Weights using induction?

Question

Given a 2 pan weighing scales. You want to 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

By using intuition, my answer would be the following weights:
1kg, 3kg, 9kg and 27kg.

The key is that you can put the weight on either side of the scale. So to weigh something that is 2 kg, you can put 3 kg on one side and 1 kg on the other. The difference is 2 kg.

Left side = Right side
Item(1kg) = 1kg
Item(2kg) + 1kg = 3kg
Item(3kg) = 3kg
Item(4kg) = 3kg + 1kg
Item(5kg) + 1kg + 3kg = 9kg
Item(6kg) + 3kg = 9kg
...
Item(37kg) = 1kg + 9kg + 27kg
Item(38kg) + 1kg = 3kg + 9kg + 27kg
Item(39kg) = 3kg + 9kg + 27kg
Item(40kg) = 1kg + 3kg + 9kg + 27kg

In general, use weights that are 3^0, 3^1, 3^2, 3^3, ... 3^n

With 1 weight, you can weigh up to 1kg
With 2 weights (1kg and 3kg) you can weigh up to 4kg
With 3 weights (1kg, 3kg and 9kg) you can weigh up to 13kg
With 4 weights (1kg, 3kg, 9kg and 27kg) you can weigh up to 40kg
With 5 weights (1kg, 3kg, 9kg, 27kg and 81kg) you can weigh up to 121kg.

Answer 2

1111 gets you to 4
add a nickel weight gets you to 9
add a dime weight gets you to 19
add a 20 weight gets you to 39
what about that 40?