Questions
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Best
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Find the hidden number by asking "is it this or lower?" — but the answerer is allowed to lie once per round, and never picks a number until you corner them.
A number is hidden somewhere in the shown range. Tap any number to ask one question: is the hidden number that value or lower? The answer comes back as Yes or No, and the board updates itself.
The answerer is allowed to lie, but only once in the whole round. So a single No does not rule a number out — it only spends that number's one lie. A number that contradicts two answers is gone for good.
Each number on the board shows its own state: full brightness means it has agreed with every answer so far, a dimmed number means it has already used up its lie, and a struck-out number is impossible. You never have to keep count yourself.
The round finishes the instant exactly one number is still possible — there is no guessing step. Three rounds run back to back over 1–10, 1–20 and 1–40, and your score is the total number of questions, so fewer is better.
The most important rule of this game is the one that is easiest to miss: nothing is hidden in advance. The answerer keeps every number that is still consistent with what they have said, and picks the answer that leaves them the most room. That means hunches are worthless and there is no luck at all — the only thing that matters is what your question leaves behind in the worse of the two cases.
The practical way to think about it is to score the board. Give a bright number two points, because it can still absorb a lie, and a dimmed number one point, because it cannot. Add it all up and you have the size of your remaining problem. Every question splits that total into a Yes half and a No half, and you will always be handed the bigger half, so you want the two halves to come out as equal as you can make them.
At the start of a round, when nothing has been struck yet, that rule is exactly ordinary bisection: cut the range in half. It stops being ordinary the moment some numbers are dimmed. A dimmed number is worth half of a bright one, so the balance point drifts away from the middle of the count and towards the middle of the weight. Counting numbers instead of weight is the single most common way to waste a question here.
Two endgame shapes are worth recognising on sight. If every surviving number is dimmed, no one can lie any more and the rest is a plain binary search. If exactly one bright number is left alongside some dimmed ones, one well-chosen question usually finishes the round, because the bright number is the only thing that can survive a contradiction. Perfect play is six questions for 1–10, seven for 1–20 and eight for 1–40, so twenty-one is the number to beat.