The Seemingly IMPOSSIBLE Guess The Number Logic Puzzle

Alice and Bob are secretly told consecutive numbers, but neither knows the other person's number. They cannot communicate or plan a strategy in advance. They get $1 million each if one of them can guess the other's number correctly. The clock is ticking, how should they best play the game to win?

My blog post for this video

Source of puzzle

Connect on social media. I update each site when I have a new video or blog post, so you can follow me on whichever method is most convenient for you.

My Books

[jeffreycanfield]

why are Alice and Bob always in such crazy situations?

[chrisninety]

That sounds like an absolute banger of a game show. Two contestants sitting there in complete silence for twenty minutes, until Alice goes "does Bob have 21", and then everyone just sort of goes home.

Still better than Deal or No Deal.

[mickeyrube]

Man, they weren't kidding when they said Alice and Bob would have to have perfect reasoning. Jeez...

[anythinganyway]

Host to Alice: the number im giving you is
Bob: RIP Alice I guess we aren't winning the $1 Million

[MegaMGstudios]

me watching thumbnail:
thats easy!

me watching video:
ERROR 404: brain.exe not found

[SunnyGoodbye]

I got this one, but only because I've seen similar riddles in the past, such as the green eyes riddle and the "how many trees" video you had.

[mittfh]

Of course, your solution depends on them having been given reasonably small numbers - the game show's producers are likely to lose patience if they're forced to wait more than a few minutes/rings (15? 30? 60?) and either force a guess or send them both home without any winnings.

[wookiebush]

I have seen a lot of puzzles that use the same kind of solution, and I can tell you they are flawed and wont work. In the problem, each person is given a number and told the other person has one more or one less. In the solution, you state that the players come to a conclusion based on the possibility that they could have ANY number. You cannot come to a logical conclusion starting off with the statement "if bob/alice had 1" if it is an impossibility for them to have 1. In the example above, Alice had 20 and she is told that Bob has 19 or 21, therefore she would NEVER logically start with the assumption that bob has 1 and start the count from there. There are only 6 possible numbers for each person to wait on. From Alice's point of view, she knows that bob HAS to have either 19 or 21, so she can logically draw the conclusion that if Bob has 19, he knows alice has 18 or 20 and if bob has 21, alice either has 20 or 22. From Bob's point of view, he knows Alice HAS to have either 20 or 22, so he can logically assume that if Alice has 20, she knows Bob has 19 or 21 and if Alice has 22, Bob has either 21 or 23. This means that between the two of them, the lowest number that would be possible between them is 18, so it is a logical impossibility to include the number 1 or 2 or so on.

[ttris]

Eeeeeasy,
Bob says
And when alice screams of happiness, everything is good. If not he ads ...Two to the So 22 and they win

[leefisher]

This is NOT a 100% perfect strategy. You see, Alice is given the value of 994 duodecillion, 775 undecillion, 945 decillion... ... ...

[Uebeltank]

If you have number N, wait N minutes and guess that the other person has N+1.

[GeoDetective]

This is one of those "We silently agree to wait for the right moment" puzzles. Like "Oh, the other person did not guess yet. Then he does not have 1!" But you know this already before the game begins.

[Ruskettle]

Give them seven digit numbers. Then they have to guess.

[michaelspence]

I came to the same conclusion with entirely different reasoning. They know the number is one away from each other and only have the clock to coordinate. They need to invent a rule that works in both cases (their number higher or their number lower). They can only communicate with each other by guessing or not guessing. If they are perfect logicians they each independently (and without coordinating beforehand) invent the following rule: Wait until the clock ticks a number of times equal to the number I have, and then guess one number higher. If the other person's number was one lower, they would have guessed already, so it must be one higher.

[nykout]

If Chris Hansen participated in that puzzle, he would have known that the other person "was told 18"

[MrLosarath]

This is the first puzzle of yours that I was actually able to solve! I felt so accomplished, then I realized that I solved it because of another video on your channel with a puzzle a lot like this.

[tharfagreinir]

Another way to describe the solution is that they have a perfect system for each of them to discover that they have the lower number. As the clock can be used to count, all that Alice has to do is to count all the way up to her own number and by then she knows that Bob has the higher number, because he's staying silent so far as his number hasn't come up yet. The same applies to Bob from his perspective. They're collectively waiting for the counter to go up to the lower number and whoever has that shouts out the higher number when the lower number comes up. It's really a very simple principle but of course it only works for relatively small numbers in practice.

[nomelehT]

This is practically the same as the tree puzzle you had, so pretty simple

[silverdragon]

I feel like the 50% thing is better than this time-consuming confusing thing

[gcewing]

Like some others here, I'm not convinced that the proposed solution is purely logical. An assumption is made that a strategy exists, and that it is the *only* strategy that can be arrived at by reason alone, therefore Alice and Bob will both find it if they reason perfectly.

The proposed strategy certainly works, but it's not the only one. Generally the problem is to encode a positive integer N using some positive integer R of rings. The solution R = N is arguably the "simplest" or "most obvious", but those are subjective judgements, not logical conclusions. There are other strategies -- for example, R = 2N would also work.

To argue that Alice and Bob are *guaranteed* to both arrive at the R = N strategy, you would somehow have to prove that there is *no other* line of reasoning that would lead to a different strategy. I'm not sure what such a proof would even look like. Given that infinitely many working strategies obviously exist, it seems similar to proving that a theorem is true but can't be proven. Where's Kurt Gödel when you need him...