Question

2023 maths challenge: J5 Factor Cards:

a) if the card h the largest available number is moved to the score pile at each turn in a 20-game, what will be the score?
b) Show steps that will produce a score of more than 100 points in a 20-game.
c) Explain why every 20-game ends with 8 or fewer cards in the score pile.
d) What is the maximum score for a 20-game? Explain why it is the maximum.

Answer 1

Answer: All I know is that a is 68

And B is kinda easy. Just choose 19 first, then 15, then 18, the 20, then 16, then 14, then 12. And BTW you need the explaining image.

And D is 124 :ddddd

Answer 2

a) 68

b) see the second attachment

c) 4 cards cannot be used, and the number of discards is at least the number of score cards, leaving 8 maximum score cards

d) 121, uses all of the highest possible score cards

Explanation:

You want to answer various questions regarding the J5 Factor Cards game.

a) Score using highest

The first attachment shows the Score pile, the Central pile, and the Discard pile for a game in which the highest eligible card is chosen at each turn. Those cards are 20, 18, 16, and 14, so the score is ...

score = 20 +18 +16 +14 = 68

b) Score > 100

The second attachment shows the Score, Central, and Discard piles for a game with a score of 121. After the 7th turn, the score is 107. On each row of the table, the numbers in the Score pile are listed in the order chosen, the most recent choice being listed last.

Choosing 19, 9, 15, 10, 20, 18, 16, 14 will give a score of 121.

c) Number of Score cards

According to the rules, a card cannot be added to the score pile unless there is at least one other card that is a divisor of it. All the divisor cards are moved to the Discard pile, so the number of discards must be at least equal to the number of score cards.

Only one of the 4 prime numbers greater than 10 can be scored, since 1 is the only other divisor of a prime, and it will be discarded on the first turn. Excluding those primes, there are 17 cards, of which half or more must be discarded. Hence the maximum number of score cards is 8.

d) Maximum Score

The divisors of the numbers 1–20 are 1–10. With careful selection, the numbers on the discard pile can be restricted to 1–8. If we use the maximum possible prime (19), then we cannot use 17, 13, or 11. The number 12 has so many divisors that choosing it will exclude numbers that total more than 12. The numbers 1–20 total 210, so not scoring the numbers 1–8, 11, 12, 13, and 17 means the maximum score is 210 -89 = 121.

The maximum total of scorable numbers is 121.

__

Additional comment

The algorithm for part (b) is to choose the highest eligible number that has the lowest number of divisors remaining on the Central pile.

The third attachment shows the rules of the game.

<95141404393>