## Minimum number of weights------------------------------------------------------------------------------------ *Choose a minimum number of weights to be able to measure all weights between 1-40 kg.*
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To be able to measure 40 kg, sum of all weights chosen must be 40 kg. We can choose 40 weights each of 1 kg. This will enable us to measure all weights between 1 to 40 kg. So, there at least is a solution - 40 weights each of 1 kg.
If we have one weight of 20 kg and 20 weights of 1 kg, we can still measure all values. => number of weights required is 1 + 20 = 21
The 20 weights of 1 kg are used both above and below 20 kg and their range can again be halved. We can have 1 weight of 20 kg as before and 1 weight of 10 kg and 10 weights of 1 kg. => no. of weights required = 1 + 1 + 10 = 12
Half again the number of 1 kg weights => no. of weights required = 1 + 1 + 1 + 5 = 8
Again half => no. of weights required = 1 + 1 + 1 + 1 + 2 = 6
Thus, 6 weights are needed having weights 20, 10, 5, 3, 1, 1
If weight range is 1-N, then weights required are: N/2, N/4, N/8 ... 1, 1 (This last one weight helps to reach 2k from k) =>
Number of weights required are:
1 kg weight is still a must. To measure 2 kg, we can either have 2 kg or 1 kg +1 kg or 3 kg -1 kg weight combinations. Off course, 3-1 is better, because we can then measure, 1, 2, 3 and 4 kg weights.
To measure 5 kg, we can do it as 5 or 1+4 or (3-1)+3 or 3+2 or 6-1 or 7-(3-1) or 8-3 or 9-(3+1) Best is to choose last combo so we will get all weights from 1 to 13.
So to measure 1 to 13 kg weights, we need 1, 3, and 9 kg weights.
To measure 14, we need x-13=14 --> x=27 kg weight.
This seems to imply that a series of 1,3,9,27 ... 3^r can measure all weights between 1 and 1+3+9+27...3^r = (3^r-1)/2 |

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