‘Fore!’ cried Peter, driving his ball straight down the fairway.

‘Fore!’ cried Clarry, slicing his ball into an inconsiderately placed clump of trees.

‘Fore!’ cried Jim, and watched unbelievingly as his ball curved unerringly towards the only patch of water in sight.

‘Fore!’ cried Rey, missing the ball completely.

Some (undisclosed number of) hours later the gallant golfers were enjoying the delights of the 19th, and reflecting on the game.

‘You know,’ said Jim, who had an analytical mind, ‘at each of the holes 1 through to 18, we had four ‘Fores’! I wonder if each of the numbers 1 to 18 can be expressed in terms of four 4s.’

Rey took up the challenge. ‘Well, [(4 + 4)/4] x 4 = 8, and (44 + 4)/4 = 12,’ he said. ‘In fact,’ he continued shortly, ‘if I am allowed to use 4!, I can find an expression for each number.’

Noting that 4! (4 factorial) stands for 4 x 3 x 2 x 1 = 24, can you do as well as Rey?

Hint 1

List the possible ways in which the 4s can be combined.

Hint 2

Make a list with rows labelled 1 through to 18. Fill in as many gaps as you can, starting with Rey’s answers. Try to get a feel of the size of your expected answers.

Solution

Here is a list of possible answers. There are likely to be several possible expressions for many of the numbers.

1 = 44/44; 2 = 4/4 + 4/4; 3 = (4 + 4 + 4)/4; 4 = 4 + 4 x (4–4); ;

5 = ((4 x 4) + 4) /4; 6 = (4 x 4)/4 + 4; 7 = 4 + 4 –(4/4); 8 = (4 x 4)--4 –4;

9 = 4 + 4 +(4/4); 10 = (44 – 4)/4; 11 = (4! +4! – 4)/4; 12 = (44 + 4)/4;

13 = (4! + 4! + 4)/4; 14 = (4!/4) +4 + 4; 15 = (4 x 4) – (4/4); 16 = 4 + 4 + 4 = 4 ;

17 = (4 x 4) + (4/4); 18 + (4! + 4! + 4!)/4.

Extensions

1. We can ask how many of the numbers of the sequence 1, 2, 3, ... we can construct using a given number of 2s, 3s, 4s ... . The problem changes if we allow say up to four 4s.

2. A similar problem is to use combinations of the digits in the current year to construct 1, 2, 3, ... . Success may depend on the year!

Hint 1

Hint 2

Solution

Extensions