Weight puzzle?

Question

Given a 1 pan weighing scales. You wantto be able to weigh objects weighing 1 to 40kg in whole numbered kg units.
What is the minimum number of weights you need to have and what are they(hint think induction!)

And then if you can find this solution try and generalise the solution to weigh larger amounts(theres your induction again :D!)

Answer 1

Neat. You need 1, 3, 9, 27

Put weights on each side:

1: 1 on left, nothing on right
2: 3 on left, 1 on right
3: 3 on left
4: (3,1)()
5: (9)(3,1)
6: (9)(3)
7: (9,1)(3)
8: (9)(1)
9: (9)()
...
13: (9,3,1)
14: (27)(9,3,1)
...
40: (27,9,3,1)

The general rule is you need powers of 3, and the reason is that each weight can be counted on the left 0 times, 1 time, or -1 times.

Neat.

Answer 2

1, 2, 4, 8, 16, 32 will weigh up to 63 kg in 1 kg increments
20, 10, 5, 2, 2, 1 will weigh up to 40 kg in 1 kg increments

In general, {1, 2, 4, 8, ....., 2^(n-1)} units will provide (2^n) - 1 incremental measurements, which is the most efficient. A "standard" weight set comprised of 1, 2, 2, 5, 10, 10, 25, 50, however, is easier to use for people unfamiliar with powers of 2. A 50 lb weight or a 25 kg weight are the largest that can handled one-handed, so in practice, the progression has to be abandoned in favor of multiple 25 kg or 50 lb weights (32 kg or 64 lb in the binary progression)

Answer 3

a cup?