Staircase patterns

Hello there mathematicians, welcome back. Now, you might have been working on a problem we set you recently, which said if this is case one and this is case two and case three and case four.

What might the twelfth, how many squares might there be in the twelfth case and how could we work it out?

And some of you noticed a pattern, yes, where, ah ha, there's an increase each time. Look if we start here with case one, there's one block, and when we get to case two, there's the one block and two more. And in case three, there's the three blocks and three more, I'll leave it there so I can see it. See this, ah ha. And in case four there's the four. There's the blocks that we have in case three and another four.

Yes, so we have in fact a growing pattern, don't we? So in case one, some of you worked it out like this where you said in case one, there's one, and in case two, there's three, and in case three, there's six and in case four there's six and four makes ten. And we want to go six, six, eight nine, ten, eleven, twelve. I'll just squeeze them into my table.

And so what we worked out that, is that you add, yeah, so, the distance from here to here is we added 2, then, increased by 3, then increased by 4, so the next one would be an increase. Yes, by 5. It's a growing pattern, isn't it? So that would be 15. And then you would increase by 6 to be 21 and then you would increase by 7, yes, to be 28 and then it increased by 8.

Yes, so 28 and 2 is 30 and 6 more is 36. Then you have to add 9. Ah ha, 45 'cause if you added 10 it would be 46 and then get rid of 1. Add 10 more to get 55, ah ha, then add 11 to make, uhm 66 and then you add 12 and that is 78.

Yeah, so. We could make a bit of a table. Oops. I made this very wonky. Ah ha and I was thinking, well, that's one way that some of you used to work out how many blocks there would be in the twelfth case of the pattern. So that's one way you might have worked out what the twelfth term might look like. Now as I was working this out, I started to think about, well, notice some other things that I could see.

Yeah, the first thing I started to notice actually is that there's something special about these numbers, look. I just move these across a tiny bit. What can you see about the shape that they form? Ah ha, they form triangles. Yes, and in fact if I made the fifth case which would have five on the bottom row. And then a row of four. And then three, ah ha. And then two. And then one more.

Case five, and yes, because that was ten this should be fifteen. So how could we check that?

Yes, so we know this structure is ten. So, look, there's a row of four, a row of three, a row of two and a row of one. So that has to be ten and then five more so it's fifteen. But what I started to notice is that the're triangular numbers. And so, this made me wonder about what would be the next triangular number in our sequence. But I also, as I was trying to think of more efficient strategies to work out what case twelve would be. I think I notice something and I'm just going to put them back into the staircase position these triangular numbers were presented to us in before.

Like a reverse staircase. Let's move that one down a bit so you can see it. Is, I think I notice something cool. Look, if I take the triangular number here at one and move it across to there it makes a square number. Look, and if I take this triangular number of three and move it to this triangular number of six. It also makes a square number and so now what I wondered is if I take any two consecutive triangular numbers, will it always make a square number? Oh my gosh, that sounds like a fun maths investigation. So it's over to you mathematicians.