Hey, this is Presh Talwalkar.
Alice and Bob are on a game show.
Each is secretly told a whole positive number.
They are told the two numbers are consecutive, but neither knows the other person's number.
For example, if Alice is told 20 she does not know if Bob was told 19 or 21.
And if Bob is told 21 he does not know if Alice was told 20 or 22.
The point of the game is to guess the other person's number, but there are rules to the game.
Alice and Bob cannot communicate with each other and they are not allowed to plan a strategy either.
The two are in a room where a clock rings every minute.
After the clock rings either player can call out a guess of the other player's number or they can both stay silent.
Here's a clock that rings every minute on the minute.
The game continues until Alice or Bob makes a guess.
After the first guest is made the game ends.
Alice and Bob win 1 million dollars if the guess is correct and they lose and get nothing if the guess is incorrect
How can Alice and Bob best play this game to win?
Each knows the other is perfect at logical reasoning.
Can you figure it out? Give this problem a try and when you're ready keep watching the video for the solution.
At first it seems like Alice and Bob can do no better than random chance.
If Alice is told 20 for instance there is no way to know if Bob has 19 or 21.
But since Alice can limit Bob's number to two possibilities, she can at least have a 50% chance of guessing correctly.
Bob has the same issue: if he was told the number n,
then he cannot be sure if Alice was told n minus 1 or n plus 1. If Bob guesses between two possibilities
then he also has a 50% chance of guessing correctly.
Acting individually Alice and Bob have a 50% chance of guessing the other person's number and winning the game.
Remarkably they can do much better. There is a method they can win
100% of the time if they use logic perfectly.
One key detail is the two are given positive
consecutive numbers. When Alice gets the number n, she usually has to consider if Bob has n minus 1 or n plus 1.
But suppose Alice gets 1.
Bob could have 0 or 2
But 0 is not a positive number so Bob must have 2.
(Search your feelings Alice, you know it to be 2.)
A player that gets 1 will know the other person has 2 and
since the player is sure the player will guess on the very first ring on the clock,
they will win 1 million dollars for sure.
This was from Alice's perspective
But of course if Bob got 1 he would also figure out that Alice has 2 on the first ring of the clock.
But what about larger numbers? We can continue the logic. Suppose Alice gets two and the first ring of the clock passes.
Bob could have 1 or 3.
But if Bob had 1 he would have guessed on the first ring of the clock.
So Bob does not guess after one ring then Bob must have 3.
Alice realizes this after the first ring and she will guess on the second ring of the clock.
In this way Alice and Bob will win 1 million dollars for sure.
The logic would be similar if Bob got 2 and the first ring of the clock passes he would know that Alice had 3.
So now what about larger numbers? We can continue the logic inductively.
Suppose Alice gets n, and the clock rings n minus one times with no guess from Bob.
If Bob had n minus one he would have guessed on the n minus one ring of the clock.
Since Bob did not guess then Bob must have had the larger number n plus one.
Alice therefore guesses that Bob has n plus one after the clock rings exactly n times.
They will win the game for sure and get 1 million dollars.
Bob would of course reason similarly.
So in summary Alice and Bob can always win the game
when they're given two numbers n and n plus 1.
The person who gets the smaller number n
will always guess after the clock ring exactly n times that the other player has n plus 1.
This will be correct, and they will always win the game.
Did you figure it out? Thanks for watching this video. Please subscribe to my channel. I make videos on math and game theory.
You can catch me on my blog Mind Your Decisions,
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