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

Stage: 3 Challenge Level:

You will need a calendar for this problem; you can print one off here.

Choose any three by three square of dates on a calendar page.

Add the numbers in the four corners.

Try a few examples. What do you notice?

Can you explain why this happens?

Now try adding the numbers in each row, column and diagonal that passes through the centre number.

What do you notice?
Can you explain your results this time?

Now try adding the numbers in the top and bottom rows, and the left and right columns.

What do you notice?
Can you explain your results?

Finally...

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.

Here is an example:

Find the total for the three numbers circled. What do you notice?

What if you choose different numbers to circle?
What if you choose a different three by three square?
Can you explain your findings?

Extension:

How would your results change if you chose a 4 by 4 square?
What would happen if you chose a rectangle rather than a square?