Solutions to Autotheist's Problem Page
Problem 1 (Physics)
One bright sunny day you take a rowboat into the public swimming pool.
In your boat there is a large boulder which you have been using as an anchor.
In the middle of the swimming pool, you drop the anchor overboard. But
before you do this, you wonder what will happen to the water level in the
pool. Will the water rise, fall or remain unchanged?
Solution
The water level lowers. Do not worry if you did not get the answer
right, legend tells that even great minds and Nobel prize winning physicists
were tricked by this problem. When the boulder is in the boat, the requirement
that the boat float is that it displace a volume of water equal to is weight,
and the boulder's weight. When the boulder lands at the bottom of the pool,
it displaces its own volume of water, and the boat continues to displace
its weight in water. Since the density of the rock is greater than the
density of the water, the volume of water with equal weight is greater
than the volume of the rock, and so the water level falls.
Problem 2 (Mathematics)
You meet an eccentric millionaire one day, and he tells you that he loves
to weigh things with his golden balance scale (there are two pans, he puts
the object on one side and balances it with a set of weights). The weights
he uses are solid gold, and he has a weight for each integer, (ie a 1 gram
weight, a 2 gram weight,...) up to his special 1 kg weight. You convince
him that he does not need so many weights, and he offers to give you each
weight that he does not need. However he is not very good at math, and
asks you to select the smallest set of weights needed to weigh every weight
from 1 gram to 1000 kg. What weights do you leave him?
Solution
You leave weights 1,2,4,8,16,32,64,128,256,and 512. The
solution is determined using binary numbers. In order to weigh 1g, you
must leave him a 1g weight. In order to weigh 2g, you must leave the 2g
weight. But to weigh 3g requires only the 1g and 2g weights. Then for 4g
you leave a 4g weight, but you can keep the 5g,6g,and 7g weights. You leave
an 8g weight, and can take 9-15. Leave the 16g weight and take 17-31. ...
Leave the 512g weight, but take 513-1000. In fact, this set of weights
will allow him to measure up to 1.023 kg.
