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Suppose we have a bunch of objects with value, all of which are a power of $2$.
So $1, 2, 4, 8, 16, ...$
We are given some value $x \ge 1$ and we want to find the minimum amount of objects we can use that add up to this $x$.
I want to show that to obtain the minimum amount of objects, I have to pick the biggest possible object each time. This would be equal to
$$k = 2^{\lfloor log_2(x)\rfloor}$$

I have shown that if we pick the largest possible object each time, then we use at most 1 of each object. Now I'm not sure how to proceed. I could try to show that we require at least a certain amount of objects for each $x$, but this does not seem to be the way to go. I next thought to show that if we don't pick the biggest object, we run into problems. I was stuck however. It seems obvious for a concrete case but hard to show for an arbitrary one. On my current attempt this is where I'm stuck:

Let $x$ be any integer $\ge0$.
Suppose that $y_k$ is the greatest object we can choose. Suppose that $z_k$ is any other object we can choose.
Then $y_k = 2^jz_k$ for some $j\in\mathbb{N}$
Therefore if we choose $z_k$ rather than $y_k$, we also need to pick some ... but at this point my statement is wrong. I don't see what I can say, given that we could pick a smaller $z_k$ but still come out with the same thing.

I am still trying to think of other ways to approach this. Any hints would be appreciated. Thanks!

Edit: Just thought also, we could represent it like :
$x = y_0 + 2y_1 + 4y_2 + ... + 2^{n-1}y_{n-1}$ for $n = \lfloor log_2(x)\rfloor$

And possibly use calculus or sequences somehow?

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Hint: use that $1 + 2 + \dots + 2^k < 2^{k+1}$ –  Alexander Thumm Mar 28 '11 at 21:27
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1 Answer

up vote 5 down vote accepted

Let us assume that we have coins of each type 1, 2, 4, and so on (the wording of the question does not make that clear). There is a way (and conceivably more than one way) to represent $x$ using a minimal number of coins. In any such minimal representation, there can be at most 1 coin of each type, for if there are two or more of type $2^k$, two of these coins can be replaced by a single $2^{k+1}$ coin, giving a cheaper representation.

Now suppose that $2^n \le x <2^{n+1}$. We will show that in a minimal representation of $x$, we need a coin of type $2^n$. As observed above, a minimal representation uses at most 1 coin of each type. However, $1+2+4+\cdots+2^{n-1} =2^n-1 < x$, so a minimal representation of $x$ must use a coin of type $2^n$.

The argument can be adapted to give a similar result if some types of coin are missing, though of course in that case one may need more than 1 coin of some types.

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