How can I solve the following: I have a 4x4 board such that every square must be an integer (note that negative integers and zero are allowed). The square has 30 subsquares Ai, and I will denote the sum of its elements by s_i. Is it possible to show that integer can be choosen such that {1,2,…,24} is a subset of {s_1,…,s_{30}}?

For example, if we have square

```
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
```

Then it sums are 1,1,…,1 (16 times), 4,4,4,…,4 (9 times), 9,9,…,9 (4 times) and 16 (one time). But now for example the number 2 is not a sum of elements of any square.

In general case, the sum of elements of squares of

```
a b c d
e f g h
i j k l
m n o p
```

are

```
a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, a+b+e+f, b+c+f+g, c+d+g+h, e+f+i+j, f+g+j+k, g+h+k+l, i+j+m+n, j+k+n+o, k+l+o+p, a+b+c+e+f+g+i+j+k, b+c+d+f+g+h+j+k+l, e+f+g+i+j+k+m+n+o, f+g+h+j+k+l+n+o+p, a+b+e+f+g+h+i+j+k+l+m+n+o+p
```

so is it possible to pick 24 of those numbers such that those numbers are 1,2,…,24 in some order.