Thread: Number Loop

I am looking for what I see as a really complicated thing, however it is probably really simple to calculate.

Anyway, I'm looking for a way to calculate how many different ways there are to arrange 4 different numbers in a loop (this itself would be the factorial of 4), but so that rotating the loop doesn't effect it (meaning it cannot be 24 because a value such as 1234 would be the same as 4123).

Anyway here is the digraph I was just wondering how many different ways there are to arrange the numbers.


The order of the numbers doesn't matter however the order cannot be repeated twice.
So far I have the following orders but I'm not sure if there are anymore:

• 1>2>3>4>
• 1>2>4>3>
• 1>3>2>4>
• 1>3>4>2>
• 1>4>3>2>
• 1>4>2>3>

I'm sorry if this is in the wrong section or if it's too easy for you to calculate but I have been working on it for hours and can't seem to find an easy way to work it out mathematically without just going through possible options and then discarding repeated ones.

Hi Lawrence,

Since rotation of the numbers doesn't matter, let's assume the 1 is at the upper left hand corner of the square (assuming a "square table" like in your diagram). We then have 3 numbers we can place in the remaining 3 locations in any order. This can be done in 3! ways.

So the total number of arrangements is 3! = 6.

Perfect, thanks for explaining how to mathematically explain the way to determine that there are only 6 different ways to position them.

The "3!" refers to the factorial of the 3 correct?

Yes, 3! is factorial 3.

3! = 1 x 2 x 3.