You might think that if the sequence of prices doesn't grow very fast, then using the Thue-Morse sequence is okay. Here is the sequence of prices that I specifically constructed for this purpose: 5,4,4,4,3,3,3,2,2,2,2,1,1,0,0,0. At the end we get a sequence that I decided to call the Fibonacci fair-share sequence. He also knows that all the genuine coins weigh the same and all the fake coins have different weights, and every fake coin is heavier than a genuine coin. He has a balance scale without weights that he can use to compare the weights of two groups with the same number of coins. The strategy is to compare one coin against one coin.
The rule is: every time a turn in the Thue-Morse sequence switches from A to B, the value goes down by 1. If the prices grow faster than a power, then the sequence doesn't work either. This value is bigger than the value of all the leftovers together. After that we can continue and flip the whole thing: ABBBAABAAABB. What is the smallest number of weighings the collector needs to guarantee finding at least one genuine coin? If the scale balances, we are lucky and can stop, because that means we have found two real coins.
They have a 2-seat scooter which rides at 25 miles per hour with 1 rider on it; or, at 20 miles per hour with 2 riders. How fast can all three of them make it to Bob's house? Replaced "the same distance from" with "halfway between" to eliminate the possibility of the plumber living in the yellow house.
They are going to Bob's house which is 33 miles away.
I also wrote a very difficult puzzle called Murder at the Asylum. Whenever I see a problem where the question talks about digits that can be in any order, the first tool to use is the divisibility by 9. She wouldn't use my washing machine, because she didn't have such a thing in Russia. I posted one essay about the puzzle and another one describing its solution.
This is a monstrosity about liars and truth-tellers. The why question, is very important in mathematics. It took me many years to start asking why people did this or that. Every time I came back from work, she complained that she was tired. I promised her that I'd do the laundry myself when there was a sufficient pile. The second puzzle, 50/50, is considered one of the most difficult hunt puzzles ever.
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.
Alice has the advantage as the first person to choose.
If both of them have the same value for each piece, then the Thue-Morse sequence might not be good either.If you expect the difficulty, you might appreciate the fantastic beauty of this puzzle. 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.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.Many people suggest using the Thue-Morse sequence instead of the alternating sequence of taking turns. If this is the order of paying for things, the sequence gives advantage to the second person.