Final answer:
a) The score in a 20-game will be the sum of the largest available number in each game, which can be calculated using the formula for the sum of the first n natural numbers. b) A score of more than 100 points can be achieved by using a card with a number greater than 13 as the largest available number. c) Every 20-game ends with 8 or fewer cards in the score pile because all the cards in a suit will have been used after 13 games. d) The maximum score for a 20-game is 91 because there are only 13 cards in a suit.
Step-by-step explanation:
a) In a 20-game, if the card with the largest available number is moved to the score pile at each turn, the score will be the sum of the largest available number in each game. If the largest available number is n, then the score will be n + (n-1) + (n-2) + ... + 1. This is the formula for the sum of the first n natural numbers, which is given by the formula n*(n+1)/2. So, the score will be n*(n+1)/2.
b) To produce a score of more than 100 points in a 20-game, the largest available number must be greater than 13. For example, if the largest available number is 14, then the score will be 14*(14+1)/2 = 105. So, by using a card with a number greater than 13 as the largest available number, a score of more than 100 points can be achieved.
c) Every 20-game ends with 8 or fewer cards in the score pile because in each game, the card with the largest available number is moved to the score pile and cannot be used again. Since there are only 13 cards in a suit, after 13 games, all the cards in a suit will have been used and there will be no more cards left to move to the score pile. Therefore, the maximum number of cards in the score pile at the end of a 20-game is 13 * 4 = 52 - the total number of cards in a deck.
d) The maximum score for a 20-game is achieved when the largest available number is 13. Using the formula from part a), the score will be 13*(13+1)/2 = 91. This is the maximum possible score because there are only 13 cards in a suit and after 13 games, all the cards in a suit will have been used and there will be no more cards left to move to the score pile.