Pico Fermi Bagel
A guessing game where the answers tell you how close you are, but never where.
I am thinking of a number made of three different digits. You guess a number, and I reply with one word for each digit you got something right about:
- Fermi
- A digit that is right, in the right place.
- Pico
- A digit that is in the number, but somewhere else.
- Bagels
- Nothing. None of those digits are in it at all.
What makes it hard is that I never say which digit earned which word. Two Fermi and a Pico tells you a great deal about how close you are and nothing whatsoever about where to look — you have to get that by comparing one guess against the next.
The digits are always different from each other, and a leading zero is allowed, so 047 is a perfectly good three-digit answer.
I have picked 3 different digits. Leading zeros are allowed.
How to think about it
There are 720 three-digit numbers with no repeated digit. A guess that could only ever come back one way teaches you nothing; a guess that splits what is left into several roughly equal groups teaches you the most, whatever the answer turns out to be. It is the same instinct as guessing the middle in Higher or Lower, applied to a space you cannot line up in order.
This is why opening with three disjoint guesses — 012, 345, 678 — is a reasonable start. Between them they touch nine of the ten digits, so the replies tell you roughly which digits are in play before you spend any effort on arrangement. Separating which digits from in what order is most of the game.
At three digits the counter above shows exactly how many numbers are still consistent with everything you have been told. Watching it fall is a better guide to whether a guess was any good than whether it happened to be right.