Gift Coupons

Number Theory Level 4

Raju’s father gave Raju 10 envelopes and 1000 gift coupons. He said to Raju, “Split all the coupons into the 10 envelopes in such a manner that no matter how many coupons I ask for (from 1 to 1000), you can give me one or more of the envelopes containing the exact number of coupons I ask for. If you can do this, you can keep the 1000 gift coupons." Raju found a way and arranged the envelopes in increasing order of the number of coupons inside them.

How many coupons are there in the 1 0 th 10^\text{th} envelope?


The answer is 489.

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Venkatachalam J by Venkatachalam J , 43, India

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1 solution

Venkatachalam J
Dec 28, 2017

Let the nine envelops with the coupons in increasing order is 1, 2, 4, 8, 16, 32, 64, 128, 256.(powers of 2)

Sum of coupons in nine envelops is 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 =511.

Using this nine envelops we can easily give coupons from 1 to 511 with one or more envelops.

Therefore, the number of coupons in the remaining envelop is 1000 - 511=489.

Hence the 10th envelop will have 489 coupons.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Fun problem, though I would suggest rewording a little, as it was somewhat confusing what constituted the ability to produce "one or more of the envelopes containing the exact number of coupons [asked for]". I would suggest wording closer to "such that if I choose any number between 1 and 1000, you can hand me a number of envelopes that all contain that number of coupons in total". Just a suggestion.

Thanks for putting this problem up!

Will van Noordt - 3 years, 5 months ago

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