This is problem six for grades 11 and 12 on the 2025 Mathkankaroo. The shapes shown below are the first three shapes of a sequence. How many dots make up the fifth shape in the sequence? So yeah, this is the sequence that goes on. Our first answer choice A, 72, B, 74, C, 76, D, 78, and E, 80. Think about it for a bit, you know, pause the video. All right, so there are a few ways we could go about doing this. We could like A, try to generate a formula. Like a very explicit, like I plug in this step of the sequence and I get this exact output for the number of dots. That might be a little bit challenging though. So we want to keep B in the back pocket, a recursion. And that's basically like given like state N, I can just generate N plus one. And technically both of these generate the entire sequence. One might just be a little bit more work, like require a little more cleverness in getting. And the other one would be a little more work computationally. So let's start with A. There's one very clever insight you can get here. And that is, well, I'll draw it out because I quite like it. So in this first drawing here, we can draw sort of this, like these dots. In the next one, we can draw the dots maybe a little bit faster. And like alarm bells should kind of be ringing here. If we like draw two of these things out in cyan, that's a two by two square. And then in gray, that's another two by two square. So here there are two, two by two squares. Here in cyan, there's a three by three square. And in gray, there's a three by three square. So here there are two, three by three squares. And like if we continue this pattern upwards, I'm not going to draw each dot. But you see what I'm going for here. There are two, four by four squares. So really, what we should sort of be going for is the formula. Like we get this consistent pattern. There are two of each types of square, and they grow in size every time. So it should be two times n plus one, because this is step one. This is step two. This is step three squared. So if we plug five into this, we should get the answer 72. Let's keep that in the back pocket. What we can also do is try to generate a recursion. And we can do that by sort of like fitting in where any previous step would have gone. I would not fully recommend doing this, because it gets graphically like very intensive very quickly. But I do want you to take notice that this section here is completely identical to the step that came before it. And then we're adding on like two rows of increasing size every time. And we do the same thing here. Same section as the one before. But there are these extra two rows that, again, keep increasing in size every time. So I want you to try to generate your own recursion that generates each thing, assuming this pattern holds. Eventually, though, you'll see that it is 72. So that should be our answer. A, 72.

dot sequence

pattern formula

recursive approach

shape progression

mathematical solution