See what a gem I came across. It’s a pity such kind of problems are scarcely seen at math competitions for students.
Problem 6. Zeus and Hades take turns writing numbers on an infinite board. Starting by Zeus, who writes the number , each player can either copy the number previously written, or divide this number by and write it. After an infinite amount of turns, the numbers written on the board form a sequence . The players then compute the sum
Zeus wins if this sum converges to a rational number. Otherwise, Hades wins.
Determine if any of the players has a winning strategy, and if so, determine who has it.
Proposed by Éric Hernández.
At each move Hades can make two choices. Vaguely, it means the set of different series, he can impact to be constructed, is non countable, but the rational numbers are. So, perhaps he could escape from all of them. Of course, it’s nonsense, it’s just intuition that turns out to be true. The strategy of Hades is simple. He enumerates the rationals as and at each step he takes care to ensure the series cannot converge to See that the same approach applies for any countable set of real numbers instead of the rational ones. Note also, there is no number theory here or something that plays with specific nature of the rationals. We only play we the freedom of two choices Hades has to evade consecutively each rational number.
Hades wins. He makes a list of all rational numbers and for each he makes a series of moves to ensure the series cannot converge to The first rule he complies to is as follows.
i) Hades can write the same number he receives from Zeus at most consecutive times. The third time he is obliged to halve the number.
Further, for Hades does the following procedure. He halves the number Zeus writes, waits for his opponent’s move and suppose Zeus writes some number . Let the partial sum of the series at this moment is Hades compares to
1st case. Hades makes the sum of the series to surpass He writes Zeus writes a number that is at least Hades again Zeus writes at least and finally Hades adds (the third time he should halve the number). Thus, we came with a sum and it’s guaranteed the series cannot converge to
2nd case. Hades makes a series of moves that guarantees . He writes and halves every subsequent number Zeus puts. Hence, the sum increases with no more than Apparently it cannot surpass . Hades applies to this strategy till the written number drops to After that, he forgets about and proceeds with the next number . The sum of the series cannot exceed Indeed, complying only to i), starting from we cannot add more than