Fill the 4x4 grid with numbers to make eight arithmetic progressions

Question

In the 4x4 grid below, fill each empty square with a number such that the numbers in each row form an arithmetic progression when read from left to right. Similarly, the numbers in each column form an arithmetic progression when read from top to bottom.

(An arithmetic progression is a sequence in which each term after the first is obtained from the previous term by adding the same constant which might be negative, zero or positive. For example, 3, 5, 7, 9 are the first four terms of an arithmetic progression.)

Inspiration: 2016 Gauss Contest

Answer 1

This seems to be the completed grid:

I got this by trial and error, but here's a more mathematical explanation:

Let a be the difference between each number in the first column. We can start by filling the grid in with the obvious:

From here, we can write the difference of the bottom row in terms of a; it's 25- (27+3a) = -2-3a. We can then fill in the bottom row in terms of a:

We can now find the third column's difference in terms of a. It's (23-3a)-45 = -22-3a. So we can now fill in the third column (remember, we're subtracting this difference because we're working upwards):

And we now know the difference in the second row, that being 55-(67+3a) = -12-3a. So we fill in the second row, once again working backwards:

This gives us

27+a = 91+9a, so a=-8.

From here, it's easy to fill in everything we've solved for already:

...and then the rest of the grid.