Associated Topics || Dr. Math Home || Search Dr. Math

Minimum Set of Weights Puzzle

```
Date: 10/18/2001 at 23:58:12
From: Jhonen
Subject: Minimum Set of Weights Puzzle

Dear Dr. Math,

What is the minimum number of weights needed for a scale that will be
able to weigh objects from the weight of one pound to 100 pounds,
inclusive, at one-pound increments?

It seems as if the answer should be four, because of Bachet's
Conjecture, and the fact that a set of four weights exists satisfying
the conditions of the problem for other maximum numbers of similar
size, such as 40. I'm a bit stumped as to how to proceed to find a
conclusive proof for the number of weights needed, or the actual
minimum set of weights in order to solve this problem.

Sincerely,
Jhonen
```

```
Date: 10/21/2001 at 00:41:45
From: Doctor Schwa
Subject: Re: Minimum Set of Weights Puzzle

Hi Jhonen,

For each weight, there are three things you can do: put it on the left
pan, the right pan, or not on the balance at all.

So, if you have n weights, there are 3^n things you can do with them.

One of those things is not putting any weights on the scale, which is
good if you want to weigh a 0-pound object, so really there are only
3^n - 1 arrangements.

Then, for each arrangement there's also its mirror image (where all
the weights are switched to the opposite pan of the scale), so there
are at MOST (3^n - 1)/2 arrangements of n weights.

That's enough to prove that 4 weights can weigh at most 40 different
things ... 40 is really the upper limit for 4 weights.

With a fifth weight, you should be able to get up to (3^5 - 1)/2 =
121 pounds.

For some clues in the other direction (that is, how to figure out
what the weights should actually be), see the Dr. Math archives at

Three Weights
http://mathforum.org/dr.math/problems/macdougall12.7.97.html

or, if that doesn't help you, feel free to write us back!

Enjoy,

- Doctor Schwa, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Permutations and Combinations
High School Puzzles

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search