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?
(In reply to
question by broll)
After thinking it over I'm assuming the notation is "the shortest unique string that ends in an X." Thus 6 is represented by XX because it is divisible by 2 and 3 and no other numbers before it can say the same.
(By contrast, 8 would be XOXOOOX, since XOX has already been used by 4.)
My solution follows, stop reading here if you don't want to read it yet:
My guess at the answer is that Jonah was thinking of the number 80, which would be represented by the 9-character string XOXXOOXOX. Alice, looking for the smallest integer divisible by 2, 4, 5, 8 and 10 but not by 3, 6, 7 or 9, incorrectly guesses that the answer is 40. While it is true that 40 meets that condition, the integer 40 in this notation would more simply be represented by the string XOXXOOX. The integer 80 needs the additional two characters at the end, which otherwise seem redundant, to differentiate it from 40.
EDIT:
Actually I think I missed one. 50 would be represented by the 9-character string XOOXOOOOX. So perhaps Jonah was thinking of 50 and Alice incorrectly guessed 10 (which would more simply be represented by XOOX).
Edited on February 16, 2024, 6:49 pm
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Posted by tomarken
on 2024-02-16 18:40:08 |
re: question
