How many cards can be taken at most while satisfying a certain rule?

[songoku]

Homework Statement

Please see below

Relevant Equations

Not sure

The answer is 33 (based on the answer key).

At first, I thought Paul can just take all the 100 cards on 1st draw but since the answer is 33, obviously this thought is wrong. So I assume that the rule (2n + 2) must always be satisfied for each turn and all the cards must be taken (no card remained).

I try all the possible combinations:
Paul = 1 card → Allen = 4 cards
Paul = 2 cards → Allen = 6 cards
Paul = 3 cards → Allen = 8 cards
Paul = 4 cards → Allen = 10 cards
Paul = 5 cards → Allen = 12 cards
Paul = 6 cards → Allen = 14 cards
Paul = 7 cards → Allen = 16 cards
Paul = 8 cards → Allen = 18 cards
Paul = 9 cards → Allen = 20 cards
Paul = 10 cards → Allen = 22 cards
Paul = 11 cards → Allen = 24 cards
Paul = 12 cards → Allen = 26 cards
Paul = 13 cards → Allen = 28 cards
Paul = 14 cards → Allen = 30 cards
Paul = 15 cards → Allen = 32 cards
Paul = 16 cards → Allen = 34 cards
Paul = 17 cards → Allen = 36 cards
Paul = 18 cards → Allen = 38 cards
Paul = 19 cards → Allen = 40 cards
Paul = 20 cards → Allen = 42 cards
Paul = 21 cards → Allen = 44 cards
Paul = 22 cards → Allen = 46 cards
Paul = 23 cards → Allen = 48 cards
Paul = 24 cards → Allen = 50 cards
Paul = 25 cards → Allen = 52 cards
Paul = 26 cards → Allen = 54 cards
Paul = 27 cards → Allen = 56 cards
Paul = 28 cards → Allen = 58 cards
Paul = 29 cards → Allen = 60 cards
Paul = 30 cards → Allen = 62 cards
Paul = 31 cards → Allen = 64 cards
Paul = 32 cards → Allen = 66 cards

Then I tried several combinations but the maximum I can get is 32 cards:
a) Paul = 31 cards, Allen = 64 cards. Then Paul = 1 card, Allen = 4 cards → Total Paul's cards = 32 cards

b) Paul = 30 cards, Allen = 62 cards. Then Paul = 2 cards, Allen = 6 cards → Total Paul's cards = 32 cards

c) Paul = 29 cards, Allen = 60 cards. Then Paul = 3 cards, Allen = 8 cards → Total Paul's cards = 32 cards

d) Paul = 28 cards, Allen = 58 cards. Then Paul = 4 cards, Allen = 10 cards → Total Paul's cards = 32 cards

e) Paul = 27 cards, Allen = 56 cards. Then Paul = 5 cards, Allen = 12 cards → Total Paul's cards = 32 cards

f) Paul = 26 cards, Allen = 54 cards. Then Paul = 6 cards, Allen = 14 cards → Total Paul's cards = 32 cardsDo I even interpret the question correctly? Thanks

 

[.Scott]

After Paul takes a card, Allen only takes 1 card - if that card is available. So if Paul takes card #1, Allen must take card number 4. Paul could have taken card 4, but he wants to save it for Allen. Possible cards for Paul to take without ending the game are 1 through 49, minus the ones he saves for Allen.

 

[songoku]

After Paul takes a card, Allen only takes 1 card - if that card is available. So if Paul takes card #1, Allen must take card number 4. Paul could have taken card 4, but he wants to save it for Allen. Possible cards for Paul to take without ending the game are 1 through 49, minus the ones he saves for Allen.

Ah so I did misinterpret the question.

Thank you very much for the explanation and help .Scott

 

[Prof B]

Two parts of the question are worded strangely:

• "Paul and Allen take the card"
• "there are certain cards for Allen to take, but not for Paul"

I wonder if the question was translated incorrectly. As I understand the game, Paul can take 34 cards.

 

[songoku]

Two parts of the question are worded strangely:

• "Paul and Allen take the card"
• "there are certain cards for Allen to take, but not for Paul"

I wonder if the question was translated incorrectly. As I understand the game, Paul can take 34 cards.

I just realize after reading your reply I also got 34 cards.

My attempt:
a) I started from the highest number Paul can take without ending the game (by ending the game I mean Allen can not take any card), which is card number 49

Then I decrease the number until the number Allen has to take is 50, so:
2n + 2 = 50
n = 24

This means Paul can take card number 24 to 49 → 26 cards

Paul can not take card number 11 to 23 because the game will end.

b) Then I started from card 1:
Paul takes number 1, Allen takes number 4
Paul takes number 2, Allen takes number 6
Paul takes number 3, Allen takes number 8
Paul takes number 5, Allen takes number 12
Paul takes number 7, Allen takes number 16
Paul takes number 9, Allen takes number 20
Paul takes number 10, Allen takes number 22
Paul takes number 11, Allen can not take any card so the game ends

Total cards Paul can take = 34 cards

Paul takes card number: 1, 2, 3, 5, 7, 9, 10 , 11 and 24 to 49 → 34 cards
Allen takes card number: 4, 6, 8, 12, 16, 20, 22 and all even numbered cards from 50 to 100 → 33 cards
Remaining cards: 13, 14, 15, 17, 18, 19, 21, 23 and all odd numbered cards from 51 to 99 → 33 cards

Not sure whether my interpretation is wrong or the answer key is wrong