# How many triangles can you see?

**URL:** <https://boards.straightdope.com/t/how-many-triangles-can-you-see/955870>\
**Category:** Factual Questions\
**Created:** [December 11, 2021, 5:06am UTC](https://boards.straightdope.com/t/how-many-triangles-can-you-see/955870 "2021-12-11T05:06:52Z")\
**Posts on this page:** 2\
**Page:** 1

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**Author:** ![Bullitt](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bullitt/32/5725_2.png) [@Bullitt](https://boards.straightdope.com/u/Bullitt)\
**Post date:** [December 11, 2021, 5:06am UTC](https://boards.straightdope.com/t/how-many-triangles-can-you-see/955870/1 "2021-12-11T05:06:52Z")

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The FQ thread created earlier today (**[Geometry buffs: Pls explain this “How many triangles” puzzle](https://boards.straightdope.com/t/geometry-buffs-pls-explain-this-how-many-triangles-puzzle/955860)**) reminded me of a mathematics competition I participated in years ago. One question was similar, and I’ll pose it here.

(First, my sketches aren’t great. Please assume the triangles I sketched are all equilateral.)

The question goes like this —

If this is a triangle of side length = 1,

[[Album] imgur.com ![](https://i.imgur.com/dWPBH1j.jpeg?fb "imgur.com") ](https://imgur.com/a/3xzZ5QT)

where you can see 1 triangle,

and if this is a triangle of side length = 2,

[[Album] imgur.com ![](https://i.imgur.com/5Bypqci.jpeg?fb "imgur.com") ](https://imgur.com/a/KgXwAI3)

where you can see 5 triangles,

then how many triangles can you see for a triangle whose side length = 10?

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**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\
**Post date:** [December 11, 2021, 6:17am UTC](https://boards.straightdope.com/t/how-many-triangles-can-you-see/955870/2 "2021-12-11T06:17:04Z")

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315  
[1,5,13,27,48 - OEIS](https://oeis.org/search?q=1%2C5%2C13%2C27%2C48&language=english&go=Search)

I computed the first 5 and cheated my way up to 10 by searching the OEIS.

It’s tricky because the upside-down triangles are not easy to fit. The small triangles just go up with n^2, and there’s an easy formula for all the right-side-up triangles, but mid-size ones are more difficult.

There is a formula in the link above, but it uses the floor() function. It’s usually not easy to come up with explicit formulas containing those.