Question 15
Alice and Bob team up to play a game. At the start of the game, the two of them are dropped at two positions on a train whose total length is $1$ kilometre. The train is closed and completely dark, so unless they are at the front or the rear of the train, they do not know their own position, and they have no way of knowing their teammate's position either. The game gives each of them a device, and the left half of the device displays:
- which direction the holder is currently facing (so he/she can choose to move towards the front or the rear of the train);
- the total distance the holder has moved so far;
- whether the holder is currently touching the front of the train;
- whether the holder is currently touching the rear of the train.
The right half of the device displays the information shown on the left half of the teammate's device. The game ends at the instant the two of them meet (their positions coincide).
Assume that before the game starts Alice and Bob are fully informed about the train and all functions of the device, and can discuss their strategy together beforehand. Find the smallest real number $x$ such that Alice and Bob have a strategy guaranteeing that, no matter where the two of them are placed at the start of the game, the sum of the distances the two of them have moved when the game ends does not exceed $x$ kilometres.
- which direction the holder is currently facing (so he/she can choose to move towards the front or the rear of the train);
- the total distance the holder has moved so far;
- whether the holder is currently touching the front of the train;
- whether the holder is currently touching the rear of the train.
The right half of the device displays the information shown on the left half of the teammate's device. The game ends at the instant the two of them meet (their positions coincide).
Assume that before the game starts Alice and Bob are fully informed about the train and all functions of the device, and can discuss their strategy together beforehand. Find the smallest real number $x$ such that Alice and Bob have a strategy guaranteeing that, no matter where the two of them are placed at the start of the game, the sum of the distances the two of them have moved when the game ends does not exceed $x$ kilometres.