If the average of 7 numbers is 23, the sum is 7 * 23 = 161.
The median is 23, so we can write the numbers in increasing order as _, _, _, 23, _, _, _
Call the smallest x. Then, the largest is equal to 4x + 15.

We want to maximize the range, which means we want to maximize the difference between the largest and the smallest.

Let's write the list again. Now we have x, _, _, 23, _, _, 4x + 15

We are constrained by the fact that the numbers can only add up to 161. We want as much of this as possible to go to the largest number, so it can grow, and increase the range.

So, we want to minimize the other numbers. Since x is the smallest number, the numbers between x and 23 must be at least x. Add, the numbers between 23 and 4x+15 must be at least 23.

So, the list should be x, x, x, 23, 23, 23, 4x + 15

Add it up and you get 3x + 3*23 + 4x + 15 = 161
7x + 69 + 15 = 161
7x + 84 = 161
7x = 77
x = 11
So, 4x+15 = 44+15 = 59
And the range, largest - smallest is [spoiler]59 - 11 = 48.[/spoiler]