# Geometry buffs: Pls explain this "How many triangles" puzzle **URL:** \ **Category:** Factual Questions\ **Created:** [December 10, 2021, 10:09pm UTC](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860 "2021-12-10T22:09:48Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![lobotomyboy63](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lobotomyboy63/32/2900_2.png) [@lobotomyboy63](https://boards.straightdope.com/u/lobotomyboy63)\ **Post date:** [December 10, 2021, 11:14pm UTC](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860/21 "2021-12-10T23:14:45Z")
> [@lobotomyboy63](#): > > But look at the center. You can make two more there. And one for the overall. That’s 21. Wait, I double counted those in the center and the overall. 18. ---
**Author:** ![Miller](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/miller/32/481_2.png) [@Miller](https://boards.straightdope.com/u/Miller)\ **Post date:** [December 10, 2021, 11:19pm UTC](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860/22 "2021-12-10T23:19:57Z")
If you label each section of the drawing, 1 2 3 across the top, 4 5 6 across the middle, and 7 8 9 along the bottom, the triangles I count are: 1 2 3 1 2 2 3 1 2 3 1 4 1 4 7 2 5 2 5 8 3 6 3 6 9 1 2 4 5 1 2 4 5 7 8 2 3 5 6 2 3 5 6 8 9 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 Which is 18. ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [December 11, 2021, 4:00am UTC](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860/23 "2021-12-11T04:00:11Z")
The answer is 200. Now guess what base I’m using. ---
**Author:** ![Atamasama](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/atamasama/32/12961_2.png) [@Atamasama](https://boards.straightdope.com/u/Atamasama)\ **Post date:** [December 11, 2021, 4:32am UTC](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860/24 "2021-12-11T04:32:47Z")
That would be 18 in Base 3, right? ---
**Author:** ![Riemann](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/riemann/32/3133_2.png) [@Riemann](https://boards.straightdope.com/u/Riemann)\ **Post date:** [December 11, 2021, 4:34am UTC](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860/25 "2021-12-11T04:34:24Z")
That depends what base you are writing “18” in. ---
**Author:** ![Atamasama](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/atamasama/32/12961_2.png) [@Atamasama](https://boards.straightdope.com/u/Atamasama)\ **Post date:** [December 11, 2021, 4:38am UTC](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860/26 "2021-12-11T04:38:19Z")
Base 10. But then what base is the 10 in…? ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [December 11, 2021, 7:38am UTC](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860/27 "2021-12-11T07:38:04Z")
> [@Thudlow\_Boink](#): > > I object to the word “guess” in the headline. Agreed. That would not be suitable here for a FQ question. It should have been posted in IMHO. ---
**Author:** ![Isilder](https://avatars.discourse-cdn.com/v4/letter/i/cc9497/32.png) [@Isilder](https://boards.straightdope.com/u/Isilder)\ **Post date:** [December 11, 2021, 7:57am UTC](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860/28 "2021-12-11T07:57:47Z")
Well, 4 in a “window” count. 3 triangles to form the window, and the window itself is triangular so add it. If the rule was “overlapping not allowed” people would still say 4, because of the window exception to the overlapping rule. Is there a “transparent edges not allowed” rule ? 🙂 ---
**Author:** ![DorkVader](https://avatars.discourse-cdn.com/v4/letter/d/73ab20/32.png) [@DorkVader](https://boards.straightdope.com/u/DorkVader)\ **Post date:** [December 11, 2021, 1:15pm UTC](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860/30 "2021-12-11T13:15:13Z")
I got 26, starting with viewing it as a drawing of a 4 sided pyramid, then shifting my perspective to that of a flat, overlapping bunch of triangles. But I did break the rule about guessing and counted. ---
**Author:** ![kayaker](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/kayaker/32/441_2.png) [@kayaker](https://boards.straightdope.com/u/kayaker)\ **Post date:** [December 11, 2021, 1:20pm UTC](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860/31 "2021-12-11T13:20:05Z")
> [@Riemann](#): > > I think @Grrr’s answer is correct, because he is the only one who followed the instruction to _guess_ how many triangles. All the rest of you are _counting_ There are an infinite number of triangles, that’s my guess. Seems like math puzzles often wind up with infinity somehow, so there. ---
**Author:** ![RivkahChaya](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/rivkahchaya/32/8919_2.png) [@RivkahChaya](https://boards.straightdope.com/u/RivkahChaya)\ **Post date:** [December 11, 2021, 1:30pm UTC](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860/32 "2021-12-11T13:30:10Z")
18 18 is a sacred number in Hebrew (well, in Judaism), because it is the value of the word life-- every word has a numeric value, since letters are reused as digits. I wonder if someone deliberately constructed the triangle so the answer would be 18. ---
**Author:** ![DorkVader](https://avatars.discourse-cdn.com/v4/letter/d/73ab20/32.png) [@DorkVader](https://boards.straightdope.com/u/DorkVader)\ **Post date:** [December 11, 2021, 3:10pm UTC](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860/33 "2021-12-11T15:10:17Z")
The article linked says the correct answer is 18, but the only rule stated is you must guess, so I disagree, it’s a 3d drawing of a pyramid or a flat drawing of overlapping triangles depending on perspective based on the problem as stated in the article. Is there another article somewhere that has the rules actually stated that would limit the correct number to 18? Also, dammit, I try not to get sucked into these ambiguous gotcha/ football moving kind of things. ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [December 11, 2021, 6:20pm UTC](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860/34 "2021-12-11T18:20:02Z")
Each pixel is actually a triangle surrounding another triangle surrounding another triangle until you get down to turtles. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [December 11, 2021, 7:01pm UTC](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860/35 "2021-12-11T19:01:21Z")
Triangle Man hates Particle Man. ---
**Author:** ![BigT](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bigt/32/12044_2.png) [@BigT](https://boards.straightdope.com/u/BigT)\ **Post date:** [December 11, 2021, 7:18pm UTC](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860/36 "2021-12-11T19:18:08Z")
> [@DorkVader](#): > > it’s a 3d drawing of a pyramid or a flat drawing of overlapping triangles depending on perspective based on the problem as stated in the article. While such a drawing may _represent_ a 3D shape, the drawing itself is 2D. So I would not consider a 3D interpretation valid. Sure, some people talk about the triangles “overlap,” but that doesn’t mean they’re 3D shapes. It’s just a way to describe how line segments (and portions of line segments) may be reused in other triangles. They still all exist on a 2D plane. I do understand that there are similar puzzles that only work if you assume the object is 3D, but those tend to ask you to count an object that could be assumed to be 3D. If you asked how many triangles are in a pyramid, a valid answer would be “an infinite number,” as there are an infinite number of triangular cross-sections. ---
**Author:** ![CookingWithGas](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cookingwithgas/32/485_2.png) [@CookingWithGas](https://boards.straightdope.com/u/CookingWithGas)\ **Post date:** [December 12, 2021, 1:59am UTC](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860/37 "2021-12-12T01:59:12Z")
Here are the 18 that I get. [![](https://dl.dropboxusercontent.com/s/aiodgjqz7sban4p/triangles%2018.jpg?dl=0) ](https://dl.dropboxusercontent.com/s/aiodgjqz7sban4p/triangles%2018.jpg?dl=0) ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [December 12, 2021, 12:15pm UTC](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860/38 "2021-12-12T12:15:38Z")
> [@BigT](#): > > I do understand that there are similar puzzles that only work if you assume the object is 3D, but those tend to ask you to count an object that could be assumed to be 3D. Which leads immediately to puzzles like: How many squares in this picture? [![](https://upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Schlegel_wireframe_8-cell.png/240px-Schlegel_wireframe_8-cell.png) ](https://upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Schlegel_wireframe_8-cell.png/240px-Schlegel_wireframe_8-cell.png) When you’ve figured that out, try these (again, how many squares?): [![](https://images1.sw-cdn.net/cdn-cgi/image/quality=85,gravity=auto,format=auto,fit=scale-down,width=1920/product/picture/710x528_31543185_16383546_1598121364_1_0.jpg) ](https://images1.sw-cdn.net/cdn-cgi/image/quality=85,gravity=auto,format=auto,fit=scale-down,width=1920/product/picture/710x528_31543185_16383546_1598121364_1_0.jpg) [![](https://img2.cgtrader.com/items/1947745/d52a2e0046/large/vortex-tesseract-3d-model-stl-amf-3mf.jpg) ](https://img2.cgtrader.com/items/1947745/d52a2e0046/large/vortex-tesseract-3d-model-stl-amf-3mf.jpg) ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [December 12, 2021, 8:28pm UTC](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860/39 "2021-12-12T20:28:37Z")
The first one is 24 squares (and 32 edges), assuming it’s meant as a representation of a 4-d hypercube. The second one is 80 squares, assuming it’s meant as a representation of 5-d. The third one appears to be merely 3-dimensional, with an easily-counted 36 squares, unless I’m missing something. ---
**Author:** ![Beelzebubbly](https://avatars.discourse-cdn.com/v4/letter/b/b9bd4f/32.png) [@Beelzebubbly](https://boards.straightdope.com/u/Beelzebubbly)\ **Post date:** [December 13, 2021, 2:22am UTC](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860/40 "2021-12-13T02:22:21Z")
Entirely too many. ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [December 13, 2021, 2:40am UTC](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860/41 "2021-12-13T02:40:31Z")
> [@Beelzebubbly](#): > > Entirely too many.   There’s plenty more where those came from. 6-d cubes next? Then 7-d? Oh, wait. This thread is about _triangles!_ Okay, how many triangles here? [![](https://pages.uoregon.edu/koch/pictures/4d/tetra4.jpg) ](https://pages.uoregon.edu/koch/pictures/4d/tetra4.jpg) It’s a 4-d “tetrahedron”. Like cubes, these can also exist in any number of dimensions. Apparently the term for an arbitrarily-high-dimension “tetrahedron” is a n-simplex. How many triangles in a 5-simplex? A 6-simplex? There’s a pattern to these that’s very straightforward and many readers of this Board are probably very familiar with it, even if you don’t know it! [Previous page](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860.md?page=1) [Next page](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860.md?page=3)