B How to pick some numbers out of 13 integers, by a 4 digits code

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The discussion revolves around deciphering a code made of up to four digits that corresponds to selecting numbers from the first 13 integers. Participants explore methods to connect the code to specific subsets of integers, noting that the sum of selected integers can match the last two digits of the code. Additionally, they consider using modular arithmetic to derive insights from the codes provided. A key suggestion involves using binary representation to map the four-digit codes to the subsets of integers. The conversation seeks further hints and strategies to advance the problem-solving process.
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How to pick some numbers out of 13 integers, by a 4 digits code
This is the question I read in a Math quizzes book:
- you have the first 13 integers from 1 to 13
- someone gives you a code made of at most 4 digits (0 to 9)
- you decipher the code and tell him exactly which numbers he wanted from the list each number can be picked only once

For example.
case 1 He gives you 8191 and you tell he thought all the 13 integers from 1 to 13.
case 2 He gives you 6145 and you tell he thought only 1, 2, 13.
And so on.

I've not come up with any solutions yet. The only sparse ideas I have are:
case 1 if you sum the first 13 integers you get 91, so the last two digits of 8191
case 2 if you sum 1, 2, 13 you get the same result summing the digits of 6145

Also,
8191 mod 3 = 13
6145 mod 2 = 3

Beside that , I have no other clues how to move forward.
I would be grateful for some hints how to go on. Thank you very much.
 
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Hint: binary.
 
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If might help to enumerate how many possible outputs there are. From a pure counting perspective, is it even possible to make a map from 4 digit codes to subsets of integers between 1 and 13?
 
Office_Shredder said:
If might help to enumerate how many possible outputs there are. From a pure counting perspective, is it even possible to make a map from 4 digit codes to subsets of integers between 1 and 13?
Definite. Could fit 1/4 - 1/5 of the further ones required for a 14 element set, as well.
 
robertphy said:
I would be grateful for some hints how to go on.
Hill said:
Hint: binary.
This is a very good hint: how are you getting on?
 
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