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

[robertphy]

TL;DR

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.

 

[Hill]

Hint: binary.

 

[Office_Shredder]

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?

 

[hmmm27]

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.

 

[pbuk]

I would be grateful for some hints how to go on.

Hint: binary.

This is a very good hint: how are you getting on?