### Card Trick 2

Can you explain how this card trick works?

### Happy Numbers

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

### The Patent Solution

A combination mechanism for a safe comprises thirty-two tumblers numbered from one to thirty-two in such a way that the numbers in each wheel total 132... Could you open the safe?

# 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:
• 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.
For each one, students can use algebra to prove any relationships they discover.

### 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.