## Monday, 24 June 2013

### Four 4s problem

With the summer holidays approaching, here's an engaging diversion for a long train journey or a long wait in an airport departure lounge. I remember first coming across this problem when I was about [( 4 × 4) - (4 ÷ 4)] years old. I see from an internet search that it is still a much-discussed problem; so readers of this blog will have to resist the temptation to look up solutions elsewhere!

The challenge is to see how far you can get in constructing whole numbers, starting with 1, each using up to four digits 4, but no other digits.

You are allowed to use standard mathematical symbols: +, –, ×, ÷, √, for example. So √4 is a way of writing 2. And the [( 4 × 4) – (4 ÷ 4)] written above is a way of writing 15.

You can use two of the 4s to make 44. You can use decimals without a leading zero (for example, .4) and powers, as long as you don't use any other digits. So, for example, you could also make 15 by writing (√4)⁴ – (4 ÷ 4).

Very useful is the mathematical symbol for 'factorial', which is an exclamation mark. So, for example, 4! = 4 × 3 × 2 × 1 = 24, and 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720.

So, yet another way of making 15 might use 6! (720) and divide this by 48. To get the '6' we can use (4 + √4) and to get the 48 we can use (4! × √4). This gives the following very complicated way of constructing the number 15: (4 + √4)! ÷ (4! × √4).

So, here's a start ...
1 = 4 ÷ 4
2 = √4
3 = 4 – (4 ÷ 4)
4 = 4
5 = 4 + (4 ÷ 4)
6 = 4! ÷ 4

So, over to you ... you should find it fairly plain sailing up to 50. Now, remember, no cheating. Googling 'four 4s' is not allowed!