1. ## Chocolate store

At the window of a chocolate store there are 100 different types of chocolates, each having a label with its price, all of them different from each other.
Is it possible to rearrange all the labels so that the sum of the prices of any group of 1 to 99 labels is different from the sum of the original prices?

2. ## Re: Chocolate store

Originally Posted by Alderamin
[FONT="]At the window of a chocolate store there are 100 different types of chocolates, each having a label with its price, all of them different from each other.[/FONT]
[FONT="]Is it possible to rearrange all the labels so that the sum of the prices of any group of 1 to 99 labels is different from the sum of the original prices?[/FONT]
yes

for arrangement 1 simply let the price be the chocolates index.

for arrangement 2 reverse this so chocolate 1 now costs 100 and chocolate 100 now costs 1, chocolate k goes from k to 101-k.

Now sum the first 99 elements of each.

you get 4950 for the first arrangement and 5049 for the 2nd.

If you had to sum from 1 to 100 then all arrangements of the labels would yield the same total.

3. ## Re: Chocolate store

Maybe I need to rephrase this: We want the sum of labels of ANY group consisting of 1 or 2 or 3 or... up to 99 labels to be different from the sum of the original prices for the same chocolates.
For example, if we have a group consisting of the 1st, 4th and 15th, the sum of the original prices would be 20 but after the reversal the sum of the (repositioned) labels with be 283. How do we ensure that this will happen for ALL subsets of 1 and up to 99 elements?