64 = 65 Geometry Paradox

[rookie1ja]

64 = 65 Geometry Paradox - Back to the Geometry Puzzles

Where does the hole in second triangle come from (the partitions are the same)?

The same principle - moving the same parts - allows creating objects 64, 65 and 63 squares big. This geometric fallacy is also known as '64 = 65 Geometry Paradox'.

This old topic is locked since it was answered many times. You can check solution in the Spoiler below.

Pls visit New Puzzles section to see always fresh brain teasers.

64 = 65 Geometry Paradox - solution

It looks like a triangle, because a thick line was used. Hypotenuse of the composite triangle is actually not a straight line – it is made of two lines. Forth cusps are where the arrows point (c9, l6).

The 64 = 65 paradox arises from the fact that the edges of the four pieces, which lie along the diagonal of the formed rectangle, do not coincide exactly in direction. This diagonal is not a straight segment line but a small lozenge (diamond-shaped figure), whose acute angle is

arctan 2/3 - arctan 3/8 = arctan 1/46

which is less than 1 degree 15' . Only a very precise drawing can enable us to distinguish such a small angle. Using analytic geometry or trigonometry, we can easily prove that the area of the "hidden" lozenge is equal to that of a small square of the chessboard.

[Guest]

yeay go me i got it right...for once

[Guest]

actually these fit in the category of optical illusions

In the first diagram, the red and blue triangle have different angles. W/o using trig, you can see the slope of the red triangle is portrayed as 2/5, while the blue triangle is portrayed as 3/8. In the picture with emboldened lines, the eye can't see the difference between .4 and .375, but it is there. If the angles were the same, the blue triangle would intersect halfway between the first 2 graph points, and in the second portion of that diagram, the blue shouldn't actually cover the 8 spaces spanned by the green and light blue figures. The hole comes from the fact that you cannot calculate the total area of the first triangle as (13*5)/2=32.5, because in actuality, keeping the angles to be the same, it's (12.5*5)/2=31.25. In the second picture, to keep the angles the same, you would have to add the areas because the blue wouldn't complete a triangle, so blue=7.5*3/2=11.25, green=8,lblue=7,red=2*5/2=5, total is magically 31.25

The second diagram is the same principal, the angles that are being matched between the 2 symetrical orange pieces and the 2 symetrical green pieces don't match. Again, .375 vs .4, the unseen gap is what equates to the differing apparent areas.

[Guest]

if that is a hole, then technically it would not be a triangle, therefore it is impossible to answer it.

[Guest]

I loved this problem.

I didn't actually solve it, but I knew something was fishy about the hypotenuse of the composite triangles, because it didn't intersect the grid lines accurately. Drawing endless designs on my dad's engineering tablets when I was a kid made me recognize that, but I chalked it up to "the computer doesn't always draw diagonal lines as clearly as it does vertical and horizontal lines". Yeah right. Too bad I was too lazy to make a precise drawing of my own.

I did figure out that both drawings were inaccurate, since the area under the line should have been = 32.5. In the top diagram the total area of the 4 blocks = 32, and in the bottom diagram the total area of the 5 blocks = 33. Together, the 2 composite triangles has the expected total area of 5 X 13 = 65.

So, I guess the area of the diamond (or lozenge, as you call it) in the top diagram gets excluded because we can't visually detect it, but it is included in the area of bottom diagram because we clearly see the extra square. Or is it just coincidental that the area of the 2 composite triangles equals 65?

Puzzle Nut

[Guest]

They are not a triangle. The red triangle two side are 2 and 5 (6:15), the blue one are 3 and 8 (6:16). so they have different angle. so they are fake triangel when they compond together. you can found from the last third of top.

[Guest]

Rubbish. The correct answer has nothing to do with angles. The empty space is due to the fact that the blue triangle overlaps part of the space it occupied before the partitions were moved around. The overlapped area amounts to one square slot.

e1 and e2 are overlapping the area that c6 and c7 occupy. In other words, both e1 and c6 ended up on k6, and both e2 and c7 ended up on k7. The sum of the area occupied in k6 and k7 by the blue triangle adds up to one square slot; and hence the "hole" in m8.

[Guest]

Rubbish. The correct answer has nothing to do with angles. The empty space is due to the fact that the blue triangle overlaps part of the space it occupied before the partitions were moved around. The overlapped area amounts to one square slot.

e1 and e2 are overlapping the area that c6 and c7 occupy. In other words, both e1 and c6 ended up on k6, and both e2 and c7 ended up on k7. The sum of the area occupied in k6 and k7 by the blue triangle adds up to one square slot; and hence the "hole" in m8.

hey, honestly i dunno what u mean. can u please explain in clearer words? i dont get where you are getting there grid points and such. Thank you ^^

[Guest]

hey, honestly i dunno what u mean. can u please explain in clearer words? i dont get where you are getting there grid points and such. Thank you ^^

There's no overlap. Focus on the 3x1 rectangle in the middle of the 13x5 grid. A true line from the corners of the 13x5 grid from (0,0) to (13,5) would not touch either corner of that 3x1 rectangle-- the bottom-left corner of the 3x1 area falls slightly below the line, and the top-right corner falls slightly above it. The rectangular figures from the two diagrams each touch one of the corners of that 3x1 section in the middle of the grid.

[Guest]

They are not a triangle. The red triangle two side are 2 and 5 (6:15), the blue one are 3 and 8 (6:16). so they have different angle. so they are fake triangel when they compond together. you can found from the last third of top.

..that's right the two triangles are not similar.If they was: 5/8 could be equal to 2/3 ...but it's not & cause the difference is too low we could not suppose that it's existing ...at first time

thanks for the lesson!

You all are great bariniaks!

[Guest]

turquoise piece (total of 7 sq units) combined green piece (total of 8 sq units) in picture one is 3x5=15 (7 turqoise + 8 Green), in the second pic they take up a space of 2x8=16...the missing piece isnt actually a missing piece, because the turquoise and green partitions didnt change, the sq units required to fill the 2x8 space simply isnt enough.