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

Since, the square is composed of 9 numbers,

3 numbers will mark its length, and 3 numbers will mark its breadth.

(as both are equal in square)

three numbers will be consecutive and the next three numbers will be the same days of the next week (∴ they will also be consecutive)

and the next three numbers will be the same days of the next week (∴ they will also be consecutive)

Let the smallest number be x.

∴ next number = x + 1

∴next number = x + 3

Next week, same day

Number will be x + 7

And next number to it = x + 7 + 1 = x + 8

Next number = x + 8 + 1 = x + 9

Next week, same day

Number will be x + 7 + 7 = x + 14

Next number = x + 14 + 1 = x + 15

Next number = x + 15 + 1 = x + 16

Sum of all the nine numbers = 90(given)

⇒ x + x + 1 + x + 2 + x + 7 + x + 8 + x + 9 + x + 14 + x + 15 + x + 16 = 90

⇒ 9x + 72 = 90

⇒ 9x = 90-72

⇒ 9x = 18

⇒

⇒x = 2

x + 1 = 3

x + 2 = 4

x + 7 = 9

x + 8 = 10

x + 9 = 11

x + 14 = 17

x + 15 = 18

x + 16 = 19

Hence, the required numbers are 2,3,4,9,10,11,17,18and 19.