## Problem

Place the numbers 1, 2, 3, 4, and 5 into the boxes of the diagram below to form a cross-shaped matrix, so that the sum of the three numbers in the horizontal row is equal to the sum of the three numbers in the vertical column.

## Analysis

Based on the characteristics of the diagram, the number in the center box is shared by both the horizontal row and the vertical column. To make the sum of the three numbers in the horizontal row equal to the sum of the three numbers in the vertical column, we can first determine the middle number, and then ensure that the sum of the numbers on the left and right is equal to the sum of the numbers above and below.

## Solutions

(1) If the center box is filled with 1, then the remaining numbers are 2, 3, 4, and 5. Since 2+5=3+4, there are 8 possible ways to fill the boxes as shown in Diagram 1.

(2) If the center box is filled with 2, then the remaining numbers are 1, 3, 4, and 5. However, these four numbers cannot form a pattern like â–¢+â–¢=â–¢, so the center box cannot be filled with 2.

(3) If the center box is filled with 3, then the remaining numbers are 1, 2, 4, and 5. These four numbers can form the combination 1+5=2+4, so there are 8 possible ways to fill the boxes as shown in Diagram 2.

(4) If the center box is filled with 4, then the remaining numbers are 1, 2, 3, and 5. However, these four numbers cannot form a pattern like â–¢+â–¢=â–¢, so the center box cannot be filled with 4.

(5) If the center box is filled with 5, then the remaining numbers are 1, 2, 3, and 4. These four numbers precisely form the combination 1+4=2+3, so there are 8 possible ways to fill the boxes as shown in Diagram 3.

final answer: there are 24 ways to fill up the boxes.