I got this problem from Oriel Maxime, who was inspired by the Turing Machine game. I slightly simplified the wording.

I solved it and wrote up my solution.

Alice and Bob are considering codes consisting of three digits, each 1–5.

Alice has selected a code.

Alice emails Bob the following four pieces of data:

• A: whether the middle digit is less than, equal to, or greater than 4;
• B: whether the code consists of three identical digits, two identical digits with another different, or three different digits;
• C: whether the sum of the digits is even or odd; and
• D: how many 3s are in the code.

Bob replies: "Now I know the code. Good thing you sent me all four of those; I couldn't have been certain of the code if you hadn't sent them all."

What is the code?

The code is 111.

Fact B says the code is three identical digits. Fact A says the middle digit is less than 4, reducing the possibilities to 111, 222, and 333. Fact C says the sum is odd, limiting the possibilities to 111 and 333. Finally, fact D says there are no 3s in the code, guaranteeing that the code is 111.

If fact A hadn't been given, 555 would have been another possibility.

If fact B hadn't been given, there would have been lots of other possibilities, including 115.

If fact C hadn't been given, 222 would have been another possibility.

If fact D hadn't been given, 333 would have been another possibility.