# Fun with Factorials

So recently I saw this on Twitter

I think I’ve actually seen two versions of the same idea in the last week or so. If you haven’t figured it out yet, the answer is 5!, not 5. As in, 5 factorial (5 x 4 x 3 x 2 x 1 = 120).

Hilarious (or not…). I found it kind of dumb, but it got me thinking that it might be fun to find a version of this that works both ways. ie the answer can be worked out to be 5, or 5!

For that to happen, the question needs to be done in one of two ways. In other words, some kind of order of operations problem, which means we need a multiplication or division in there somewhere. The challenge has been set!

So whilst I was sat in the car park waiting for my son’s sports stuff to finish, I opened the glove box, where my handy emergncy maths kit resides… a book and a pencil.

I decided to go with something along the lines of a + b x c , so that it could be (mis)calculated as (a + b)c rather than correctly as a + bc.

The key is that both answers would work – ie one gives me an answer of 5, and the other 5! (120).

Time for some awesome algebra. Click the image below to be able to read it.

by rearranging and fiddling around a bit, I’ve finally got expressions for b and c in terms of only a.

The only restriction on ‘a’ is that it cannot be 115, otherwise we’re dividing by zero to calculate ‘b’, and nobody wants that!

So now we need a number for ‘a’. I could put anything in, but for the question to look nice, the simpler the numbers the better.

You can see that ‘c’ is going to be a decimal unless ‘a’ is negative. I don’t want ‘a’ to be negative, otherwise the simplicity of the question will be gone. So I want to make sure that ‘c’, if it has to be a fraction, is a simple one. Well 1/2 is probably the simplest fraction there is, and to get that, I need ‘a’ to be equal to 230.

So now we have a = 230, and c = 0.5

Put the value of ‘a’ into our formula for ‘b’, and we get a nice round ‘-220’. The only way i can get that to be positive is if I sacrifice the simplicity of my fraction for ‘c’, but having a negative is no big deal for ‘b’ rather than ‘a’, because it’s not at the start of the question. It just means the question will be a subtraction and a multiplication rather than an addition and a multiplication. Fine by me. The final workings are below (again, click for enlargement)

Time to make it look pretty at the end: