by Nathan Day (@nathanday314)

In today’s Year 8 lesson I discovered a startling fact: **the number of cuboids you can make from 𝑛 cubes is equal to the number of non-prime factors of 𝑛.**

Not long later, I discovered an even startlinger fact: **this isn’t true.** At least, not for all 𝑛.

I decided this would make for an interesting investigation, interweaving ideas about volume, factors, primes and systematic counting.

I would suggest starting with the questions:

- How many cuboids can be made with 12 cubes? (Not counting rotations.)
- How many factors of 12 are not prime? (Remembering that 1 is not prime.)

Pupils can then find the same two properties of different numbers, at first aimlessly, later with the goal of finding numbers that give 1 cuboid, 2 cuboids, 3 cuboids etc.

Pupils will find that the two quantities are always equal (unless they are unlucky with their choices), and will be able to generalise to some nice results, e.g.

- Prime numbers always give 1 cuboid and 1 non-prime factor,
- The product of two primes (distinct or non-distinct) always gives 2 cuboids and 2 non-prime factors,
- $2^𝑛$ seems to give 𝑛 cuboids and 𝑛 non-prime factors,
- The structure of the prime factorisation is all that matters, for example 12 and 45 will give the same number of cuboids/non-prime factors as they are both the square of a prime multiplied by a different prime.

However, the two quantities are not always equal! **[Spoilers below images.]**

Below is my summary of the two key properties, and some prompts for guiding the investigation.

This can be given as much or as little structure as you feel appropriate.

**[They aren’t equal for 𝑛 = 36, 48, 60, 64,..]**