Question

В клетки таблицы 4×4 вписаны числа (возможно, нецелые) так, что для любой клетки сумма чисел в клетках, соседних с ней по стороне, равна 23. Чему равна сумма всех чисел в таблице?

Answer 1

Problem Analysis

We are given a 4x4 table with numbers (possibly non-integers) written in each cell. The sum of the numbers in any cell and its adjacent cells (horizontally or vertically) is equal to 23. We need to find the sum of all the numbers in the table.

Solution

To find the sum of all the numbers in the table, we can start by considering the cells in the middle of each row and column. Since each cell is adjacent to four other cells, we can write the following equation for each of these cells:

Cell Value + Left Cell Value + Right Cell Value + Upper Cell Value + Lower Cell Value = 23

Let's label the cells in the table as follows:

A B C D
E F G H
I J K L
M N O P

Based on the equation above, we can write the following equations:

1. B + A + C + F = 23 2. F + B + G + J = 23 3. J + F + K + N = 23 4. N + J + O + M = 23

Now, let's solve these equations to find the values of the cells.

Solving the Equations

We can solve these equations by substituting the values of the adjacent cells into each equation. Let's solve them one by one:

1. B + A + C + F = 23 - Since A and C are not adjacent to any other cells, we can assume their values to be 0. - Therefore, the equation becomes: B + 0 + 0 + F = 23 - Simplifying, we get: B + F = 23

2. F + B + G + J = 23 - We already know that B + F = 23 from the previous equation. - Therefore, the equation becomes: 23 + G + J = 23 - Simplifying, we get: G + J = 0

3. J + F + K + N = 23 - We already know that G + J = 0 from the previous equation. - Therefore, the equation becomes: F + K + N = 23 - Simplifying, we get: F + K + N = 23

4. N + J + O + M = 23 - We already know that G + J = 0 from the previous equation. - Therefore, the equation becomes: N + O + M = 23 - Simplifying, we get: N + O + M = 23

From equations 3 and 4, we can see that F + K + N = N + O + M = 23. This means that the values of F, K, and O can be any numbers as long as their sum is equal to 23.

Finding the Sum of All Numbers in the Table

Now that we have determined the values of some cells, we can find the sum of all the numbers in the table.

Let's consider the sum of the numbers in each row:

- Row 1: A + B + C + D = 0 + B + 0 + 0 = B - Row 2: E + F + G + H = 0 + F + 0 + 0 = F - Row 3: I + J + K + L = 0 + J + K + 0 = J - Row 4: M + N + O + P = M + N + 0 + 0 = M + N

Therefore, the sum of all the numbers in the table is equal to B + F + J + (M + N).

Since the values of F and J are not determined, we cannot find the exact sum of all the numbers in the table. However, we can express it in terms of B, M, and N.

Conclusion

The sum of all the numbers in the table is equal to B + F + J + (M + N), where B, M, and N can be any numbers as long as their sum is equal to 23.

Please note that the values of F and J are not determined, so we cannot find the exact sum of all the numbers in the table.