I like to ask trainees what their favourite number is in mathematics, on their first day. This inevitably goes the same way every year.

T1: “seven”

Me: “why?’

T1: “i don’t know”

T2: “three – because i have three children”

T3: “nine – because i live at 9 blah blah close”

I have a purpose to this task – two in fact. The first is that this is a question children ask maths teachers all the time. They assume we all sit studying numbers all day (a few of us genuinely do!) and that we have determined which are the ‘best’ numbers, and (perhaps paradoxically more interestingly) which numbers are less interesting. The whole idea of interesting and less interesting numbers is encapsulated nicely by the interesting number paradox which states

“all natural numbers are interesting. The “proof” is by contradiction: if there exists a non-empty set of uninteresting numbers, there would be a smallest uninteresting number ā but the smallest uninteresting number is itself interesting because it is the smallest uninteresting number, thus producing a contradiction.”

There’s a twitter account somewhere that capitalises on this idea, but I can’t remember it. If you know it let me know in the comments.

Anyway I’m digressing, the SECOND purpose of the exercise is to supply trainee maths teachers with cool answers so that they’re prepared for when they get asked the question “What is your favourite number?” or “which number is the best?”. If we give them some single digit thing and sentimental reasons, it’s a little unsatisfying for a curious mind isn’t it? I think so. So if you’re reading this and in desperate need for a more interesting number for those rainy days, here are a few of my favourites:

2520 – The lowest common multiple of 1,2,3,4,5,6,7,8,9,10

10^{3003} – known delightfully as a millinillion

Graham’s Number

1729 – Ramanjuan’s number (the smallest number that can be expressed as the sum of two cubes in two different ways – 1729 = 1^{3} + 12^{3} = 9^{3} + 10^{3}

18446744073709551615 – the solution to the ‘rice and chessboard’ problem (fair enough, this might be a little hard to recall on the spot… but practice makes perfect!). This problem dates back to around 1000 AD, i’ll blog about it one day. It basically involves putting one grain of rice on square one, then doubling the amount of rice for each subsequent square, and finding the sum.

44488 – the first of five consecutive happy numbers. A happy number is defined by the following process:

“Starting with any positive integer, replace the **number** by the sum of the squares of its digits in base-ten, and repeat the process until the **number** either equals 1 (where it will stay), or it loops endlessly in a cycle that does not include 1.Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers (or sad numbers).”

625

- 625
^{1}=**625** - 625
^{2}= 390**625** - 625
^{3}= 244140**625** - 625
^{4}= 152587890**625** - 625
^{5}= 95367431640**625** - 625
^{6}= 59604644775390**625** - 625
^{7}= 37252902984619140**625**

Let me know more interesting numbers / your favourite number (and why)Ā in the comments. š

16. The perimeter and area of the same square only equal at this number, I think.

Pi

343 because (3+4)^3 = 343. Helps get another cube number into the memory of students! Palindromic numbers are always cool forthe kids, too.

128 because it is 2^7

i because it is useful, beautiful and incredibly clever and solves a lot of previously insolvable problems.