## Triangular number equations via pictures: solutions

Here are some solutions to my previous post. However, they are almost certainly not the only solutions! If you have other cool ways to visualize any of these (or any other triangular number equations) feel free to post in the comments — either a link to an image if you have one, or just a description if you don’t have a good way to make an image.

$3T_n + T_{n-1} = T_{2n}$

$3T_n + T_{n+1} = T_{2n+1}$

$(2k+1) T_n + T_{kn - 1} = T_{(k+1)n}$

$T_n T_k + T_{n-1} T_{k-1} = T_{nk}$

Assistant Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.
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### 10 Responses to Triangular number equations via pictures: solutions

1. Fergal Daly says:

I didn’t actually attempt them before you posted your solution but the solutions are beautiful. I wonder is there a nice way of turning each picture into an algebraic proof.

• Brent says:

Well, algebraic proofs for these are all pretty simple: just plug in the formula for triangular numbers and do some algebra. I don’t know whether there’s a way to see a connection between the pictures and the specific algebraic manipulations required; there might be!

• jd2718 says:

If we flipped the odd triangles to the center, eg on the first row flipped the red ones down and brought the blue up, at least we have a suggestion.

There are 3 blue triangles, each reaching halfway up the “perimeter” of the big one to meet the next, and leaving a gap, one smaller, in the center. Maybe that’s a subtraction I’m looking at?

Jonathan

2. Excellent solutions Brent. And nice graphics too.

3. By the way, I have included you in my blogroll.

4. Tom says:

Very pretty! The final solution is particularly impressive. Congrats on recognising that pattern amidst your M&M’s 🙂

BTW, what program are you using to generate these graphics??? I’ve been hunting around for some free software which could quickly generate pretty mathematical drawings, and these drawings do look quite impressive.

• Brent says:

I was wondering if someone would ask that. =) The short answer is that I used software I wrote myself, which I’ve been working on in fits and starts over the past two and a half years. I hope to make it more widely available soon (free and open source), at which point I’ll probably write about it. The short version is that it is a domain-specific language for describing pictures and diagrams, embedded within the Haskell programming language. To make simple diagrams doesn’t really require knowing much (if any) Haskell; you just use the supplied primitive shapes, transformations, and ways of combining diagrams. But when you want to start doing more sophisticated things you have the entire power of Haskell at your disposal.

• Tom says:

Wow, that’s pretty cool. Yet another reason to learn Haskell… 🙂