I had lunch with László Babai yesterday, and he asked me the following problem. I changed it up a little to make it less findable online (though I did not find the problem in this variant).
Say you have shakshuka with a friend every Monday at the same restaurant. The meal is always the same, but the price can vary every week; the restaurant adjusts it to market prizes for the many vegetables that enter the dish (shakshuka literally means something like mixed stuff from the market). There is no way of knowing what the prizes next week will be, and it can even be that the restaurant pays you a little for taking overdue vegetables from their hand.
Clik here to view.
Now your friend is quite gentlemanly, and every second week, he allows you to look at the check, and decide whether you want to pick up the bill, or let him do it, with the understanding that you alternate in the week after.
You are not such a polite person, and try to save money in a sneaky way. What should be your strategy?
Here, the strategy works even if your friend knows the strategy. And even if he has set the prices in advance for eternity.
It is one of those cases where your gutfeeling is correct.
