It’s a maths question thread!

[RevolutionofourTime]

Liam’s house has a 12 step staircase. He can go down the stairs 1 or 2 steps at a time. For example, 1, 2, 1, 1, 1,...1.

How many different ways can Liam go down the stairs?

I’ve worked out that there’s 1 way L can go down in 1s only (so the number of steps he takes is 12), 1 in 2s only (so L takes 6 steps), and 11 if using all 1s except for one 2. But I’m stumped for the remaining permutations, i.e. if L takes 2, 3, 4, or 5 2-steps.

Does it matter which order he takes them in?

If he takes the top 4 steps in ones and then the bottom 8 steps in twos, is that the same as taking the top 8 in twos and the bottom 4 in ones?

If the order doesn't matter, and you are simple concerned with the number of twos, then I assume the answer is 7 different ways. The maximum number of twos is 6. The minimum number of twos is 0. Every number in between is possible. He can take 0,1,2,3,4,5 or 6 twos.

If the order is important, then it is much trickier.

If this is for your child I would get them to write out all of the ways by hand first to visualise it. And only when written out start to think about what they notice about it. Or maybe use counters? Red ones for one step and blue for 2 steps.
Try not to focus too much on what you notice, more of what they notice. 12 steps is a lot tho, you'll get bogged down by keeping the 2s together and separated. Start with less steps, find a way then work out for 12.

But if you just want the answer then click on the link.
nrich.maths.org/1step2step/solution

Brilliant, thank you. I was trying to remember how to do binomial coefficients, but maybe that’s a bit advanced for year 6? I think the teacher wants them to see the Fibonacci sequence.

Thank you!

No worries.