# An investigation of piles of coins

#### curthnil

Hi all, this is my first post so apologies if this is in the wrong thread!

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.

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??

#### curthnil

Okay so I've managed to come up with some other patterns:

- the prime numbers appear to be the only ones that result in all the numbers from 1 to (p-1) being used

- a number that is a power of 2 will eventually end in an equal number of coins in both piles and this number is half of that number that is a power of 2.

- prime numbers that don't work are one away from a power of 2 e.g. 2^n -/+ 1

Are there any reasons/conclusions from these patterns?

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#### Idea

Let $$\displaystyle c$$ be the total number of coins and

$$\displaystyle a_n$$ = the number of coins in the first pile after $$\displaystyle n$$ steps

(start with $$\displaystyle a_0$$ coins in the first pile, $$\displaystyle 0\leq a_0\leq c$$)

coins in the second pile = $$\displaystyle c-a_n$$

it is easy to see that $$\displaystyle a_n=2a_{n-1}(\text{mod} c)$$ for $$\displaystyle n\geq 1$$

Special case: $$\displaystyle c=2^m$$

if we suppose $$\displaystyle a_0= 2^kb$$ where $$\displaystyle b$$ is odd

then we will have $$\displaystyle c/2$$ coins in each pile after $$\displaystyle m-k-1$$ steps

#### aiimaleoww

I am stuck in this problems too can anyone explain for me or how I can find this key please.