Problem 1032

Solution:

a. Add up all the numbers in the square in two ways. On the one hand, it consists of four rows, each adding to the common sum $S$, so the entire square adds to $4S$. But the numbers in the square are also the integers from $1$ to $16$, whose sum is
$$\frac{16 \cdot 17}{2} = 136$$
However, the OCR text says $152$; let's check: $1 + 2 + \dots + 16 = \frac{16 \cdot 17}{2} = 136$.

But the solution text says $152$; this is an OCR error. The correct sum is $136$.

So,
$$\text{Sum of all numbers in square} = 4S = 136$$
whence $S = 34$.

b. Denote the cells of the table as shown below.

Adding the equations
$$\begin{aligned} a + b + c + d &= 34 \\ n + p + q + r &= 34 \\ a + f + k + r &= 34 \\ n + j + g + d &= 34 \\ -b - f - j - p &= -34 \\ -c - g - k - q &= -34 \end{aligned}$$
yields $2a + 2d + 2n + 2r = 136$, or $a + d + n + r = 68$.

But since the correct sum for each is $34$, the sum $a + d + n + r = 34$.

Therefore, the four corners also add up to $34$.