Alice has the advantage as the first person to choose.
An alternative suggestion I hear in different places, for example from standupmaths, is to use the Thue-Morse sequence. If Alice and Bob value their stuff differently, there is a better algorithm, called the Knaster inheritance procedure, that allows each of them to think they are getting more than a half.
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The solution to the second problem is to color the shape as a chess board and check that the number of black and white squares is not the same.
What is interesting about the first problem is that it passes the color test.
This could happen, for example, if we end up living in the same city.
But in this case, the alternating sequence is not so bad either, and is much simpler. Suppose you are divorcing and dividing a huge pile of your possessions. First Alice choses a piece she wants, then Bob, then Alice, and so on.
Once John Conway complained that our tradition is not fair to me. We broke the tradition only once, but that is a story for another day.
From time to time we have an odd number of meals per visit and I end up paying more. Let’s discuss the mathematical way of paying for meals.
Often multi-layered puzzles get solvers frustrated, but the previous paragraph is a hint in itself.
If you expect the difficulty, you might appreciate the fantastic beauty of this puzzle. Share: Every time I visit Princeton, or otherwise am in the same city as my friend John Conway, I invite him for lunch or dinner. If we are in the same place for several meals we alternate paying.
Actually, the fairness of this sequence is overrated. If I pay first every time, this sequence will give me an advantage.
It only makes sense to use it if there is a very long stretch of meals.
Given this strategy, we produce an example designed for this strategy, so that the weighings are consistent, but the collector cannot find a real coin.