## A Puzzle Courtesy of Dick Hess

Puzzle. Alice and Bob divide a pie. Alice cuts the pie into two pieces. Then Bob cuts one of those pieces into two more pieces. Then Alice cuts one of the three pieces into two pieces. In the end, Alice gets the smallest and the largest piece, while Bob gets the two middle pieces. Given that both want to get the biggest share of the pie, what is Alice’s strategy? How much can she get?

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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

π’ 80% , 20%
π‘ 40% , 40% , 20%
π’ 40% , 20% , 20% , 20%
π° A: 60% , B: 40% The Best

π’ 50% , 50%
π‘ 50% , 49% , 1%
π’ 50% , 24.5% , 24.5% , 1%
π° 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