The Coin Problem!

We know that the minimum possible value for $n$ is $1 + 2 + 3 + 4 + 5 = 15.$

Now if $n = 15,$ since there is only one possible distribution of coins everyone will know how many coins the others have received.

For $n = 16$ the only possible distribution of coins is $(1,2,3,4,6),$ so again all will know the distribution.

For $n = 17$ the only possible distributions are $(1,2,3,4,7)$ and $(1,2,3,5,6).$ In both cases both Peter and Daniel will know what the others have received. (For example, in Peter's case, if he gets $7$ coins he knows that there is only one valid distribution of $10$ coins to the $4$ others, and if he gets $6$ coins he knows that there is only one valid distribution of $11$ coins to the $4$ others.)

For $n = 18$ we know that the maximum number of coins that Peter can receive is $8.$ The possible distributions are then $(1,2,3,4,8), (1,2,3,5,7)$ and $(1,2,4,5,6).$ In each of these cases Peter, (and Peter alone), will know how the remaining coins are distributed to the others.

Now for $n = 19,$ look at the distribution $(1,2,4,5,7)$ and how it plays out for each person:

• From Peter's perspective there are two possible distributions, namely $(1,2,4,5,7)$ and $(1,2,3,6,7).$

• For Daniel, it could be $(1,2,4,5,7), (1,2,3,5,8)$ or $(1,3,4,5,6).$

• For Calvin, it could be $(1,2,4,5,7)$ or $(1,3,4,5,6).$

• For Brian, it could be $(1,2,4,5,7), (1,2,3,6,7)$ or $(1,2,3,5,8).$

• For Aaron, it could be $(1,2,4,5,7), (1,2,3,6,7), (1,2,3,5,8)$ or $(1,3,4,5,6).$

So for $n = 19,$ there exists a distribution in which none of the people can be certain of what all the others have received without being given further information. Thus the smallest possible value of $n$ is $\boxed{19}.$

Edit: I misunderstood the question, the solution is discussed below.

The only way each person cannot tell the number of coins anyone else has is if there are multiple choices for the number of coins each of the others could've got.

The easiest way to start is by concluding the Brian cannot get only 2 coins as then he could instantly tell that since Aaron must have fewer coins than him he must only have 1 coin. If Brian has 3 coins, Aaron could have 1 or 2 coins.

We can continue this for each person so Calvin gets 4 coins as Brian could have 2 or 3 coins and so on.

In the end we get this: Brian gets 3 coins Calvin gets 4 coins Daniel gets 5 coins Peter gets 6 coins

We can now say Aaron should get 1 coin as it is the smallest possible number and if he got 2 coins he could deduce from the original number of coins what everyone else got.

Therefore we can see the answer is a total of 1+3+4+5+6=19.

Edit: I misunderstood the question, the solution is discussed below.