Measuring Rice 1
Logic
Level
3
You are given a regular balance, a 1 g weight and enough bags[can be used to contain rice]. You want to measure 1 kg of rice accurately from a pile using these. What is the least number of times you have to use the balance?
7
8
9
10
11
12
15
1000
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First 1g weight can be used to measure 1g rice.
Then put that rice in a bag. Now use 1g weight and 1g rice to measure 2g rice and put it in another bag.
Now use 1g weight, 1g rice bag and 2g rice bag to measure 4 gram rice and put it in another bag.
This way continue to measure to get 8g, 16g, 32g, 64g, 128g, 256g, 512g.
If you add all the weights of rice it will be 1023g. But we want 1000g. So remove 1g, 2g, 4g, 16g bags (total 23g).
The rest of the bags' rice add up to 1000g = 1kg. [512+256+128+64+32+8=1000]. So only 10 weighing is needed.
INTUITIVE METHOD:
Turn the number 1000 to binary. 1000 in binary is 1111101000. Now we measure weights of powers of 2, 10 times to get 1g to 512g. Assign position 0th to 9th to these bags respectively. Now let every digit of binary of 1000 from right to left have 0th to 9th position assigned. Separate the bags according to the binary digits that is, if the same position binary digit as the bag is 1, then they are kept together and 0s are kept separately. The bags in 1's side will add up to 1000. The binary of 1000 has 10 digits, so the number of weighing needed is 10.
The reason it works is, every time we measure the rice we get a rice bag with 2 times the weight of previous weight. So we always get bags which has equal rice of power of 2. Every number can be written using summation of different powers of 2. Which powers should be taken for the summation to be our required number is a question. Every digit of binary number has place value of power of 2 and the summation of the place values gives the decimal number it(the binary) represents. As these two are similar, we can use the binary to find out which powers of 2 is needed. The power which has 1 multiplied with it is needed and the power which has 0 multiplied is not. This can be felt with heart if you know how to convert any number of different number system into decimal.