# Best Tutor Interviews | Maths | Finding The Cube Root

Updated: Aug 5, 2021

We continue to share highlights from our tutor recruitment process in our Best Tutor Interview series. Today's post looks at a particularly impressive Maths lesson observed by Toby.

One of the most enjoyable 10 minute __Maths lessons__ I’ve had raised mental arithmetic to the level of a party trick. The applicant began by asking me to pick any one or two digit number I liked and then, without telling him what that number was, use a calculator to cube it. (That is, to times the number by itself twice. So 3³ = 3 x 3 x 3 = 27)

This done, I read out the answer. 12,167. After about four seconds of calculation, the applicant told me that this was twenty-three cubed, 23³. I was baffled. How had he done it? Learning the answer to this was how I spent the next ten minutes.

Like all mental arithmetic, it’s not nearly as difficult as it might first appear; essentially it’s a question of learning by heart a few results, just as you once learned your times-tables as a child.

The easiest part is figuring out the final digit of the cube route, so let’s start there. If we look at the numbers 1-10 cubed, we notice and interesting and useful fact.

1³ = 1 × 1 × 1 = 1

2³ = 2 × 2 × 2 = 8

3³ = 3 × 3 × 3 = 27

4³ = 4 × 4 × 4 = 64

5³ = 5 × 5 × 5 = 125

6³ = 6 × 6 × 6 = 216

7³ = 7 × 7 × 7 = 343

8³ = 8 × 8 × 8 = 512

9³ = 9 × 9 × 9 = 729

10³ = 10 × 10 × 10 = 1,000

The useful and interesting fact is that if you look at the list of cube numbers above (1, 8, 27, 64 etc., etc.), each of them ends in a different digit. That means that when presented with a given cube number, we know immediately the last digit of its cube route. So when I told the applicant (who had memorised the table above) that the cube number was 12,16**7**, he knew straight away that the cube route was _**3** (because **3** cubed = 2**7**). Conversely, if I had told him to find the cube route of, say, 110,59**2**, he would have known at once that the answer was _**8** (because **8** cubed = 51**2**).

The next step, a little more difficult, is how to fill in those blanks by figuring out the first digit of the cube route. Fortunately, there are no more equations for us to set to memory. Or rather, there are, but they are almost identical to the ones above. All we have to do is to add on a few zeroes. Which means if you have learned the cube numbers above, you have effectively already learned the ones below too.

10³ = 10 × 10 × 10 = 1,000

20³ = 20 × 20 × 20 = 8,000

30³ = 30 × 30 × 30 = 27,000

40³ = 40 × 40 × 40 = 64,000

50³ = 50 × 50 × 50 = 125,000

60³ = 60 × 60 × 60 = 216,000

70³ = 70 × 70 × 70 = 343,000

80³ = 80 × 80 × 80 = 512,000

90³ = 90 × 90 × 90 = 729,000

100³ = 100 × 100 × 100 = 1,000,000

What this second list tells us, is how to find the ‘tens’ value of our cube route. To go back to the initial example, 12,167, we can see that this is greater than 8,000 (or 20³) but smaller than 27,000 (or 30³). So our cube route must fall between 20 and 30. Given that we’ve already worked out that it ends in 3, we can say with confidence that the answer is 23. If we apply this to my second example, 110,592, we see that this value is greater than 64,000 (or 40³) but smaller than 125,000 (or 50³). So our answer, ending in 8, must be between 40 and 50, and it therefore 48.

At first, this might all seem terribly complicated. But try asking a friend to type in a two-digit number into a calculator, and cube it. Ask them what the result is and see if you can work out what the cube route is, while having the above tables in front of you. Practise this enough times, and you’ll be able to do it without referencing the tables. Perform this trick at a party, and everyone will think you’re __Einstein__.

__Blog Post Crafted by Toby__

*Toby is one of our top Maths and English tutors, as well as our *__Chief Assessor__*, and he also works on our *__Admin Team__*. His proudest moment was when he tied Christopher Lee’s (AKA Saruman’s) bow-tie! *

*Toby is in charge of *__recruitment of new tutors__*. He conducts interviews with prospective tutors and assesses their lessons to get a feel for whether they have the teaching style we're looking for.*