Investigation into Factors and Volumes

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.

Summary Summary

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