The Balance
Puzzle: You have a balance and need to
weigh objects. The weight of each object will be between 1 and 40
pounds inclusive and will be a round number. What’s the fewest number
of weights that you need to be able to balance any of these
objects?
(If you think you have the answer, see the Answer before you
see the Hint. The answer is just the number of weights, not
what they are.)
Show Answer
Show Hint
Hint: Your weights don’t
all need to go on the same side of the balance.
Show Solution
Solution: You need
weights of 1, 3, 9, and 27 pounds (powers of three). If you
want to weigh an object of, say 32 pounds, then on
one side of the balance you put the object to be weighed
along with the 1 and 3 pound weight. On the other
side you put the 9 and 27 pound weight. Both sides
add up to the same weight and the balance is horizontal,
confirming that the object was 32 pounds.
If your best answer was 6 weights, using the binary
scheme of 1, 2, 4, 8, 16, and 32 pounds, then your solution
was able to weigh objects up to 63 pounds. This is more
than was required (40 pounds), so you know that your
solution is not optimal.
Update 9/21/2007: Ken Belcher came up with a clever way
to use four weights to weigh items up to 81 pounds. You
double the weights above so that you have 2, 6, 18, and
54 pound weights. You then use the above scheme if your
test object has an even weight. If it has an odd weight,
you test the two even weights around it to show that it's
greater than one and less than the other. Since test objects
always have weights that are round numbers, you know it must
be the odd number between the two even ones.
~ See all puzzles ~