Four Fours

The Four-Fours challenge is a classic. It is ideal to play on a long car journey and it is good practice for mental arithmetic.

The idea is to use four fours, and simple arithmetic operations to generate all the integers (the counting numbers) starting from zero.

The rules can differ, depending upon your tastes - but without enough options there are some numbers that are impossible.

Here’s some starting rules:

• Addition, Subtraction, Multiplication and Division

• Brackets are allowed

• Digits can can concatenated (e.g. you can have 44).

• All the fours must be used.

• Powers can be used, e.g. 4⁴ = 4 × 4 × 4 × 4 = 256

Examples

• 0 = 4 + 4 - 4 - 4 = 8 - 8

• 1 = (4 × 4) ÷ (4 × 4) = 16 ÷ 16

• 2 = (4 × 4) ÷ (4 + 4) = 16 ÷ 8

• 3 = (4 × 4 - 4) ÷ 4 = (16 - 4) ÷ 4 = 12 ÷ 4

• 4 = 4 × (4 - 4) + 4 = 0 + 4 

• 5 = (4 × 4 + 4) ÷ 4 = 20 ÷ 4

And so it goes on.

How far can you go?

Flexibility

What other mathematical operations will you invoke? Will you allow factorals (e.g. 4! = 4 × 3 × 2 × 1 = 24)

What about double factorals, i.e. n!! = n × (n - 2) × (n - 4)…..

So 6!! = 6 × 4 × 2 = 48

Then triples factorals, i.e. n!!! = n × (n - 3) × (n - 6)…..

There is a dodge at the bottom of this post that gives you every number.

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Extension

We have to be careful about which operations are allowed; it is possible to make the challenge trivial.

If you find a trick that does this, then I would have a catchall rule of ‘the honour system’. If it feels like a cheat, then it is.

Here is one such cheat that allows you to get any number by stacking square root signs within a logarithm.

Four 4s trick

A solution without honour!