We start with the stet $\left\{1,4,32,128,256\right\}$ Is it possible to reach the set $\left\{512,32,16,16,2\right\}$? With given rules

Question

We start with the stet $\left\{1,4,32,128,256\right\}$. In each step we may divide one number by $2$ and multiply another number by $2$. We may repeat this step as many times as we want. Is it possible to reach the set $\left\{512,32,16,16,2\right\}$?

My solution was to let $(1,4,32,128,256)=(a,b,c,d,e)$ to help organize the moves.

1. Divide e and multiply a by $2$ to get $(2,4,32,128,128)$
2. Divide e and multiply b by $2$ to get $(2,8,32,128,64)$
3. Divide c and multiply b by $2$ to get $(2,16,16,128,64)$
4. Divide d and multiply b by $2$ to get $(2,32,16,64,64)$
5. Divide d and multiply a by $2$ to get $(4,32,16,32,64)$
6. Repeat step $5$ to get $(8,32,16,16,64)$
7. Repeat step $1$ to get $(16,32,16,16,32)$
8. Repeat step $1$ to get $(32,32,16,16,16)$
9. Repeat step $1$ again to get $(64,32,16,16,8)$ 10.Repeat step $1$ to get $(128,32,16,16,4)$
10. Repeat step $1$ to get $(256,32,16,16,2)$ Thus can't reach the desired goal.

Now is there a easier way to go about problems like this instead of going through various steps?

Answer 1

Note that the permitted operation preserves the product of the $5$ numbers. And since $1\cdot4\cdot32\cdot128\cdot256=2^{22}$ and $512\cdot32\cdot16\cdot16\cdot 2=2^{23}$, it is not possible to transform the first list into the second.

Answer 2

No. The sum of the $\log_2$ values is unchanged by the given operation and it doesn't match in your two sets.

Answer 3

The starting set has the following indices for the powers of two $\{0, 2, 5, 7, 8\}$. Looking at whether they are odd $o$ or even $e$ we have $\{e, e, o, o, e\}$.

The number of even indices is 3, i.e. there are an odd number of even indices.

The operation will either change $(o,o) \rightarrow (e,e)$, $(e,e) \rightarrow (o,o)$, or $(e,o) \rightarrow (e,o)$. In each case, there will still be an odd number of even indices after applying the operation..

For the target set, the indices of two are $\{9, 5, 4, 4, 1\}$, we have the pattern of index parities as $\{o, o, e, e, o\}$. I.e. there are an even number of even indices.

Therefore we cannot reach the target set with the given operation.