# Can you solve it? The birthday birthday problem

Join in the celebrations

UPDATE: Read the solution here

Hi guzzlers,

My Monday Puzzle column celebrates its third birthday this week. What better way to mark the occasion than a puzzle about a birthday!

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?

Today’s puzzle comes from the USA Mathematical Talent Search, a free maths competition for US schoolchildren, organised by the Art of Problem Solving, which develops amazing educational resources for motivated maths students.

Some readers may also recognise the puzzle as a homage to the Cheryl birthday problem, which I wrote about in April 2015 and whose viral success was the reason I started the Monday Puzzle in the first place. So it’s a circular birthday cake...

I’ll be back at 5pm with the answers, and please NO SPOILERS.

UPDATE: The solution can be read here.

Maybe suggest new versions of this type of puzzle instead.

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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