- #1

songoku

- 2,302

- 325

- 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