KöMaL 2024 B5420

Ádám, the famous conman signed up for the following game of luck. There is a rotating table with a shape of a regular 13-gon, and at each vertex there is a black or a white cap. (Caps of the same colour are indistinguishable from each other.) Under one of the caps 1000 dollars are hidden, and there is nothing under the other caps. The host rotates the table, and then Ádám chooses a cap, and take what is underneath. Ádám's accomplice, Béla is working at the company behind this game. Béla is responsible for the placement of the 1000 dollars under the caps, however, the colors of the caps are chosen by a different collegaue. After placing the money under a cap, Béla a) has to change the color of the cap, b) is allowed to change the color of the cap, but he is not allowed to touch any other cap. Can Ádám and Béla find a strategy in part a) and in part b) , respectively, so that Ádám can surely find the money? (After entering the casino, Béla cannot communicate with Ádám, and he also cannot influence his colleague choosing the colors of the caps on the table.)