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# Calendar Capers

Why do this problem?

In this problem, students are invited to explore in a new way a very familiar object - a calendar. Numerical patterns lead to some surprising results which can be explained using the power of algebra.

Possible approach

*This problem is structurally similar to Crossed Ends.*

Students will need access to calendars - you can print off a calendar here.

"Choose any 3 by 3 square of dates anywhere on the page, and add the numbers in the four corners."

Invite students to share their results - ask them to describe the square they chose, and what their answer was. They may choose to describe their square in terms of the top left corner, or the number in the middle, so decide as a class on a common reference point.

"Is there a way to work out the total of the four corners without adding them all up?"

Students should notice a relationship between the total and their reference point.

"How can we be sure our relationship will always hold?"

"What happens to a number when we move to the left? To the right?"

"What happens to the number when we move up? Down?

If appropriate, suggest to students that they could call their reference point $n$ and label all other dates in terms of $n$. This gives them the tools they need to prove algebraically that their rule will always hold.

There are a few more challenges suggested in the problem:

### Possible Extension

Pair Products invites students to make sense of a multiplicative relationship using algebra, and would make a good follow-up to this problem.

### Possible Support

Students could start by exploring the relationships in a two by two square.

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Age 11 to 14

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

Why do this problem?

Possible approach

Students will need access to calendars - you can print off a calendar here.

"Choose any 3 by 3 square of dates anywhere on the page, and add the numbers in the four corners."

Invite students to share their results - ask them to describe the square they chose, and what their answer was. They may choose to describe their square in terms of the top left corner, or the number in the middle, so decide as a class on a common reference point.

"Is there a way to work out the total of the four corners without adding them all up?"

Students should notice a relationship between the total and their reference point.

"How can we be sure our relationship will always hold?"

"What happens to a number when we move to the left? To the right?"

"What happens to the number when we move up? Down?

If appropriate, suggest to students that they could call their reference point $n$ and label all other dates in terms of $n$. This gives them the tools they need to prove algebraically that their rule will always hold.

There are a few more challenges suggested in the problem:

- Add the numbers in each row, column and diagonal that passes through the centre number
- Add the numbers in the bottom rows and the left and right columns
- Circle any number on the top row and put a line through the other numbers that are in the same row and column as your circled number.

Repeat this for one of the remaining numbers in the second row.

You should now have just one number left on the bottom row; circle it.

Add together the three circled numbers.

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!

Choose two digits and arrange them to make two double-digit numbers. Now add your double-digit numbers. Now add your single digit numbers. Divide your double-digit answer by your single-digit answer. Try lots of examples. What happens? Can you explain it?