# Did you solve it? The birthday birthday problem

The solution to today’s puzzle

In my puzzle blog earlier today I set you the following problem:

Ariel, Balthazar and Chastity are great mates, genius logicians and they always tell the truth. Neither Ariel nor Balthazar know the day or the month of Chastity’s birthday, so she decides to tell them in the following way:

First, she says out loud, so both Ariel and Balthazar can hear her: “The day (of the month) of my birthday is at most the number of the month of my birthday.”

Then she whispers the day to Ariel and the month to Balthazar

Ariel says “Balthazar cannot know Chastity’s birthday.”

Balthazar thinks a bit, then says: “Ariel also cannot know Chastity’s birthday.”

They carry on like this, each saying these exact same sentences in turn until Balthazar announces: “Both of us can now know what Chastity’s birthday is. Interestingly, that was the longest conversation like that we could have possibly had before both figuring it out.”

When is Chastity’s birthday?

Solution

There are 12 months in the year. So, the maximum number for the month of Chastity’s birthday is 12. Since the day of the birthday is at most the number of the month of the birthday, the maximum number for the day must also be 12. In other words, both Ariel and Balthazar are whispered a number between 1 and 12.

We are told that the day of the birthday is at most the number of the month. This is the same as saying that the day can be any number less than or equal to the month.

Ariel knows the day of the birthday, Balthazar the month. The only way for Balthazar to also know the day would be if the month is January, since then he could deduce that her birthday was January 1. But for Ariel to be able to say that Balthazar cannot not know the birthday, he must know that Balthazar does not have January. How does he know that? Well, he must have a number bigger than 1. Since if he had a 1, he would be able to deduce nothing about what Balthazar has. But if he had anything bigger than 1, Balthazar cannot have January since the day is always less than the month. So, we can eliminate 1 for Ariel, and 1 (January) for Balthazar.

Now to the next statement. The only way for Ariel to know the month of the birthday would be if he had been told 12, since he would be able to deduce that the month would also be 12, or December. But following the above reasoning, for Balthazar to know that Ariel does not have 12, he must have a month with a number 11 or under. So we can eliminate both 12 and 12 (December).

If this exchange were to happen again, we can eliminate February and 2, and November and 11.

For this to continue for as long as possible it will continue until Ariel eliminates 6 and 6. Once that happens the only solution left is July 7, (7/7)

I hope you enjoyed the puzzle. Thanks again to the Art of Problem Solving for suggesting it. I’ll be back in two weeks.

I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.

I’m the author of several books of popular maths, including the puzzle books Can You Solve My Problems? and Puzzle Ninja. I also co-write the children’s book series Football School.

Alex Bellos

The Guardian

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