How Many Participants Can Have Unique Task Scores in a Math Competition?

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In a math competition with six tasks where participants can score 6, 5, 3, or 0, the challenge is to determine the maximum number of participants that can have unique task scores while ensuring that for every pair of participants, there are two tasks where their scores differ. A proposed solution suggests that the answer might be 4^5, but the reasoning behind this needs clarification. The hint provided encourages listing all possible score combinations, although this method is questioned for its effectiveness. Ultimately, a mathematical proof is required to validate the proposed solution and determine the correct maximum number of participants.
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I don't know how to solve this task:

Participants of math competition are solving six tasks. For each task
you can get one of marks - 6,5, 3 or 0 .
It transpired that for each pair of participants we can indicate two
tasks , that in each of them participant A got different mark from
participant B.
Delimit the highest number of participants for which this situation is
possible.

Could anybody help me?

Thanks in advance
 
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Hint: List all the possibilities. But I don't think that is effective...
 
Last edited:
I think that the answer is 4^5 but i don't know how to prove it.
 

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