### The One Sided Balance Beam Problem

A one sided balance beam actually has two sides but one side is
reserved for fixed weights that come with the set, and the other side
is reserved for the object to be weighed. There are 4 weights which
will enable one to weigh any whole number of grams from 1 t0 15. What
are they?

We can proceed by taking the smallest possible weight and working
up.

The only way to measure a 1g weight would be if one of the weights
were 1 gram.

The smallest number of grams that you could not measure with this
weight would be a 2 gram weight, so we need a 2 gram weight as well.

With a 1g and 2g weight we can measure 1, 2, or 3 grams. The last
weight is weighed by putting the 1g and 2g weights together. The
smallest weight we cannot measure with these weights would be a 4 g
weight. So we heed a 4 gram weight.

With these weights, we can measure any number of grams up to 7. 7
grams is measured by putting all three weights together. The smallest
number of grams that we cannot yet measure is 8 grams. Our fourth and
final weight will be an 8 gram weight. With is and the 7 grams from
the previous weights, we can measure up to 15 grams.

Notice that the weights we used, 1, 2, 4, and 8 grams are all
powers of 2. Moreover, the next weight we will need, 16 is also a
power of 2.

The powers of 2 are used in base 2. Very simply, if there is a 1
in a place in the base 2 representation of a number, put the weight
that corresponds to that place on the scale. If there is a 0 in that
place, do not put that weight on the scale.

There is a combinatorial aspect to this problem. How many
different configurations of weights are there? With each weight,
there are 2 choices: to put it on or leave it off. So with 4 weights,
there will be 24 = 16 configurations. That is enough to weigh any
number of grams from 0 to 15.

This illustrates a well known formula from algebra:

1 + 2 + 2^{2} + 2^{3} + . . . +
2^{n-1} = 2^{n} - 1
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The 2 Sided Balance Beam
Problem

Groupwork