Currently I am reading a series of book by Martin Gardner, the one I am working on is "The colossal book of Mathematics". Knowing that this man is hail as the greatest Math-Magician of the 20th Century, I am still surprised by his rather magical tricks.
The following problem is inspired by a Martin Gardner's problem I have read last February.
Two new-weds are traveling to a group of island that has $2019$ islands in total. Some of the islands are connected by boats, some are not. Any island is linked to at least one other island. The couple plays a game. The husband picks an island they will start and the couple will travel there by airplane. From then on, they will take turns to choose the next island they can travel to using boats and which they have not visited before. Who cannot move anymore loses the game.
Prove that: No matter how the wife moves, as well as how the islands are connected, the husband always has a winning strategy.
This problem is just inspired by math tricks, but it is a serious problem. The answer to this may contain graph theory.