Written by Francis Heaney
For now, we need only consider the first six statements. Sigmund Freud cannot be the liar. If Sigmund Freud is the liar, then, by statement 2, The Blob does in fact lie all the time. In other words, The Blob is the liar. But we don't have two liars.
Suppose Sigmund Freud is the truthteller. Then statement 3 is incorrect. (The truthteller cannot claim that the liar doesn't lie all the time.) So The Blob is lying. The Blob can't be the liar, since we know he doesn't lie all the time from statement 2. He's not the copier, since he would copy Sigmund Freud's true statement. Therefore he must be the alternator (choosing to start with a lie).
Therefore Jeanne Dixon's first statement (that The Blob sometimes lies) is truthful; she would then be the copier (starting with the truth), and Mary Shelley the liar. Jeanne Dixon second statement would be a lie (copying The Blob's false statement), and The Blob's second statement would be true (alternating with his original lie). So far we have, as statements go: true, true, false, false, true. This is followed by Mary Shelley's statement, which is therefore false, and Mary Shelley is the liar, so this is consistent, so we have one possibility for the four entities: The Blob = alternator, Mary Shelley = liar, Sigmund Freud = truth-teller, Jeanne Dixon = copier.
But what if Sigmund Freud is either the alternator or the copier?
This makes statement 3 true. (Sigmund Freud can make statement 2 falsely about the liar and truthfully about everyone else.) That means The Blob is telling the truth. So The Blob is not the liar. So The Blob doesn't lie all the time, which means statement 2 is true.
Let's try to figure out if Jeanne Dixon is lying in statement 1. If so, then, by statement 1, The Blob in fact never lies. The Blob must be the truthteller. Since statement 2 is true, Sigmund Freud must be the alternator. (If he were the copier, he'd copy Jeanne Dixon's lie in statement 1.)
Now some cases. Jeanne Dixon is either the copier or the liar. Suppose Jeanne Dixon is the copier and Mary Shelley is the liar. That means Jeanne's second statement is true, copying The Blob's truthful statement. So the first six statements are: false, true, true, true, true, false. But that means Mary Shelley's statement that four statements have been true so far is actually true! So that is not the case. If Jeanne Dixon is the liar and Mary Shelley is the copier, then Jeanne Dixon's second statement is a lie, and Mary Shelley copies The Blob's truthful second statement, giving us: false, true, true, false, true, true -- which is consistent. This gives us a second possibility: The Blob = truth-teller, Mary Shelley = copier, Sigmund Freud = alternator, Jeanne Dixon = liar.
But Jeanne Dixon may be telling the truth in statement 1. The first three statements are then all true, which means that the liar must be Mary Shelley (since The Blob, Sigmund Freud, and Jeanne Dixon can apparently utter true statements). From 1 and 2 being true, we know The Blob can make both true and false statements. Therefore The Blob is not the truthteller. For this hypothetical we are assuming that Sigmund Freud is either the alternator or the copier, and we know Mary Shelley isn't the truthteller. Therefore Jeanne Dixon is the truthteller.
Now, if Sigmund Freud is the copier and The Blob is the alternator, then the truth-values of the first six statements are: true, true, true, true, false, false -- leading once again to a contradiction in that Mary Shelley's statement about there being four true statements so far is true, but Mary Shelley is the liar. So Sigmund Freud can't be the copier. If Sigmund Freud is the alternator and The Blob is the copier, though, the first six statements come out as: true, true, true, true, true, false -- which leads to no contradiction. So we have a third possibility: The Blob = copier, Mary Shelley = liar, Sigmund Freud = alternator, Jeanne Dixon = truth-teller.
So here are our three possibilities:
Now let us consider The Blob's hypothetical. In possibility 1, Mary Shelley could say "I am the alternator"; she would be lying. In possibility 2, Mary Shelley would be copying Sigmund Freud's previous statement, which is a lie (as Sigmund Freud is the alternator, and his previous statement was truthful), so, again, she could make the statement; she would be lying. In possibility 3, as in possibility 1, she could make the statement, again, as a lie. So if Mary Shelley were to make that statement, it would not help us distinguish between the three possibilities. Therefore The Blob's statement that, if Mary Shelley were to say such a thing, we would not have enough information to solve the puzzle is true.
The Blob in possibility 1 could not make a true statement there, being an alternator who had told the truth in his previous statement. Neither could The Blob in possibility 3 make a true statement there, because he is a copier following Sigmund Freud's statement, which must be false (as Sigmund Freud is an alternator whose previous statement was true). The only possibility that remains is possibility 2, in which The Blob can clearly tell the truth because he is a truth-teller. Therefore The Blob is the truth-teller, Mary Shelley is the copier, Sigmund Freud is the alternator, and Jeanne Dixon is the liar.