## A Puzzle Courtesy of Dick Hess

1. #### Kai:

Alice takes at least 60% under the correct strategy.

Alice cuts the pie into 80-20. Bob has to cut the piece of 80 otherwise Alice takes more than 80.

Assume the 2 pieces are x and 80-x, where x x/2 and 80-x > 20 when x <= 40. 80-x is the biggest piece.

Because x <= 40, x/2 = 60.

Chatgpt to my surprise finds an even beter answer, with some help

3. #### R:

Nice max min problem. A divides the pie into equal pieces . Next b has to do the Same. A Will then divide the quarter into equal pieces. A Will have 5/8. B 3/8

A Will Always divides the middle piece in two in the last step. If she is greedy in the First step, b Will divide the biggelt part in equal pieces

🟢 80% , 20%
🟡 40% , 40% , 20%
🟢 40% , 20% , 20% , 20%
🔰 A: 60% , B: 40% The Best

🟢 50% , 50%
🟡 50% , 49% , 1%
🟢 50% , 24.5% , 24.5% , 1%
🔰 A: 51% , B: 49% The Worst

5. #### CyberK:

I also got the answer 0.6, though proving that was a bit tedious, I wonder if someone has a shorter / more elegant argument…
Say Alice divides pie as A, 1-A, and A>=0.5. I split it into 4 cases:
1) A>=0.8 which means A/4 >= 1-A – the best Alice can do is 1-A/2, so this gives best case 60% for A=0.8.
2) 2/3<=A= 1-A but A/4 < 1-A, Bob can split into A/2, A/2 and 1-A ensuring that Alice cannot get more than 3A/4 < 60%.
3) 0.6<=A<2/3. Here A/2 < 1-A. Alice can't get more than 55%.
4) A<0.6 – Bob can split this into (A, 1-A, 0), now Alice cannot get more than A+0 < 0.6, and we already know Alice can get this much with A=0.8. Bob can probably do better here…

@R If Alice divides into equal pieces, why does Bob have to do the same? Bob can split into (0.5, 0.5, 0), and now Alice can't get more than 0.5.

5/8