Question 1

We can find interesting number patterns in a calendar using simple algebra.

Step 1 — Cross-shaped grid sum

Let us consider a cross-shaped grid of five numbers on the calendar. Let c be the number in the center of this cross.

The number directly above c is always 7 less than c. So, it is $c - 7$. The number directly below c is always 7 more than c. So, it is $c + 7$. The number to the left of c is always 1 less than c. So, it is $c - 1$. The number to the right of c is always 1 more than c. So, it is $c + 1$. The five numbers in the cross are $c - 7$, $c - 1$, $c$, $c + 1$, and $c + 7$. Let us find the sum of these five numbers.

$\text{Sum} = (c - 7) + (c - 1) + c + (c + 1) + (c + 7)$

$= c - 7 + c - 1 + c + c + 1 + c + 7$

$= 5c + (-7 - 1 + 1 + 7)$

$= 5c + 0$

$\boxed{\text{Sum} = 5c}$

This means the sum of the five numbers in a cross-shaped grid is always 5 times the center number.

Let us check this with an example from the August 2025 calendar. Consider the cross with the center number 8. The numbers are 1, 7, 8, 9, 15. Their sum is:

$1 + 7 + 8 + 9 + 15$

$= 40$

Using our trick, 5 times the center number (8) is:

$5 \times 8$

$= 40$

Both methods give the same sum, 40.

Step 2 — 3x3 grid sum

Let us consider a 3x3 square grid of nine numbers. Let a be the number in the top-left corner of this grid.

The numbers in the first row are $a$, $a + 1$, and $a + 2$. The numbers in the second row are $a + 7$, $a + 8$, and $a + 9$. The numbers in the third row are $a + 14$, $a + 15$, and $a + 16$. The center number of this 3x3 grid is $a + 8$. Let us find the sum of these nine numbers.

$\text{Sum} = a + (a + 1) + (a + 2) + (a + 7) + (a + 8) + (a + 9) + (a + 14) + (a + 15) + (a + 16)$

$= (a+a+a+a+a+a+a+a+a) + (0+1+2+7+8+9+14+15+16)$

$= 9a + 72$

We can factor out 9 from this sum.

$= 9(a + 8)$

$\boxed{\text{Sum} = 9(a + 8)}$

This means the sum of the nine numbers in a 3x3 grid is always 9 times the center number ($a + 8$).

Let us check this with an example from the August 2025 calendar. Consider the 3x3 grid starting with 10 in the top-left corner. The numbers are: 10, 11, 12 17, 18, 19 24, 25, 26 The center number is 18. Their sum is:

$10 + 11 + 12 + 17 + 18 + 19 + 24 + 25 + 26$

$= 162$

Using our trick, 9 times the center number (18) is:

$9 \times 18$

$= 162$

Both methods give the same sum, 162.

Step 3 — 1x3 horizontal grid sum

Let us consider a horizontal grid of three numbers. Let a be the number on the left.

The three numbers in this grid are $a$, $a + 1$, and $a + 2$. The middle number of this grid is $a + 1$. Let us find the sum of these three numbers.

$\text{Sum} = a + (a + 1) + (a + 2)$

$= a + a + 1 + a + 2$

$= 3a + 3$

We can factor out 3 from this sum.

$= 3(a + 1)$

$\boxed{\text{Sum} = 3(a + 1)}$

This means the sum of the three numbers in a 1x3 horizontal grid is always 3 times the middle number ($a + 1$).

Let us check this with an example from the August 2025 calendar. Consider the horizontal grid with numbers 28, 29, 30. The middle number is 29. Their sum is:

$28 + 29 + 30$

$= 87$

Using our trick, 3 times the middle number (29) is:

$3 \times 29$

$= 87$

Both methods give the same sum, 87.

Answer

(i) For a cross-shaped grid with center c, the sum of the 5 numbers is 5c. (ii) For a 3x3 grid with top-left number a, the sum of the 9 numbers is 9(a + 8), which is 9 times the center number. (iii) For a 1x3 horizontal grid with left number a, the sum of the 3 numbers is 3(a + 1), which is 3 times the middle number.