red and blue triangles do not have the same angle of elevation. Therefore, when placed end to end, like they have been, the hypotenuse edges do not give a straight line. If you look closely you can see this.
This alters the area because the angle of elevation for the red triangle is less than that of the bluish triangle. So in the first main triangle the hypotenuse lines are jutting inwards, in the second triangle they're jutting outwards. the difference in areas between these two triangles is approximately one square. This square is shown in the bottom triangle as an extra space.

Answer 4

Calculate the area of white space in each:

Top shape:
0.5 * 5 * 2 = 5
2 * 8 + 0.5 * 3 * 8 = 16 + 12 = 28
Total area = 33

Bottom shape:
0.5 * 8 * 3 = 12
5 * 3 + 0.5 * 5 * 2 = 15 + 5 = 20
Total area = 32

There's your missing unit. The 'triangle' isn't actually a triangle if you calculate the gradients; the bottom shape bulges out more into the white space than the top one.

Answer 5

The square comes from the dark green and red triangles, as on the second one it is steepened by a very small amount, and then the black line is a little thicker so that it still looks normal on the grid lines, and the black lines go into the colour, so it doesn't take up any of the newly created square.

The reason it didn't work when you tried it was because they didn't really use the SAME shapes to make the second triangle as the first, just ones which were ROUGHLY SIMILAR and looked similar.

Answer 6

look closely at the triangles. The sides are not whole numbers as they are illustrated in the diagrams.

The 2 perpendicular sides of the whole figure is 5:13
The small triangle "looks" to be 2:5
The big triangle is "looks" to be 3:8

Mathematically, it is impossible. The whole figure is 5:13. The small triangle and the big triangle cannot be 2:5 & 3:8. They should be 1.92:5 and 3:7.8. They only "look" like they are perfectly fit to the grid. In fact, they are not.

Answer 7

Okay, this is quite easy to answer with a diagram and not so easy without.

The two triangles are different shapes. Neither one's hypoteneuse (longest side) is straight. The bottom one bulges where the blue and red parts meet, and the top one bends inwards where they meet. The difference in area between the bulge and the indent is 1 square and that is why there appears to be a square missing in the bottom triangle.

Answer 8

Forget the paper. Look at the where the interchanged red and green triangles overlay the same graph squares... they don't fall in the same place.

When you draw and cut the picture, you are assuming the same fit to the grid, which isn't the case, it's an optical illusion.

Answer 9

The hypotenuse on the top triangle is slightly concave i.e. bends inwards, and the small amount of area that is lost due to this, accounts for the missing square in the bottom triangle.

Answer 10

Easy. The hypotenuse is angled. The two larger triangles does not have the same top-bottom line ratio as the total triangle, giving a non-linear hypotenuse that is angled inwards in the top picture and outwards in the bottom picture.