Answer: Alice has the prize
<u>Step-by-step explanation:</u>
![\boxed{\begin{array}{c|c|c||l}\underline{Alice}&\underline{Bob}&\underline{Carol}&\underline{Prize}\\T&T&F&\text{Alice}\\T&F&F&\text{no possible answer}\\T&F&T&\text{no possible answer}\\F&T&T&\text{no possible answer}\\F&F&T&\text{no possible answer}\\F&T&F&\text{no possible answer}\end{array}}](https://tex.z-dn.net/?f=%5Cboxed%7B%5Cbegin%7Barray%7D%7Bc%7Cc%7Cc%7C%7Cl%7D%5Cunderline%7BAlice%7D%26%5Cunderline%7BBob%7D%26%5Cunderline%7BCarol%7D%26%5Cunderline%7BPrize%7D%5C%5CT%26T%26F%26%5Ctext%7BAlice%7D%5C%5CT%26F%26F%26%5Ctext%7Bno%20possible%20answer%7D%5C%5CT%26F%26T%26%5Ctext%7Bno%20possible%20answer%7D%5C%5CF%26T%26T%26%5Ctext%7Bno%20possible%20answer%7D%5C%5CF%26F%26T%26%5Ctext%7Bno%20possible%20answer%7D%5C%5CF%26T%26F%26%5Ctext%7Bno%20possible%20answer%7D%5Cend%7Barray%7D%7D)
Line 1:
Alice: Bob doesn’t have the prize (T) ... either Alice or Carol
Bob: I don’t have the prize (T) ... either Alice or Carol
Carol: I have the prize (F) ... either Alice or Bob
Alice is true for all of them!
Line 2:
Alice: Bob doesn’t have the prize (T) ... either Alice or Carol
Bob: I don’t have the prize (F) ... Bob
Carol: I have the prize (F) ... either Alice or Bob
<em>nobody satisfies all three statements.</em>
Line 3:
Alice: Bob doesn’t have the prize (T) ... either Alice or Carol
Bob: I don’t have the prize (F) ... Bob
Carol: I have the prize (T) ... Carol
<em>nobody satisfies all three statements.</em>
Line 4:
Alice: Bob doesn’t have the prize (F) ... Bob
Bob: I don’t have the prize (T) ... either Alice or Carol
Carol: I have the prize (T) ... Carol
<em>nobody satisfies all three statements.</em>
<em />
Line 5:
Alice: Bob doesn’t have the prize (F) ... Bob
Bob: I don’t have the prize (F) ... Bob
Carol: I have the prize (T) ... Carol
<em>nobody satisfies all three statements.</em>
<em />
Line 6:
Alice: Bob doesn’t have the prize (F) ... Bob
Bob: I don’t have the prize (T) ... either Alice or Carol
Carol: I have the prize (F) ... either Alice or Bob
<em>nobody satisfies all three statements.</em>