Two young math students are trying to develop a new notation for numbers for a class project. They want it to be based on completely different principles from decimal notation. Joe decides to try using a "word" of X's and O's, where the first letter tells you if the number is divisible by 2, the second letter indicates if it's divisible by 3, then 4, and so forth. An X means yes, and an O means no.

Here is how the first few numbers would be represented in Jonah's system: 1=O, 2=X, 3=OX, 4=XOX, 5=OOOX, 6=XX.

Jonah tells Alice that he has chosen a number between 1 and N, inclusive, and that he has represented it by a 9-letter word (indicating divisibility by 2 through 10). Alice, sitting across the table, looks at the word and says that the number must be A. Jonah is puzzled because that is not the number he chose. Alice, seeing his confusion, quickly realizes what the problem is, and gives him the expected answer.

What were Jonah's and Alice's numbers?

I'm probably missing something, but shouldn't 6 be something like XXOOX (divisible by 2,3,and itself, but not by 4 or 5)? But that would appear to break the problem, as all entries would have unique length.

Posted by broll on 2024-02-15 06:13:39