Bad Decision
When I told my father that one day I wanted to be so rich that I had 10000 dollars. He laughed and gave me $10000 dollars and some envelopes, he said that if I could sort the money such that I could give him one or more of the envelopes and it would total the sum of money he asked for, I could have his $10000. Unfortunately for him, I was able to sort the money this way. What is the least amount of envelopes that my father could have given me?
The answer is 14.
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1 solution
For every envelope there are two options, either I give it to my father or I do not, so it is best to think of this problem in binary. Therefore the amount in each envelope should be a power of 2. Let the highest power of 2 among the envelopes be n. Then the sum of giving all of the envelopes we have created so far will be 2 − 1 2 n + 1 − 1 . We need to find the highest power of 2 where 1 0 0 0 0 − ( 2 n + 1 − 1 ) can be written in binary with less than n + 1 digits. We create the inequality 2 n + 1 − 1 > 1 0 0 0 0 − 2 n + 1 Solving this:
2 n + 1 − 1 > 1 0 0 0 0 − 2 n + 1
2 n + 2 > 1 0 0 0 1
Considering that n is an integer it must be 12 and 2 1 4 = 1 6 3 8 4 . So now we have 13 envelopes for 2 0 to 2 1 2 and and additional envelope for the remaining amount makes 14