All sheep go BA. It's a simple two letter word and I enjoyed this problem from NRICH which imagined generating a language from these two letters.



In sheep talk the only letters used are B and A. Sequences of words are formed as follows:

The first word only contains the single letter A.

To get the next word in the sequence

1)change each A in the previous word into B
and
2)each B in the previous word into AB.… Continue

Look at this chain of numbers

6,3,10,5,16,8,4,2,1,

There is one rule for odd numbers and one rule for even numbers. If you have the right odd number rule you should get a loop back to one of the numbers in the chain.

I wonder how many chains there are? Systematically investigate the start numbers and see what happens.


Maybe you could invent your own chain rules. See what you find out.

This is a great piece of software for collaborating with students in a visual way:

http://www.dabbleboard.com/draw?b=131603&i=3&c=157f94a93a5c7f500a438abc57ac5b3c38b5bde9

Can you fill a 3 by 3 grid square so that the middle number of each row and column is the mean average of its adjacent numbers?

Eg

2 4 6
5 7 9
8 10 12

Can you work out some algebra to enable a spreadsheet to generate these puzzles?