I am well and truly stuck with this investigation and any help would be greatly appreciated.

The question is:

**Start with two unequal piles of coins. Shift enough coins from the larger pile to the smaller pile, so that the smaller one doubles in size. Continue to change the piles in this way.**

**For example if you start with 5 and 2**

5 2

3 4

6 1

...

And so on.

Investigate.

5 2

3 4

6 1

...

And so on.

Investigate.

To investigate this problem, I need to find patterns in the data, create hypotheses, conjectures, and theories. The only pattern I can find is that numbers that are powers of 2 all eventually end in numbers that are half the original power of 2, for example:

**8**

6 2

*4 4*

**16**

2 14

4 12

8 8

Are there any other patterns? Is there an overall pattern here in which an algebraic formula can be formed? Is this question even solvable??