Tim Harford (3:09)

That's Rob Eastway. He's an author friend of the program and he knows a lot of cool maths tricks, which might just make him as smart as Liam Delapp. Let's have a go. See if you can do the same. Give me the cube root of 140,608.

Rob Eastway (3:27)

Okay. I'm going to do that by a couple of amazing mathematical genius things in my head to say. The answer is going to be 52.

Tim Harford (3:37)

Wow. Okay. So you can do it. Liam Delap can do it.

Rob Eastway (3:40)

Yes.

Tim Harford (3:40)

I can't do it. Am I missing something, Tim?

Rob Eastway (3:43)

You can't do it yet.

Tim Harford (3:44)

Okay.

Rob Eastway (3:45)

You're going to be able to do this very soon.

Tim Harford (3:48)

Growth mindset. I love it.

Rob Eastway (3:49)

First of all, this trick that I'm going to explain to you only works for two digit numbers that are cubed. So the cube root is always going to be a two digit number.

Tim Harford (3:57)

Right?

Rob Eastway (3:58)

So there's two digits. We're going to have to work out here the first and second digit of the answer. So let's start with. With the first digit.

Tim Harford (4:05)

Step one, memorise the cubes of all the numbers from 1 to 9.

Rob Eastway (4:10)

So I've already told you, 3 cubed is 27. I mean, 1 cubed is 1, 2 cubed is 8 and so on. 4 cubed is 64. 5 cubed is 125. Those five are fairly easy to remember.

Tim Harford (4:21)

6 cubed is 216.

Rob Eastway (4:24)

Very good.

Tim Harford (4:24)

Because it's dice.

Rob Eastway (4:25)

I can do dice.

Tim Harford (4:26)

Excellent. Seven cubed I'm not so sure about. Eight cubed is 512.

Rob Eastway (4:31)

Correct. So you've only got two to learn because that's numbers.

Tim Harford (4:34)

Right?

Rob Eastway (4:34)

So seven cubed is 343.

Tim Harford (4:36)

Okay. Seven cubed 343 and nine cubed, 729. Okay. I will try to. Actually, I'm gonna just write them down.

Rob Eastway (4:42)

Yeah, sure, sure. Good.

Tim Harford (4:43)

Okay. Right.

Rob Eastway (4:44)

So one of the video clips shows Delap finding the cube root of 274,625. Now all you have to worry about for the first digit is the thousands. So 274. So what you do is say, okay, I know the list of cubes. What's the cube that's just below that?

Tim Harford (5:05)

Right. So 216.

Rob Eastway (5:09)

Correct. Which was what?

Tim Harford (5:10)

Which is six cubes.

Rob Eastway (5:11)

Right. Your first digit. Because it would be six.

Tim Harford (5:13)

Okay. All right. So to find the first digit, you look at the numbers in the thousands and compare it to the cubes of the numbers 1 to 9. Take whichever cube is just smaller and trace it back to its cube root and that gives you the first digit. We had 274,625. So we look only at 274. The cube just below that is 216, which is the cube of 6. So 6 is our first number. That's a bit complex, admittedly, but luckily it gets easier.

Rob Eastway (5:48)

Now for the second digit. Okay, now let's just take a little aside for a moment. Give me a number ending in four, a big number ending in four.

Tim Harford (5:55)

Okay. 560,014.

Rob Eastway (6:00)

Brilliant. Cubit. I'm joking. Okay, Because I don't care. I don't know the answer either. But I do know that the last digit is four. Because when you cube any number that ends in four, it's still four.

Tim Harford (6:13)

Right.

Rob Eastway (6:13)

When you cube any number that ends in five, guess what ends in five and six ends in six and seven ends in.

Tim Harford (6:20)

I'm going to say. Not seven.

Rob Eastway (6:21)

Correct.

Tim Harford (6:22)

Because you, because you already told me that seven cubed is 343.

Rob Eastway (6:25)

Exactly.

Tim Harford (6:25)

So, but is it always three?

Rob Eastway (6:26)

It's always three.

Tim Harford (6:27)

Okay, right.

Rob Eastway (6:28)

And three always ends in seven. So they kind of seven and three swap round and actually two and eight also swap round.

Tim Harford (6:34)

Right.

Rob Eastway (6:35)

So apart from those two, naught to nine, all just cube and ending themselves.

Tim Harford (6:39)

So the message of this is that the last digit of a cube number is always going to be predictable.

Rob Eastway (6:46)

Correct. And Delap's number was 274,625.

Tim Harford (6:51)

Okay. Right. So it ends in five and. And it starts in six and it's a two digit number, so it's 65.

Rob Eastway (6:58)

You're a genius. Or you've got a good memory, but I'm going to go with the first.

Tim Harford (7:01)

Put me up front for Chelsea. That's what I'm saying. Does this tell us something deep about maths or is it just a clever little trick?

Rob Eastway (7:08)

It's one of those little pathways into numbers and the beauty of numbers and the fact that we're spotting patterns. And if you're solving something really big and important, like Fermat's Last theorem, nice little thing to work on over the weekend. The way into it, the first step on that giant Everest of a journey is a little thing like this. Spotting a pattern and saying, maybe that helps. And it's rather beautiful. And it's a party trick.

Tim Harford (7:30)

Love it. Let us turn to something else dear to Liam delap's heart, which is Premier League goals. He scores a few. And Rob, you made a prediction in a book that you published in 2019 about Premier League goals. Just remind us what it was.

Rob Eastway (7:52)

Okay, so what I said was, here's a prediction. The number of goals in the Premier League next season will be 1000. Okay, it might be a few more than that, but it's probably right to within 5%.

Tim Harford (8:02)

Okay, so to within 5% is between 950 and 1050. Were you right?

Rob Eastway (8:09)

Well, it was all going very well until the 202223 season when the total goal record was broken and it was 1084. That was nothing compared to 202324 when the 1,246. We weren't even an amount. Yes.

Tim Harford (8:26)

So what's happened is defenses got weaker, have attacks got stronger or something else going on.

Rob Eastway (8:31)

Yep. Football has changed and in particular referees have changed. So the key thing is the 223 season is when the Qatar World cup happened. And I think it was as a result of that it became an international regulation that the.

Tim Harford (8:46)

No one was allowed to touch me on a messi or what.

Rob Eastway (8:48)

Yeah, well, no, they did. They just. Time wasting is a real problem. We must stop it. And therefore referees, you must be really tough on adding in time for not just injuries, but also things like goal celebrations and all these other things that take up time. So suddenly games became longer, quite a bit longer.

Tim Harford (9:09)

The number of minutes that a ball is actually in play increased by about 6% from the 2022-23 to the 2023-24 season from about 55 minutes to just over 58. This probably can't explain the whole change. Football's a complex game and anything from changes in tactics to differences in player quality may well have had an effect. Indeed, the goals scored increased by about 15% season to season. So it does seem something else was going on. By the way, last season I checked, it was 11, 180 goals. And I'm curious, do you know how many of those goals were scored by Liam DeLap?

Rob Eastway (9:49)

I don't.

Tim Harford (9:50)

It was 12, which is approximately 1%. So pretty good going.

Rob Eastway (9:54)

That is very impressive. And he was playing for Ipswich at the time, so even more impressive.

Tim Harford (9:59)

Our thanks to Rob Eastway, author of Maths on the Back of an Envelope. If you have any questions or comments, please get in touch at More or less@bc.co.uk or we'll be back next time. And until then, goodbye.

Grainger Advertiser (10:20)

If you're the purchasing manager at a manufacturing plant, you know having a trusted partner makes all the difference. That's why, hands down, you count on Grainger for auto reordering. With on time restocks, your team will have the cut resistant gloves they need at the start of their shift and you can end your day knowing they've got safety well in hand. Call 1-800-GRAINGER Click grainger.com or just stop by grainger for the ones who get it done.

MyFico Advertiser (10:50)

Dreaming of buying your first car or new home? Knowing your FICO score is the first step to making it real. With MyFico, you can check your score for free and it won't hurt your credit. You'll get your FICO score, full credit reports and real time alerts all in one simple app. Your credit score is more than just numbers. It's the key to building the future you've been working toward. Visit myfico.com free or download the MyFico app and take the mystery out of your FICO score.