Yesterday evening, I was merrily meandering my way around Pinterest, as you do, when a triangle-based puzzle caught my eye. I was poised to peruse this further when “stuff” started to happen (wife, son, supper, two stupid cats… need I say more) and I never got around to seeing what it was all about.

The way I remember it is that there are four identical triangles mounted side-by-side. A line is drawn from the bottom left-hand corner of the left-hand triangle to the apex of the right-hand triangle as shown in my diagram.

Let’s start by assuming these are equilateral triangles, which means all three angles are the same (that is, beta == alpha). Is there a simple formula that we can use to define the ratio of the green and blue areas?

Next, assuming we can determine such a formula, does this hold true if our triangles are isosceles (that is, beta != alpha)? I await your response with dread antici…

You can solve it visually, the green triangles fit into the white triangles drawn between the line and the blue triangles sides. Meaning that the green area is 3/2 of the area of a single triangle and the blue area is 5/2. The ratio is then 3/5. This should hold as long as the triangles are all equal.

Hi Diego — when you say “This should hold as long as the triangles are all equal,” do you mean if the triangles are all equilateral, or do you mean this is also true for isosceles triangles so long as they are all the same size?

Sorry for the lack of clarity. Yes, I meant that it should apply to isosceles and equilaterals as long as they are the same size.

A quick visual inspection will show that we can see much more blue than green. So empirically we can say that the ratio of green to blue must be smaller than 1:2, not larger.

I should have paid more attention and not being too quick. I’ll check where I’ve gone wrong. Thanks

I may have check my math, but I will go with green/blue ratio of 7/17.

Well, Diego (elsewhere in the comments) says 3:5, and you say 7:17… there can be only one winner LOL

I posted more on my solution on LinkedIn.

Cool Beans (as we French say LOL) Can you post a link here to your solution on LinkedIn?

https://www.linkedin.com/feed/update/urn:li:activity:6755992137617195008?commentUrn=urn%3Ali%3Acomment%3A%28activity%3A6755992137617195008%2C6756255264833437697%29

I just stumbled on a video solution.

https://laptrinhx.com/triangles-in-a-row-puzzle-1850471230/

It assumes equilateral triangles each with an area of six. His answer was an area of 7 out of 24, 4 x 6.

So sort of confirms my answer.

7 green versus 17 blue, where 17 comes from 24 – 7.