Threes case: Caroline David Jordan Don't Know, Don't Know, Don't Lose Don't Win Caroline saying Don't Know means that either David or Jordan (or both) has a card larger than a 2. Similarly, David saying Don't Know means that either Caroline or Jordan has a card larger than a 2. In addition, David saying Don't Know also means that if Jordan has a two, that Caroline must have a card larger than a three, else David would have said Don't Lose. If Jordan saw a pair of twos, he would have said Win since he knows he has a card greater than a two. If he saw a two and a card greater than a three, he would have to say Don't Know because his card could be a three, or King (for example). Thus he must see either a pair of threes, or a three and a two. If he sees either person has a two, he knows he beat that, and at least tied the three, so he can say Don't Lose. If he sees a pair of threes, he knows he must have a three or better, since if he only had a two then David would have said Don't Lose. So at the end of the first round, the remaining possibilities to a blind observer are: 2 3 3+ 3 2 3+ 3 3 3+ Next, Caroline says Don't Win. She knows she has a two or a three at this point, so clearly Jordan doesn't have a card larger than a three, so he must have a three. Also, David does not have a two, because otherwise she would have said Don't Lose. So David must have a three. At this point we do not know if Caroline has a two or a three. David and Jordan know, of course, but their subsequent answers, Tie for both, do not let us know as a blind observer. The game can continue from this point forever, with Caroline always saying Don't Win and the other two always saying Tie, and neither Caroline nor the blind observer will never know what card Caroline has. Fifth problem: Eight Don't Knows in a row require some amount of systematic approach to our analysis. To eliminate the top set is straightforward. Caroline saying Don't Know the first time means neither David nor Jordan have an Ace. David saying Don't Know next means neither Caroline or Jordan have a King or higher. Jordan saying Don't Know next means neither Caroline nor David has a Queen or higher. This continues, until after the eighth Don't Know, we know that Caroline and Jordan both have a 6 or lower, and David has a 7 or lower. The lower cards are slightly more complicated. Caroline saying Don't Know first means that *,2,2 is eliminated. David saying Don't Know means that 2,*,2 and 3,*,2 are eliminated. 3,*,2 is eliminated because otherwise David would know he at least tied. Jordan saying Don't Know eliminates all cases where Caroline and David both have 3 or less. That's the first round. In the second round, Caroline saying Don't Know again eliminates all the cases where David has 3 or less and Jordan has 4 or less, as well as the cases where David has 4 and Jordan has 2. David saying Don't Know then eliminates all the cases where both Caroline and Jordan have 4 or less, *and* the cases where Jordan has 2 and Caroline has 5 or less. Jordan saying Don't Know then eliminates the cases where both Caroline and David have four or less, and also the cases where Caroline has 5 and David has 3 or less. In the third round, the analysis becomes more complicated, but after the next two Don't Knows, the only remaining cases are Caroline holding 5 or 6, David holding 4, 5, 6, or 7, and Jordan holding 6; all eight cases are possible at this point. Jordan saying Win indicates that the other two must hold cards less than 6, so Caroline must hold a 5, and David might hold a 4 or a 5. Analysis of all cases: [Don't Lose,Win/Tie,Win/Tie,Win/Tie] (1) 2 2 2 [Don't Lose,Lose,Lose,Win] (12) 3-A 2 2 [Don't Win,Win,Lose,Lose] (144) 2-K A 2-K [Don't Win,Win/Tie,Lose,Win/Tie] (12) A A 2-K [Don't Win,Don't Win,Win,Lose,Lose] (144) 2-K 2-K A [Don't Win,Don't Win,Win/Tie,Win/Tie,Lose] (12) A 2-K A [Don't Win,Don't Win,Win/Tie,Don't Win,Win/Tie] (13) 2-A A A [Don't Know,Win,Lose,Lose] (11) 2 3-K 2 [Don't Know,Lose,Lose,Win] (143) A 2 3-K A 3-K 2-K [Don't Know,Don't Lose,Lose,Win/Tie,Win/Tie] (1) 3 3 2 [Don't Know,Don't Lose,Lose,Lose,Win] (10) 3 4-K 2 [Don't Know,Don't Win,Win,Lose,Lose] (121) 2-Q 2-Q K [Don't Know,Don't Win,Win/Tie,Lose,Win/Tie] (11) 2-Q K K [Don't Know,Don't Win,Don't Win,Win,Lose,Lose] (120) K 2 3-Q K 3-Q 2-Q [Don't Know,Don't Win,Don't Win,Win/Tie,Win/Tie,Lose] (11) K K 2-Q [Don't Know,Don't Win,Don't Win,Win/Tie,Don't Win,Win/Tie] (12) K 2-K K [Don't Know,Don't Know,Win,Lose,Lose] (10) 2 2 3-Q [Don't Know,Don't Know,Lose,Lose,Win] (119) 2-3 K 3-Q 4-Q K 2-Q [Don't Know,Don't Know,Don't Lose,Win/Tie,Lose,Win/Tie] (1) 3 2 3 [Don't Know,Don't Know,Don't Lose,Lose,Lose,Win] (27) 2 3 4-Q 3 2-3 4-Q [Don't Know,Don't Know,Don't Lose,Don't Win,Win/Tie,Win/Tie] (2) 2-3 3 3 [Don't Know,Don't Know,Don't Win,Win,Lose,Lose] (99) Q 2 3-J Q 3-J 2-J [Don't Know,Don't Know,Don't Win,Win/Tie,Lose,Win/Tie] (10) Q 2-J Q [Don't Know,Don't Know,Don't Win,Don't Win,Win,Lose,Lose] (98) 2-3 Q 3-J 4-J Q 2-J [Don't Know,Don't Know,Don't Win,Don't Win,Win/Tie,Win/Tie,Lose] (10) 2-J Q Q [Don't Know,Don't Know,Don't Win,Don't Win,Win/Tie,Don't Win,Win/Tie] (11) Q Q 2-Q [Don't Know,Don't Know,Don't Know,Win,Lose,Lose] (24) 4-J 2 3 4-J 3 2-3 [Don't Know,Don't Know,Don't Know,Lose,Lose,Win] (96) 2-3 4-J Q 4-J 2-J Q [Don't Know,Don't Know,Don't Know,Don't Lose,Win/Tie,Lose,Win/Tie] (1) 4 4 2 [Don't Know,Don't Know,Don't Know,Don't Lose,Lose,Win/Tie,Win/Tie] (2) 4 2-3 4 [Don't Know,Don't Know,Don't Know,Don't Lose,Lose,Lose,Win] (21) 5-J 2-3 4 5-J 4 2 [Don't Know,Don't Know,Don't Know,Don't Win,Win,Lose,Lose] (79) 2-3 J 3-10 4-10 J 2-10 [Don't Know,Don't Know,Don't Know,Don't Win,Win/Tie,Lose,Win/Tie] (9) J J 2-10 [Don't Know,Don't Know,Don't Know,Don't Win,Don't Win,Win,Lose,Lose] (77) 2-3 4-10 J 4-10 2-10 J [Don't Know,Don't Know,Don't Know,Don't Win,Don't Win,Win/Tie,Win/Tie,Lose] (9) J 2-10 J [Don't Know,Don't Know,Don't Know,Don't Win,Don't Win,Win/Tie,Don't Win,Win/Tie] (10) 2-J J J [Don't Know,Don't Know,Don't Know,Don't Know,Win,Lose,Lose] (20) 2-3 4-10 3 4 5-10 2 [Don't Know,Don't Know,Don't Know,Don't Know,Lose,Lose,Win] (74) J 2-3 5-10 J 4 3-10 J 5-10 2-10 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Lose,Win/Tie,Lose,Win/Tie] (2) 2-3 4 4 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Lose,Lose,Win/Tie,Win/Tie] (1) 5 5 2 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Lose,Lose,Lose,Win] (29) 2-3 5-10 4 4 5-10 3-4 5 6-10 2 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Lose,Don't Win,Win/Tie,Win/Tie] (2) 4 4 3-4 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Win,Win,Lose,Lose] (60) 2-3 4-9 10 4-9 2-9 10 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Win,Win/Tie,Lose,Win/Tie] (8) 2-9 10 10 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Win,Don't Win,Win,Lose,Lose] (57) 10 2-3 5-9 10 4 3-9 10 5-9 2-9 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Win,Don't Win,Win/Tie,Win/Tie,Lose] (8) 10 10 2-9 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Win,Don't Win,Win/Tie,Don't Win,Win/Tie] (9) 10 2-10 10 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Win,Lose,Lose] (25) 2-3 4 5-9 4 2-4 5-9 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Lose,Lose,Win] (54) 2-4 10 5-9 5 10 3-9 6-9 10 2-9 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Lose,Win/Tie,Lose,Win/Tie] (2) 5 2-3 5 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Lose,Lose,Win/Tie,Win/Tie] (3) 2-4 5 5 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Lose,Lose,Lose,Win] (20) 2-4 5 6-9 5 2-3 6-9 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Win,Win,Lose,Lose] (42) 9 2-3 5-8 9 4 3-8 9 5-8 2-8 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Win,Win/Tie,Lose,Win/Tie] (7) 9 2-8 9 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Win,Don't Win,Win,Lose,Lose] (39) 2-4 9 5-8 5 9 3-8 6-8 9 2-8 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Win,Don't Win,Win/Tie,Win/Tie,Lose] (7) 2-8 9 9 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Win,Don't Win,Win/Tie,Don't Win,Win/Tie] (8) 9 9 2-9 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Win,Lose,Lose] (17) 5-8 4 3-4 6-8 2-3 5 6-8 5 2 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Lose,Lose,Win] (35) 2-4 6-8 9 5 4-8 9 6-8 2-8 9 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Lose,Win/Tie,Lose,Win/Tie] (3) 5 5 3-4 6 6 2 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Lose,Lose,Win/Tie,Win/Tie] (2) 6 2-3 6 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Lose,Lose,Lose,Win] (18) 6-8 4 5 6-8 5 3-5 7-8 2-3 6 7-8 6 2 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Lose,Don't Win,Win/Tie,Win/Tie] (2) 5 4-5 5 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Win,Win,Lose,Lose] (26) 2-4 8 5-7 5 8 3-7 6-7 8 2-7 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Win,Win/Tie,Lose,Win/Tie] (6) 8 8 2-7 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Win,Don't Win,Win,Lose,Lose] (22) 2-4 6-7 8 5 4-7 8 6-7 2-7 8 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Win,Don't Win,Win/Tie,Win/Tie,Lose] (6) 8 2-7 8 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Win,Don't Win,Win/Tie,Don't Win,Win/Tie] (7) 2-8 8 8 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Win,Lose,Lose] (13) 2-4 6-7 5 5 6-7 3-5 6 7 2 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Win/Tie,Lose,Win/Tie] (1) 7 7 2 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Lose,Lose,Win] (17) 8 2-3 7 8 4-5 6-7 8 6 3-7 8 7 2-7 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Lose,Win/Tie,Lose,Win/Tie] (3) 2-4 6 6 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Lose,Lose,Win/Tie,Win/Tie] (3) 6 6 3-5 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Lose,Lose,Lose,Win] (6) 2-4 7 6 6 7 3-5 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Win,Win,Lose,Lose] (11) 2-4 6 7 5 4-6 7 6 2-6 7 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Win,Win/Tie,Win/Tie,Lose] (2) 7 2-3 7 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Win,Win/Tie,Lose,Win/Tie] (5) 2-6 7 7 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Win,Don't Win,Win,Lose,Lose] (6) 7 4-5 6 7 6 3-6 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Win,Don't Win,Win/Tie,Win/Tie,Lose] (4) 7 7 3-6 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Win,Don't Win,Win/Tie,Don't Win,Win/Tie] (4) 7 4-7 7 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Win,Lose,Lose] (2) 5 4-5 6 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Win/Tie,Win/Tie,Lose] (2) 6 4-5 6 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Win/Tie,Don't Win,Win/Tie] (2) 5-6 6 6 [Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Don't Know,Lose,Lose,Win] (2) 5-6 7 6