A square of 9 numbers is marked in a calendar. The sum of all these numbers is 90.  What are the numbers?

Given that a square of 9 numbers is marked on a calendar and their sum is 90, we can assume the numbers form a 3x3 square. Let's denote the top-left number in this square as \( n \). The numbers in the square would be arranged as follows:

\[
\begin{array}{ccc}
n & n+1 & n+2 \\
n+7 & n+8 & n+9 \\
n+14 & n+15 & n+16 \\
\end{array}
\]

To understand this:
- The numbers in each row increase by 1.
- Moving from one row to the next (downwards) increases the numbers by 7, as we assume a typical calendar where each row represents a week (7 days).

The sum of all these numbers can be expressed as:

\[
n + (n+1) + (n+2) + (n+7) + (n+8) + (n+9) + (n+14) + (n+15) + (n+16) = 90
\]

Simplify the equation:

\[
9n + (1 + 2 + 7 + 8 + 9 + 14 + 15 + 16) = 90
\]

The sum of the constants is:

\[
1 + 2 + 7 + 8 + 9 + 14 + 15 + 16 = 72
\]

So the equation becomes:

\[
9n + 72 = 90
\]

Subtract 72 from both sides:

\[
9n = 18
\]

Divide by 9:

\[
n = 2
\]

Therefore, the numbers are:

\[
\begin{array}{ccc}
2 & 3 & 4 \\
9 & 10 & 11 \\
16 & 17 & 18 \\
\end{array}
\]

These are the numbers \( 2, 3, 4, 9, 10, 11, 16, 17, \) and \( 18 \), and their sum is:

\[
2 + 3 + 4 + 9 + 10 + 11 + 16 + 17 + 18 = 90
\]

This confirms the solution.

Would you like more details or have any questions?

Here are 5 relative questions:
1. How does the sum change if the top-left number starts at a different value?
2. What happens if the square crosses a month boundary?
3. Can we extend this method to rectangles instead of squares?
4. How does this problem change if the square is selected from a calendar with non-standard week lengths?
5. What other interesting patterns can be found in calendar numbers?

**Tip:** When working with calendar grids, recognizing the pattern and increments in both rows and columns is crucial for solving related problems.

Explore how to solve problems involving the sum of numbers in calendar squares. This example calculates the sum of a 3x3 grid where the numbers increase by 1 in each row and by 7 in each column. Suitable for Grades 7-9.

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