Why Six Guesses Can Never Be Enough

Halving got you to seven guesses, every time, with no luck involved. That settles one question and leaves a bigger one sitting right next to it.

Could somebody cleverer do it in six?

You cannot answer that by being clever. Finding a better method would settle it one way; failing to find one settles nothing, because the method you have not thought of is exactly the one you cannot rule out. To say nobody can do it in six, you have to rule out every method at once — including all the ones nobody has invented yet.

That sounds impossible. It takes about ten minutes.

Stop looking at guesses. Look at answers.

Here are the rules we will count with, said out loud so there is no doubt about them. Every guess comes back exactly one of two ways: higher or lower. Nobody tells you when you land on it. At the end you say the number.

Now forget strategy completely. Whatever the guesser does, at the end of three guesses they are holding a string of three answers — something like lower, lower, higher. That string is everything they know.

So count the strings.

The first answer splits everything in two. The second splits each of those in two, making four. The third makes eight. Three guesses, eight possible strings — and that is true no matter which numbers were guessed, because the splitting happens to the answers, not to the guesses.

Now try to squeeze in a ninth number

Say the secret is somewhere from 1 to 8. Eight strings, eight numbers: it fits exactly, and that is what a good strategy does.

Add a ninth number and something has to give. There are still only eight strings and there are now nine numbers, so two different numbers must produce the same string of answers.

Sit with that, because it is the whole argument. Those two numbers send back identical answers to identical guesses. The guesser hears the same thing in both cases and ends up in the same place, holding one string and two numbers that both fit it. They can name one. It is a coin toss, not a method.

And that is true of the best guesser. Being clever changes which numbers get which string. It cannot make a ninth string appear.

The count that ends it

Each guess doubles the number of strings, so:

Six guesses make 64 strings. There are 100 numbers. Sixty-four is not enough boxes for a hundred things, so some string has to be shared, so some pair of numbers is not being told apart.

Six guesses cannot find every number from 1 to 100. Not by halving, not by being lucky, not by any method at all. Seven can, and you already have a way of doing it — so seven is not merely a method that works. It is the floor.

Sort these by hand

Six guesses handle 64 numbers. Each range below is either inside that or over it — and over it means no method exists, not that you have not found one.

Sixty-four and sixty-five land on opposite sides with one number between them. There is nothing gentle about the edge: 64 fits and 65 does not.

What a third answer would be worth

The level before this one said the two clues in the game are not worth the same amount. Counting makes that exact.

Suppose an answer could come back three ways instead of two. Then one guess makes 3 strings, two guesses make 9, then 27, 81, and 243. Five guesses would cover a hundred numbers comfortably — two fewer than you need now.

That is the real value of an answer: not how helpful it feels, but how many ways it can split what is left. An answer that only ever splits two ways is worth one doubling and never more, whoever is listening to it.

Which leaves something genuinely open, and worth chasing on paper. The hot-and-cold game does send back a second clue, and hotter-or-colder is a comparison against the point halfway between your last two guesses. So it is not free information. Whether a really careful player can turn it into a full three-way split every time, and finish in five, is a question this page has not answered — and you now own the method for working it out.