Problem with the Truth

Here are all the possibilities of the four thieves having coins (C) or not having coins (X):

Andrew: "I only have coins if every one of them have coins too." That means the following possibilities can be crossed out:

Orville: "If at least two of them don't have coins, I won't have any." Three more possibilities can be crossed out:

Nottica: "I only have coins if Andrew has some too." Two more possibilities can be crossed out:

Xorax: "I only have coins if Orville and Nottica both either have coins or none at all." No new possibilities can be crossed out.

In summary, there are only two possibilities left:

In both cases, Xorax is guaranteed to have coins.

First we note that all four statements are true. Let us consider all the cases as follows.

Case 1: If Andrew has coins

$\begin{array} {|r|c|c|c|c|} \hline & \text{Andrew} & \text{Orville} & \text{Nottica} & \text{Xorac} \\ \hline \text{Statement 1}: & \blue{\text{Yes}} & \blue{\text{Yes}} & \blue{\text{Yes}} & \blue{\text{Yes}} \\ \hline \text{Statement 2}: & & \blue{\text{Yes}} & & \\ \hline \text{Statement 3}: & & & \blue{\text{Yes}} & \\ \hline \text{Statement 4}: & & & & \blue{\text{Yes}} \\ \hline \end{array}$

• Andrew: I only have coins if every one of them have coins too. This means that if Andrew has coins, all others have coins too (therefore the $"\blue{\text{Yes}}"$ underneath the four persons' names).
• Orville: If at least two of them don't have coins, I won't have any. Since the other three have coins, Orville has coins too.
• Nottica: I only have coins if Andrew has some too. Since Andrew has coins, Nottica has coins too
• Xorax: I only have coins if Orville and Nottica both either have coins or none at all. Since Orville and Nottica both have coins, Xorax has coins too.
• Conclusion: If Andrew has coins all others have coins too.

Case 2: If Andrew has no coins

$\begin{array} {|r|c|c|c|c|} \hline & \text{Andrew} & \text{Orville} & \text{Nottica} & \text{Xorac} \\ \hline \text{Statement 1}: & \red{\text{No}} & & & \\ \hline \text{Statement 3}: & & & \red{\text{No}} & \\ \hline \text{Statement 2}: & & \red{\text{No}} & & \\ \hline \text{Statement 4}: & & & & \blue{\text{Yes}} \\ \hline \end{array}$

• Andrew: I only have coins if every one of them have coins too. If Andrew has no coins, then not all others have coins.
• Nottica: I only have coins if Andrew has some too. Since Andrew has no coin, Nottica has no coin too
• Orville: If at least two of them don't have coins, I won't have any. Since Andrew and Nottica both have no coin (at least two), Orville has no coin.
• Xorax: I only have coins if Orville and Nottica both either have coins or none at all. Since Orville and Nottica both have no coin, Xorax has coins.
• Conclusion: If Andrew has no coins all others have no coin except Xorax.

Therefore, Xorax is guaranteed to have coins in all cases.

Case 1 -Everyone has coins.

Case 2 -Not everyone has coins (Some or none of them have coins)

Then Andrew doesn't have coins.

Nottica doesn't have any because Andrew doesn't have any.

Orville doesn't have any because two people don't have any.

Xorax has some coins because Nottica and Orville both have none.

Thus whatever the case, Xorax is guaranteed to have coins

• Andrew depends on the state of Nottica AND all others to have coins
• Nottica depends solely on Andrew's state to have coins
• Therefore, Andrew and Nottica are co-dependent on each other, although Andrew still have more requirements that needs fulfilling than just Nottica
• Thus, Nottica also would need to depends on the state of Orville AND Xorax to have coins
• Orville depends on the state of the majority (at least half of the four) to have coins, and Andrew's and Nottica's co-dependency ensures themselves to be his majority by their twin states
• By this, Orville made himself a part of an Andrew-Nottica-Orville trio for a newly established triplet states of the three
• Xorax depends on the state of Orville AND Nottica to have coins, but these two are already part of the triplet states so their having coins or not is decidedly the same anyway
• Xorax is guaranteed to have coins in whatever scenario of triplet states.

Start from only one have coins to everyone have coins and using the exception(s) method you will find that only Xorax is guaranteed to have coins