What's going on here? Why are people simultaneously saying Archetype sent an item and then proceeding as if he didn't?
If Archetype did not send an item, then there is probably no lying going on.
If Archetype did send an item (which is the most likely case; he said he was going to and he was town), then there is at least one scum in those that claimed to send items.
Cases:
1 claimed to send, 2 did not. In this case, both Xerxes and Yuma are scum.
2 claimed to send, 1 did not. In this case, one of Xerxes and Yuma are scum.
3 claimed to send, 0 did not. In this case, both Xerxes and Yuma are town.
I believe this is what Faust was saying. In the third case, two scum must have sent items to Sudgy (one lied and picked the one Archetype happened to send). In the second case, one is lying and one sent truthfully. In the first case, no scum sent items to Sudgy and one lied.
Did I oversimplify something or is this correct?
I think this is oversimplified. The whole range of possibilities (assuming that Arch sent something and town didn't lie) is this:
Case A: Scum sent three items
Not possible, because then sudgy would have received one item more.Case B: Scum sent two items
Case B.1: Three scum claimed to have sent items.
In this case, the third scum would have correctly guessed Arch's item, and both Xerxes and yuma are town.
Case B.2: Two scum claimed to have sent items.
In this case, one of yuma/Xerxes is scum.
Case B.3: Less than two scum claimed to have sent items.
Not possible, because then the number of received items would be greater than the number of claimed items.Case C: Scum sent one item
Case C.1: Three scum claimed to have sent items
Not possible, because then sudgy would only have received 7 items.Case C.2: Two scum claimed to have sent items
In this case, again, one of Xerxes/yuma is scum.
Case C.3: Less than two scum claimed to have sent an item
Not possible, because then the number of received items wouldn't match the number of sent items.Case D: Scum sent 0 items.
In this case, it immediately follows that one scum claimed to have sent an item and two haven't. So we get that both yuma and Xerxes are scum, and the third scum correctly guessed Arch's item.
I hope this is now correct. Please double-check, if you feel up to it.