Firstly, we arrange the known numbers in ascending order.

We have 1,2,3,4,9,12,13.
And we have an unknown.

In this case you have a total of 8 numbers.

So the median is the average of the 2 middle numbers. (The 4th and the 5th)


But we don't know what we can use as the 4th and 5th.

We find the total of the two first.

Total = 2 x 4.5 = 9

We need a total of 9 for the middle numbers.

Let's assume the unknown is not one of those middle numbers. Let's call the unknown U.

Possibilities :

U,1,2,3,4,9,12,13
1,U,2,3,4,9,12,13
1,2,U,3,4,9,12,13


1,2,3,4,9,U,12,13
1,2,3,4,9,12,U,13
1,2,3,4,9,12,13,U


For the first 3 possibilities, the middle numbers are 3 and 4. For the next 3, the middle numbers are 4 and 9.

Then the middle numbers can be 3,4 or 4,9.

But none of the pairs can give us a total of 9. So the unknown must be one of the middle numbers

Possibilities :

1,2,3,U,4,9,12,13
1,2,3,4,U,9,12,13

Now 9 cannot be one of them as it would cause the total of the two to be higher than 9.

The only possible combo is

1,2,3,4,U,9,12,13,
where the middle numbers are 4 and 9.


Since the total is 9,
the unknown will be 9 - 4 = 5.