We can first take the length of the larger square as x. This means the length of the smaller square is (x-2c), where c is the number of dots between the two squares. It will be (x-2c) because there are c dots separating the squares on both sides.
We can then formulate the expression:
(x)^2 - (x-2c)^2
This gives us the total number of soldiers in the hollow square. This uses charlie method.
Expanding and simplifying this, we get:
4c(x-c)
We can equate this to 960, as that is the total number of soldiers said in the question.
4c(x-c) = 960
c(x-c) = 240
We then find the total number of factor pairs of 240, which is exactly 10;
1 x 240, 2 x 120, 3 x 80, 4 x 60, 5 x 48, 6 x 40, 8 x 30, 10 x 24, 12 x 20, or 15 x 16
We can c is equal to the first number (the smaller number), and (x-c) is equal to the second number (the larger number). Therefore there are ten different ways that the general can arrange the 960 soldiers in a hollow circle.
A general strategy to find the number of ways of arranging the soldiers is by using the formula c(x-c), where x>c, and you must equate c and (x-c) to the factor pairs of the total number of soldiers, in that respective order.