Disco Balls

A shelf of empty seats, one per way of ordering a number’s prime factors: fill them all and you have counted the orderings by hand.

The Disco Balls board

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Grades
2–6
Subjects
Multiplying & Dividing
Kind
Puzzle
Cost
Free, no account, no ads

What it is

The number at the top has a shelf of empty seats under it, one for every different picture it can be drawn as. Tap 2, 3, 5 and 7 into a row and the row draws itself: a recipe of 2 × 3 is one big circle holding two circles, each of those holding three. When the dots come to the number, the picture takes a seat — and the same picture cannot sit down twice.

The rules

Grow the number at the top as dots, once for every order its numbers can go in.

An empty board. A 2 goes into the recipe and two blue branches grow out of the middle, each ending in a dot. A 3 goes in after it, and each branch splits into three magenta branches: six dots.2= 2×3= 6
Tap the colours to build a recipe, read from the middle out: 2 × 3 is two branches of three dots.
Two empty seats. Two times five fills the first, a tree of ten dots. Two times five is built again beside it, and the first seat flashes red and turns it away. Five times two fills the second seat: the same ten dots, on different branches.2×52×55×2
Every order gets its own seat. The same picture cannot sit down twice.

What it gives you to practise

Prime factorisation, met as a picture before it is met as a procedure. There is no tile for 6 because 6 is already 2 × 3, so a number decides the tiles for you: 72 is 2, 2, 2, 3, 3, and no other bag will do. What is left to choose is the order, and the order changes the picture completely while leaving the count exactly where it was. Filling a shelf is counting those orders by hand. Three different primes go six ways, but 12 has only three seats, because swapping its two 2s changes nothing you can see.

In a classroom

The packs run from a School Dance to the Paradise Garage, the numbers growing on the way, and a good question before anyone starts is how many seats a number will have: 30 has six, 400 has fifteen. The switch that turns the nest inside out into a branching tree is worth showing on a screen too: the same recipe, the same dots, drawn two ways, with nothing added and nothing taken away.

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The whole puzzle at one address, with every board in it. Whoever opens it stays inside it, and their progress stays in their own browser.

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Near it on the shelf