Weights on a Balance

The unknown weight is of the range 1-100 kg. Hence to determine the unknown weight, we will need a set of weights that can determine all the weights from 1 kg to 100kg (or 1kg to 99kg. If none of them matches, it implies that m is 100kg. Correct me if I am wrong).

Consider one of the weights as 67kg (I will explain later why I chose this weight). Assume we already have some set of weights that can be used to measure all the weights from 1kg to 33kg. Hence we can measure all the weights in the range (67+1)kg to (67+33) kg and (67-33)kg to (67-1)kg i.e, 68kg-100kg and 34kg-66kg. We already have 67 kg weight and a set of weights that measures from 1 kg-33kg. Thus we can measure all the weights from 1kg-100kg.

The next step is to find the set of weights for (1-33)kg. This can be done in the similar way as above. In the above scenario, I took two-third of 100, i.e, rounded off to 67. Hence the next weight can be 33 * ( $\frac{2}{3}$ ) = 22kg . Assuming that we already have a set of weights that can measure (1-11)kg, the above mentioned scenario will follow.

Next step is for (1-11)kg . 11 * ( $\frac{2}{3}$ ) is rounded off to 7kg . Remaining (1-4)kg.

4 * ( $\frac{2}{3}$ ) rounded off to 3kg . Remaining 1kg .

Hence the set of weights that can exactly measure all the weights from 1kg to 100kg is 1kg, 3kg, 7kg, 22kg, 67kg. i.e, 5 weights

Hence the answer is $\boxed{5}$ .

Eg : Assume m = 85 kg. Put m, 1kg, 3kg on one pan and 67 kg,22kg on the other.

As others have said the weights are 1, 3, 9, 27, and 81 or 3^{0}, 3^{1}, 3^{2}, 3^{3}, and 3^{4}

1,3,9,27,81 is sufficient For example you would want to weigh 64 kg, you could do 81, 9, 1 on a side and 27 on the other.

The five weights are 1, 3, 9, 27, 81. I did not prove that this was the fewest number of weight, but i could not see how to go lower.

The reason this work is because each weight is the sum of all previous weights, times 2, plus 1. Stopping when you can sum them to 100. In fact these fives weights could determine anything up to 121kg