Climbing Stairs and Keeping Count

via https://en.wikipedia.org/wiki/Penrose_stairs

Because I am an Exercise Swot, I climb the stairs from the bike park (the minus-fourth floor) to the 11th floor each day.

This is fifteen floors, which is rather a lot. From the bottom, you can look up (I recommend this — it makes you pleasantly dizzy) and think, “uh-oh.” One minute later and you’re thinking “This is a bit tiring,” and the minute after that you’re bored as well as tired.

At first you may amuse / distract yourself by counting the spelling mistakes and syntax errors in the motivational posters — six, by my reckoning. But after that you need other things to count (if you like counting) (I do).

So: Fifteen floors in my daily climb, with two flights per floor making 30 flights of stairs. 30 is a good number, because it has three prime factors: 2, 3 and 5. In other words, 30 = 2 x 3 x 5. This is good because it means that there are several different ways for me to divide my journey into smaller chunks.

And now we get to the good bit: the maths of it. Bear with me. Remember what I’m trying to do is split my journey into chunks, so that I have things to count, and so that I can tick off my progress as I go.

So, factors. The largest factors of 30 (including non-primes) are 6, 10 and 15. For me, six feels just right. “I’m a sixth of the way up,” I can tell myself, or “I’m a third of the way there”, or “only one sixth to go!” And using an even number as a factor also has the advantage that “halfway there” is on the list of options.

But.

This is all very well, but I quickly noticed that the first few sixths went by quicker than the later ones, and with good reason: The first five floors have fewer stairs per flight.

Incidentally, did you know that if you’re walking with someone and you ask them to do some maths, they’ll almost certainly (according to QI, oracle on all things trivial) stop and stand still? I confess I’m kind of proud that for most of these sums I managed to keep climbing, albeit at a slower speed and an increased rate of stupidity.

By Warren Miller (New Yorker)

Aaanyway. One of the great things about our @LateRooms building is that it has negative floor numbers (look, I’m a maths geek. These things make me happy). The first four floors are below the ground floor, and are labelled minus 4 to minus 1. So, anyway:

· The first three of these have eight stairs per flight (16 Stairs Per Floor, aka SPF).

· The fourth has 11 stairs per flight (22 SPF).

· The ground floor has 13 stairs per flight (26 SPF).

· The rest of them — so, most of them — have 12 stairs per flight (24 SPF).

When I realised this, I was unhappy — all my sums were wrong. “Halfway there” wasn’t halfway at all! It was less than halfway. But I was much happier when I spotted the fact that the overall number of stairs is still a multiple of 24, so it’s equivalent to a whole number of floors.

The fourth and fifth floors (SPF* 22 and 26) cancel each other out, because 22 + 26 = 48, which is a multiple of 24. The first three floors create between them another multiple of 24:

· 3 x 16 = 48 = 2 x 24

This is because there are three of them: it’s what makes them equivalent to two “normal” floors. Prime factors again:

· 3 x 16 = 3 x 2 x 2 x 2 x 2

· 2 x 24 = 2 x (3 x 2 x 2 x 2) = 3 x 2 x 2 x 2 x 2

Are you still with me? It’s all about the factors, man! All numbers can be reduced to their prime factors. This is fantastic. It’s really neat, not to mention useful. When you find hidden patterns in whole numbers, you can often explain it if you look at prime factors.

Via http://forums.atozteacherstuff.com/index.php?threads/prime-factorization.52496/

The pattern here is that even though each floor has 16 stairs per floor, we still end up with a multiple of 24. But it only works if you have three floors, or a multiple of three. And the reason is that 3 is not a factor of 16, but it is a factor of 24.

Those twos are important too. Because 24 = 3 x 8, and 8 is also a factor of 16.

· 3 x (8 x 2)

· = (3 x 8) x 2

· = 24 x 2

So, what? Where were we? We want to be able to split my daily climb into manageable chunks. And we’ve discovered I’m not climbing 15 floors of the same size. In fact I’m climbing the equivalent of 14 floors.

This annoys me. It annoys me because I’m not getting as much exercise as I thought I was, and because the factors of 28 (= 2 x 14, the number of flights of stairs) aren’t as nice as the factors of 30. 28 has the advantage of being splittable into quarters, but beyond that you have to go up to sevenths, and I can’t explain why but I’d rather my journey be split into six chunks rather than seven. Something to do with losing the “halfway” satisfaction, I think.

But hey, why restrict myself to flights as my base unit? Why not increase the potential number of factors by counting stairs instead of flights?

· 14 x 24 = 336

Oooh, look at all those lovely prime factors!

Via http://www.cafepress.co.uk/+teacher+mousepads

· 336 = 3 x 112 = 3 x 2 x 56 = 3 x 2 x 7 x 8 = 2 x 2 x 2 x 2 x 3 x 7=²⁴ x 3 x 7

So if I want to, I can still divide my journey into six chunks:

· 336 = 6 x 56

By the time I’d worked this out, I was at the top of my climb for that day. The following day I was distracted (“Curry!” “Cake!”) and then I couldn’t remember what a sixth of 336 was so I had to work it out again, by which time I’d travelled a significant distance and had no idea, using the new definition of a flight, what proportion of the journey I’d already covered.

So I decided to count backwards from the top. I was just arriving on the fourth floor when I realised it represented the end (or the beginning) of one of my chunks. And because I was still climbing, and some of my brain juice was distracted by leg-moving, it took me the rest of the journey to spot that yes, of course, the reason you arrive at three sixths (aka half, doh) on an actual floor (rather than in the middle of a flight, or halfway up a floor) is that

· 6 x 56 = 2 x 3 x 56 = 2 x (3 x 56) = 2 x (3 x 4 x 14) = 2 x (12 x 14)

56 is not a multiple of 12, because it doesn’t share the prime factor of 3. But if you multiply it by 6, you are multiplying it by 3 twice, which gives you two points along your journey (halfway through, and the top) where you will coincide with a multiple of 12 — so you’ll find yourself on an actual floor.

The lesson being, when you get to the 4th floor, you’re halfway there. And every 56 steps, you’ve climbed another sixth of the distance. Um. That’s kind of it, really. That’s what I learnt.

Yeah, I know. You thought there was going to be some great analogy about climbing the stairs of life. And instead, you got maths. But that’s fine, as it means you’ve given up reading by now. So I can make this really stupid face and you won’t even see.

*SPF = Stairs Per Floor