Tough geometric brain teaser

Geomenrbbraiint BB pal and former guestblogger Bruce Stewart sent along this slippery little geometry brain teaser. Click the image to see it larger.


  1. This little trick freaked me out so much in college that my world view was altered significantly for 24 hours.

    Stupid stupid me.

  2. The hypotenuse on the top is slightly convex (turquoise triangle has greater slope than red).

    Hence when they are swapped, the top is slightly concave, thus the mass is “used up” faster. Fast enough to make way for the hole, but slow enough to not be easily noticable.

    In short, the two figures are actually quadrangles; not triangles.

  3. It’s easy to see if you try to line a piece of paper up along the “hypotenuse” of the shape.

  4. More specifically, the slope of the red triangle’s hypotenuse is gentler than that of the hunter green triangle. So the top composite triangle’s ‘hypotenuse’ is indented at the join, and the bottom triangle is pushed out at the join.

    Love this puzzle, it’s a classic. And nice to have an excuse to say ‘hypotenuse’ on BoingBoing.

  5. I think if it weren’t for the thick black outlines it’d be easier to see… It’s more of a trick than a puzzle.

  6. Yes, this is a classic geometric puzzler. It was invented about 50 years ago by Paul Curry who was an amatuer magician and amateur mathematician. Very talented guy. Similar puzzles have been around for about a hundred fifty years but Curry’s is noteworthy for being especially simple to see that there’s a problem and being especially trick to solve.

  7. They are both 13 squares long, 5 squares high. Fibonacci sequence. The extra square of area is redistributed by switching the triangles. Nice!

    Now do it in three dimensions!

  8. The slope of the hypotenuse changes. You can visually see it by comparing the square three up from the missing piece.

    Basically, one of those is not a triangle.

  9. This illusion is most effective because the grid lines throw your eyes off. With the gridlines removed, the different angles on the two different triangles becomes more apparent.

  10. An old one, but a classic.

    It’s best when you have the person cut up the shapes and then put them together and try and work out where the extra space comes from. I think it took me so long the first time I saw it specifically because I was working with cut-out shapes: I just assumed that the slight difference in angles was because I cut sloppily. (I did work it out in the end though, and was chuffed).

  11. I always thought the trick wasn’t a curve or anything like that but that the bottom triangle is actually slightly larger than the upper one. In other words the empty square comes from the circumferance being slightly larger. You can really see the difference when you look where the blue/tan/red join on the hypotenuse of the bottom one and the corresponding point on the top one. The bottom intersects the blue graph lines and the top one is below it.

    One square is miniscule when spread out over the entire area

  12. A guy I used to work with made a puzzle based on this trick.

    The idea is that you can rearrange the pieces so that neither, either, or both of the 1×1 squares sitting in the bottom of the puzzle fit in the top portion of the puzzle.

  13. @ft88: That’s what people are meaning when they say that the big triangle is “concave” or “convex.” They don’t mean actually curved, just that one angles out slightly and one angles in slightly, because of this difference you mentioned.

  14. Wow, Noah! Thanks for posting the Leprechaun, I saw that YEARS ago in a Martin Gardner column, but can never find it.

  15. Don’t know if he was the original, but Charles Dodgson, aka Lewis Carroll posed a problem like this in the 19th century … (searching for references).

  16. Good puzzle. Am I the only one who felt cheated when I found out the answer? I guess I assumed since it looked straight-ish, it must be straight.

    Hey, Thad (#9): Be wary of getting linguistically pedantic, bro. “Whence” means ‘from where?’ (Though I concede from whence is pretty common, if redundant.)

    I didn’t order extra prepositions on my archaism! :P

  17. @ Strophe Am I the only one who felt cheated when I found out the answer?

    Nope: I was asking myself the exact same question a few comments before yours.

  18. Best combo of optical illusion and math problem!

    The gridlines are brilliant because they enhance the optical illusion making you think the slope is straight, but they also provide clues related to slope if you are looking closely enough. They also make it easiest to prove the trick by counting rises and runs for each of the small triangles.

    Reach all the way back to basic geometry! Rise over run! The red triangle has a slope of 3/8 and the green one has a slope of 2/5, clearly not the same.

  19. See: spiritualism vs. religion

    By proving one fix, you legitimise the other!

    Also, triangle math is not-always-applicable…

  20. It being on a grid fakes you out so you assume the lines are uhh.. lined up. So it’s actually an optical illusion, not a geometric puzzle at all.

  21. the slopes on the two triangle pieces are different so the overall shape is NOT a triangle.

  22. The heights of the components are not exactly integers although it appears that way.
    The red triangle (assuming the overall shape is 13×5) is actually about 3.07 squares high.

    The little bits add up (or subtract out).

    (If the top figure is 5 units high, the bottom figure is ~5.07 units high.)


    Grandma, stop sending me these. (and the bottom one isn’t a real triangle)

  24. Witchcraft is the only rational solution! David Pescovitz is a witch! Burn the witch! Or, if you are a Pythagorean, off the boat!

  25. Comment #33 explains this best.

    2/5 != 3/8 although they are very close.
    To put is another way 0.4 != 0.375.

  26. actually the grid just makes it easier to spot, as you can see the hypotenuse crosses the squares at different points when you compare both, thus concluding the angles are different.

  27. We don’t see it because the slight irregularity of the figures is below our normal noise level. We automatically correct for it.

  28. also notice that the point (8,3) is ON the top triangle but it is INSIDE the bottom triangle,(0,0) being the leftmost point of each triangle, so they are clearly not the same figures.

  29. Simple: the two triangles are drawn to look similar (in the geometry sense of that word) but they cannot be, unless arcsin(0.4) = arcsin(0.375)

  30. yeah, it didn’t help the low-rezzy artifacty .jpg, although maybe that was on purpose to help hide the variance in angle (that along with my fish-eye-lens glasses….)

    I thought I could waltz in with the answer to get a RISE out of you all, but I see the comments are already OVER RUN with the solution.


  31. Although “whence” technically means “from which” or “from where,” the usage of “from whence” is pretty well established; plenty of sources from the 14th century use it.

  32. Theresa wrote @45:

    We don’t see it because the slight irregularity of the figures is below our normal noise level. We automatically correct for it.

    I spotted something was off straight away, but had to count squares to be sure. The hypotenuse didn’t look straight in either one.

    I do a lot of on-paper designing things, though, and have a pretty well developed eye for lines that should line up and don’t (“The hull plates and bottom aren’t faired together quite right, that’s not really straight! D’oh!”)

  33. So it’s just all a trick. And here I thought we had invented a new math that would allow us to travel across the universe through measured triangular wormholes in space and time, from whence we could go forward.

  34. That’s easy. The dark green triangle has a slightly different angle from the red one. You can see it in how much it cuts into the squares above it.

  35. What the hell. Am I the only one out of 50-something commenters to notice that the grid itself is completely messed up within the colored shapes and that’s where the hole comes from? If you look at the grid outside the shapes it’s a normal grid made of perfect squares, but the grid “under” the shapes is all kindsa’ rectagles and double lines all flyin’ around akimbo.

  36. @ 33 – that’s how I figured it out too. Well explained.
    @ 50 – ditto. I thought – hey those triangle are filling up the squares differently.
    I’m also the person who walks around houses straightening all the pictures.

  37. I didn’t find the answer until I read the comments.

    ‘Course I’m under 40 which means that I’m of a generation that values knowing where to find the answer over how to find the answer.

  38. I swear you are all wrong! Look at my comment #53! The grid is all fudged…the triangles aren’t “curved!” What the heck!

  39. It’s not that tough IMHO– we learnt it in I guess what would be the equivalent of grade school. The cyan and red triangles are not similar triangles and so don’t form a straight line, so the combined shape is not an actual triangle for real.

  40. Also, “from where comes this hole?” strangely reminds me of “how is babby formed?”

  41. The cause of the problem is the pink circle in the middle, next to the orange prism.

  42. Nice example of how logic sometimes leads people astray. Yes the upper diagonal is the same, but just count the lenght of the segments horizontally. It is 13 segments long.

    The red part is 8 segments long next to the green part which is 5 segments long, total length 13 segments.

    Since the bottom image starts with the green triangle that is only 5 segments long horizontally, s0 we neede 8 more segments to complete the horizontal.

    Makes perfect sense to me.

  43. The gridlines are not off (well, not significantly. There may be some aliasing shift, by a pixel, but not enough to cause a unit-sized hole).

    Don’t believe me? Do what I did. Go into Photoshop, and copy the triangles from the first puzzle, and drag them onto the second. They are a perfect match!

    Just in case the other explanations are thowing you (IE: “The triangles are different”), try this:

    In the top puzzle, the hypotenuse of the entire shape (consisting of the combined hypotenuse length of the red and green triangles) seems to be pushed in a bit. If you could see a silhouette of the shape, this would be clearer.

    In the bottom puzzle, the line seems to bow outwards. This is because the green triangle truly is steeper than the red triangle.

    1. hence = from here
      thence = from there
      whence = from where
      hither = to here
      thither = to there
      whither = to where

      How did you people survive The Lord of The Rings?

  44. Here’s the fact. It’s not a curved line, and it’s not an optical illusion. All of the area of the shapes is accounted for.
    I’d have thought a lot more of you were graphic designers. I drew the shapes as vectors and placed them accordingly, when I examined the shape with a finer grid it all makes perfect sense.
    If you can use a vector programme and have a little web space somewhere, someone, please post a graphic illustration of where the extra area is accounted for, lest this post becomes a quagmire of superstition amidst normally rational BoingBoingers.

  45. When you look at how both ‘triangles’ cut the grid you know that they’re not the same.

    Hold a ruler against the longest side of the top and the bottom triangle and you’ll see that neither one is actually a true triangle to begin with.

    They’re both curved. This is a good one!

  46. not that hard – you can see there is not a straight line along the hypotenuse of the whole triangle where the two smaller triangles meet. In the top one, it’s “dented inward” so it fills the hole created by bending it out the other way.

  47. i totally felt cheated. damn grid. but then it made me want me want to listen to emerson, lake, and palmer. all was forgiven.

  48. Why is everyone blaming the grid? The grid has nothing to do with the solution. If anything, it makes finding the solution much easier, not harder, because it allows you to count gridlines and see the angle in the “hypotenuse” of the two shapes.

  49. #7 Z7Q2: nice excuse, plus remembering your comment helped me find this post again – search bb for “hypotenuse” :-)

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