# Understanding solution for this puzzle where truth-tellers and liars describe game scores

Source (Page 26-28)

Frieda was fond of visiting exotic locations. On her visit to the island of Yippiekaiyay, she was surprised that the natives of the island could be classified into one of two categories – Truth‐Tellers and Liars. The natives had a favourite game called High Cards. This game is played with four players, each holding nine cards, numbered 1 through 9. The game is played in nine rounds. On each round each player places one of his or her cards face down on a table. The four cards are then turned face up and the player having the highest card wins all of the cards in that round. These cards are collected and kept in a separate pile. If there are two or more cards tied for highest then each player takes back the card he or she played and adds it to his or her respective pile. Then the next round begins. In this manner, all cards will be won by someone. After all nine rounds, each player calculates his or her score by adding up the values of the cards in his or her pile and the player with the highest score wins the game.

Frieda spoke to Harakiri, Khiun, Teraki and Wapilipi, four players who had just concluded a game of High Cards and recorded the four statements made by each of the players.

• Harakiri: I scored more than 100. I won exactly 3 rounds. Exactly 2 rounds ended in a tie. At least 2 players won the same number of rounds.
• Khiun: I scored more than 47. I won only 1 round. Exactly 3 rounds ended in a tie. None of the scores were divisible by the square of a prime number.
• Teraki: I scored more than 12. The winner won less than 3 rounds. 4 different cards were played on each of the winning rounds. I did not win any of the rounds.
• Wapilipi: I scored less than 14. The winner scored less than 63. Exactly 1 round ended in a tie. I won exactly 2 rounds.

Who won the game?

Official solution:

Harakiri said that he won 3 rounds and there were 2 ties. To calculate his maximum possible score, he must have won 3 rounds as (9, 8, 8, 8), (8, 7, 7, 7) and (7, 6, 6, 6) and tied the 2 rounds when he played a 6 and a 5. Based on this, his maximum score would have been 98. Since he claimed that he scored more than 100, we know that Harakiri is a Liar. Wapilipi said that he won 2 rounds and there was 1 tie.To calculate his minimum possible score, he must have won the 2 rounds as (2, 1, 1, 1) and (3, 2, 2, 2) and played a 4 for the tie. Based on this, his minimum score would have been 18. Since he claimed that he scored less than 14, we know that Wapilipi is a Liar. Since Harakiri and Wapilipi are liars, all their statements are false. This means that the number of tied rounds was not 1 or 2 and must have been 3 or more. Also, “At least 2 players won the same number of rounds” is false. This means that all the players won a different number of rounds. Since there are 9 rounds, we know that Wins + Ties = 9. The only possibilities for Wins and Ties are 6 and 3 respectively, so that the number of rounds won by the 4 players are 0, 1, 2 and 3 in some order. Since we know that 3 rounds ended in a tie, we can conclude that Khiun is a Truth‐Teller. Suppose Teraki is a Liar. Then we know that Teraki would have won at least 1 round and there were 3 ties. For the minimum score, Teraki’s win would be (2, 1, 1, 1) and the cards he played for the ties would be 3, 4 and 5. So, Teraki’s minimum score would be 17. This supports that statement that he scored more than 12. We can now conclude that Teraki is a Truth‐Teller. We now know that Khiun won 1 round and Teraki won 0 rounds. Since Harakiri’s statement “I won 3 rounds” is false, we can conclude that Harakiri won 2 rounds and Wapilipi won 3 rounds. Since Teraki is a Truth‐Tellers, we know that 4 different cards were played on each winning round. Khiun has won 1 round and there are 3 ties. For maximum score, Khiun would have to win as (9, 8, 7, 6) and play 8, 7, 6 for the ties. So, Khiun’s maximum score will be 51. Since Khiun scores more than 47, his possible scores are 48, 49, 50 or 51. Since 48, 49 and 50 are divisible by the square of a prime number, Khiun’s score must be 51. Harakiri has won 2 rounds and has 3 ties. For maximum score, Harakiri wins 2 rounds as (9, 8, 7, 6) and (8, 7, 6, 5) and plays 5, 4 and 3 for the ties (as all the 7s and 6s have been used up). So, his maximum score is 68. Since the winning score is more than 63, we know that Harakiri is the winner and the possibilities are 64, 65, 66, 67 and 68. We can rule out 64 and 68 as both are divisible by 4. So, the winning score is one of 65, 66 and 67. Since Teraki has not won any round, his 9 must be tied with some one else’s 9. We can continue the analysis to get unique values for the scores of Harakiri, Khiun, Teraki and Wapilipi as 65, 51, 17 and 47 respectively, but it is not required. Harakiri won the game.

My question:

How have all the 7s and 6s been used up?

When Khiun won in his attempt his separate deck will have (9, 8, 7, 6) and the deck which he could use to play further rounds would consists of (1, 2, 3, 4, 5, 6, 7, 8). After that he had three draws due to which his separate deck would now be (8, 7, 6, 9, 8, 7, 6) and his rest of the deck would now have (1, 2, 3, 4, 5). How can I proceed from here?

• seems pretty clear each card can only be used one : These cards are collected and kept in a separate pile. Also In this manner, all cards will be won by someone wouldn't be possible if cards were played multiple times, as some card wouldn't be played. Mar 31, 2023 at 14:26
• The example in the solution was used to calculate Harakiri's maximum score. To obtain the maximum score, Harakiri will have to win 2 hands which include 2 7s and 2 6s. It was previously determined that Khiun wins a hand with one 6 and one 7 and ties with his one 6 and one 7. This accounts for all 4 7s and 6s. Thus Harakiri cannot use a 7 or a 6 for a tie and has to use 5, 4 and 3. However, this only happens if Harakiri gets the maximum score of 68. Since it is later determined that he does not, in fact, get a score of 68, this means that he can use a 7 or a 6 in a tie, among other things. Apr 2, 2023 at 6:04