17 Jan 2013

Solutions and Extensions: The Game of 24 (PMQ1)

Guided Solution to PMQ1

This is the kind of mathematical puzzle that I really like. It is very simple to state, the number of variables is very small and yet it can be excruciating to solve! Remind yourself of the question by opening the Game of 24 PMQ1 in a new tab.

The first two are, obviously, very simple.

a) (7+5)x(6-4) = 12x2 = 24

b) 5x5-(5/5) = 25-1 = 24

If you have different solutions, that's fine, so long as they really come to 24.

c) Now we come to the question that caused most headaches. Rather than just giving a solution - that's what textbooks usually do - I think it important to go through the key steps. That "Aha!" moment is important, and learning how to break out of certain mental restrictions is also a vital part of solving problems, be they maths puzzles or real-life problems. So let me go through how I found the solution.

It is quickly obvious that juggling around these whole numbers is not going to work. having 8x3=24 seems as if the solution is just round the corner, and yet there is nothing useful that can be done with the remaining 8 and 3 to yield the required 24.

I did, however, notice that 3x3=9 and so 3x3-8=1 and so that 8x3x(3x3-8)=24! Sure, I've got too many numbers, using five instead of four. But look at what happens if I divide rather than multiply: 8x3/(3x3-8) still equals 24. But if I divide the whole expression by 3 I get 8/(3-(8/3)), which is the final answer!

8/(3-(8/3)) = 24

Only the four numbers and a lot of brackets!


There are many other groups of 4 single-digit numbers that give similar solutions. The key here is that using fractions creates an extra 5th number for you out of nowhere! Just look at this to see why:

2 + 3/5 = (2*5 + 3)/5

This is not a solution to anything, but an example of what I mean by an extra number. Notice that the first expression has 3 numbers and just 2 operations; a division and an addition. If we carry out the actual sum we get the next expression which has 4 numbers and 3 operations. Count them; a multiplication, an addition and a division.

So, for those sets of 4 numbers that seem difficult to get a total of 24, try thinking in terms of fractions.

Try these two now:

3 3 7 7

4 4 7 7

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Also, feel free to discuss this and other questions below in the Comments box.

Bye for now


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