### Calendar Capers

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

### 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.

# Converging Means

##### Age 11 to 14 Challenge Level:
Take any two positive numbers and call the larger one $a_1$ and smaller $b_1$. Calculate the arithmetic mean of the two numbers and call this $a_2$, where: $$a_2 = (a_1+ b_1)/2.$$Calculate the geometric mean of $a_1$ and $b_1$ and call this $b_2$ so that: $$b_2 = \sqrt{(a_1b_1)}.$$ Suppose you start with 3 and 12, then the arithmetic mean is 7.5 and the geometric mean is 6.

Repeat the calculations to generate a sequence of arithmetic means $a_1$, $a_2$, $a_3$, ... and a sequence of geometric means $b_1$, $b_2$, $b_3$, ... where $$a_{n+1} = (a_n+ b_n)/2,$$ $$b_{n+1} = \sqrt{(a_nb_n)}.$$In the example given $$a_2 = 6.75,$$ $$b_2 = \sqrt{(45)}= 6.708\; \mbox{to 3 decimal places}.$$Calculate the first 5 terms of each sequence and mark them on a number line. Calculate a few more terms and make a note of what happens to the two sequences.

Now repeat the same calculations starting with different choices of positive values for $a_1$ and $b_1$. You should notice the same behaviour of the two sequences whatever starting values you choose. Describe and explain this behaviour.

You may like to write a short program for a calculator or computer to calculate the sequences and if so you should send in your program with your solution.