Guessing With Higher or Lower

1the hidden number

Imagine a friend picks a whole number from 1 to 100 and will only tell you 'higher', 'lower', or 'correct'. You know the game, but notice what the setup really gives you: a range of possibilities, and a source of hints that never lies. Your job is to shrink that range until only one number is left.

The game is really about shrinking a range of possibilities.

2one cut

Suppose you guess 10 and hear 'higher'. That answer is worth more than it seems. The number is not 10, but it is also not 9, not 8, not anything below 10. One reply ruled out ten numbers at once, and the hidden number must lie somewhere from 11 to 100.

One answer eliminates every number on the wrong side, not just your guess.

3two strategies

Now compare two ways of playing. Guessing 1, then 2, then 3 eliminates just one number per question, because 'higher' only rules out your guess. Guessing 10 first eliminated ten numbers. The lesson is that the position of your guess decides how much each answer is worth.

Where you guess decides how much a single answer can eliminate.

4worst case

There is a catch. If you guess 90 and the answer is 'lower', you keep 89 numbers, but if the answer is 'higher', you keep only 10. The unlucky reply leaves you with a lot of work, and you cannot control which reply you get. A guess is only safe if both possible answers leave you with a modest range.

A guess is judged by its worst possible answer, not its best.

5cut the middle

So which guess protects you best? One that splits the range into two pieces of nearly equal size, so that 'higher' and 'lower' each leave about half. Guess 50 and you keep either 1 to 49 or 51 to 100. Whatever your friend says, half the possibilities vanish. The next page asks how many such cuts it takes to reach a million.

Guess the middle: whichever answer comes, half the range disappears.