## Math puzzle Cheryl's birthday problem: Informal reasoning based and formal logic based solutions

Cheryl's Birthday Problem: Cheryl gives Albert and Bernard 10 possible dates for her birthday and also a few clues. And you have to find Cheryl's birthday.

### The math puzzle Cheryl's birthday problem

When her two new friends Albert and Bernard wanted to know her birthday, Cheryl tells them a list of 10 possible dates for her birthday but not the actual birthday.

Cheryl then tells Albert the month and Bernard the date of her birthday separately.

After knowing these clues, Albert and Bernard make three statements in sequence,

**Albert:** I don't know when Cheryl's birthday is, but I know that Bernard does not know too.

**Bernard:** At first I don't know when Cheryl's birthday is, but I know now.

**Albert:** Then I also know when Cheryl's birthday is.

**Assume** Albert and Bernard make these statements to each other. Cheryl's presence is not important any more!

Find Cheryl's birthday.

**Time to solve:** 20 minutes.

Even if you overshoot the time limit, carry on. It's not a hard puzzle and it'll be fun. Assured.

In four sections we have presented the background of the puzzle first, a brief on elementary logic analysis and then the two approaches to the solution,

**Background of Cheryl's birthday problem.****Elementary Logic and Dependency logic based on Relative knowledge.****Informal Reasoning based solution to math puzzle Cheryl's birthday problem.****Formal logic analysis based Solution to the Math Puzzle Cheryl's birthday problem.**

*You may move directly to any of the above sections by clicking its link and return by clicking on browser back button.*

### Background of Cheryl's birthday problem

From **Wikipedia page,**

"**Cheryl's Birthday** is the unofficial name given to a mathematics brain teaser that was asked in the Singapore and Asian Schools Math Olympiad, and was posted online on 10 April 2015 by Singapore TV presenter Kenneth Kong. It went viral in a matter of days. The quiz asked readers to determine the birthday of a girl named Cheryl using a handful of clues given to her friends Albert and Bernard."

You may skip the bit of theory on common logic and dependency logic next and skip straight to the solutions. To skip click **here.**

### Elementary Logic and Dependency logic based on Relative knowledge

#### Barebones elementary logic example

When I apply logic, I apply it on the basis of **my knowledge on relevant facts**, not on your knowledge or for that matter Rini's knowledge. That is the usual logic analysis that we do, by ourselves.

For **example**, when I know that "Surya grew up in Kolkata", I can say with confidence, "Surya is from West Bengal". That is **elementary logic statement** based on my knowledge of the fact that "Surya grew up in Kolkata" and also "Kolkata is in West Bengal". The first part is personal knowledge, and the second part public knowledge and both are considered to be TRUE, at least by me. The important thing here is, I have made a CONCLUSION from two pieces of FACTS. Any of the facts might have been FALSE turning my CONCLUSION also FALSE.

#### Dependency logic based on relative knowledge

In **logic puzzles, all sorts of complications are introduced**, naturally. Otherwise there would be no fun.

A special category of complexity is in what we call **DEPENDENCY LOGIC**.

When I made the statement on Surya, I analyzed the two pieces of processed final facts or statements.

But **in Cheryl's puzzle**,

You have

TWO logical persons Albert and Bernardmaking "THEIR statements" based onTHEIR knowledgeand you have to put yourself into their shoes, so to say, and think logically to decipher theIMPLICATIONof their statements.

**Your logic analysis is dependent on others' logic analysis results**. This is **DEPENDENCY LOGIC ANALYSIS** and so is obviously discomforting and unnatural to us. That's where you have your challenge.

Two approaches to the solution of the math puzzle Cheryl's birthday problem are presented,

**Informal Reasoning based solution to math puzzle Cheryl's birthday problem.****Formal logic analysis based Solution to the Math Puzzle Cheryl's birthday problem.**

*You may move directly to any of the above sections by clicking its link and return by clicking on browser back button.*

### Informal Reasoning based solution to math puzzle Cheryl's birthday problem

Among the ten dates given by Cheryl, **four months appear**, **each more than once.**

So Knowing just the month of Cheryl's birthday, Albert couldn't have known the birthday.

With the knowledge that Bernard was told only the date, Albert now analyzes THE TEN DATES GIVEN BY CHERYL.

**May has three dates:** 15 and 16 both appear more than once in the ten dates, but 19 appears ONCE only. In June out of two dates, 17 appears also in August, but 18 appears only in May. If by chance, Bernard were told the date as 18 or 19, he surely would have known the birthday immediately.

On the other hand, in July and August, the dates 14, 15, 16 and 17 each appears more than once.

So when Albert asserted confidently that **Bernard doesn't also know the birthday, it can only mean (TO YOU and also to Albert obviously),**

Albert was told the month as July OR Augustand not either of May or June.

At this point, YOU KNOW THAT possibilities are five dates in July and August.

But you are not Bernard. **He knows the date.**

When Bernard hears Albert confidently telling, "Bernard doesn't also know the birthday", he analyzes the situation based on the knowledge that,

- Albert was told the month,
- Albert is sure about Bernard not knowing the birthday, and,
- The list of ten dates.

Like you, **first he reasons that the month must be July or August.**

It is your turn now to reason.

The dates in July and August are 14th July, 16th July, 14th August, 15th August, and 17th August. This is your short-list at this point of time.

Now after analysis, Bernard declares confidently that, "Now I also know the birthday".

With this confirmation of Bernard YOU REASON THAT,

As 14th appears both in July and August, it cannot be the date for Bernard to be certain of the date (as he doesn't know the month). THE DATE CAN ONLY BE ANY OF THE THREE 16th July, 15th August or 17th August.

Your short-list is reduced to three dates.

It is now Albert's turn to reason and make his last statement. But again, you are not Albert, he knows the month, you don't know it.

After analysis now Albert declares, "Now I also know the birthday."

If the month were August, he couldn't have been confident about knowing the birthday, **as August 15th and August 17th are TWO short-listed possibilities at this stage.**

He is sure

only because, thedate is 16th of July,the single date in a month among the short-listed three possibilities.

The last reasoning belongs of course to you. You also are sure about the date now.

### Formal logic analysis based Solution to the Math Puzzle Cheryl's birthday problem

For solving any type of difficult logic puzzle, use of logic tables invariably provides great help. We will use here two tables, a **Fact table** and a **Logic status table**. The fact table will give you a clear idea of the nature of the information Cheryl had given as well as it will show you the** embedded pattern in months and dates** in the information given.

The three statements will form the remaining component of the puzzle.

When we analyze and form our conclusions at each stage, the logic status will change in the logic table. The fact table and the statements are the invariants that do not change.

#### The Fact table: Date-month table

We have put all the 6 unique dates 14, 15, 16, 17, 18 and 19 as columns and four unique months, May, June, July and August as rows. The dates given by Cheryl to her two friends are marked possible dates as "y" against the suitable cells. For example, for 15th May, a "y" is marked against 15 column and May row.

The **most important pattern of two unique dates 18th June and 19th May cannot be ignored even at this early stage** and so is colored red. This obviously will be the key to the solution.

#### The Statements: Logic statements

The key fact and the three statements made by Albert and Bernard one after the other make up the second component of the puzzle.

You have to analyze the three components—**fact table**, **statements** and **logic status table** together to arrive at **final implication at each stage**, all the time moving towards the solution.

#### The Logic table

In logic table we will record the status of logic analysis after processing implications of each statement. We will simply use **the date-month table itself as the logic table. **

**When a date is proved to be invalid by one of the statements, the corresponding entry in the table will be marked as x instead of y.** **For invalid month, it will be crossed out.** The date and month left after all the three statements are fully analyzed will be Cheryl's birthday.

#### Result after analyzing Albert's first statement: Stage 1.

At the start, Albert knows the month (you or Bernard do not know it). As he does not know the birthday, and as no month has a unique single date, month can be any of the four (to you or Bernard).

**After Albert made his first statement,** situation became more clear. *As Albert knows now that Bernard does not know the birthday*,

The

month cannot be May or June.

* These two months have dates 18 and 19 not appearing in any other month.* If Bernard knew any of these two dates he would also have known the birthday immediately. Albert's knowledge of the month being either July or August, he knows for sure Bernard does not know the birthday.

*In these two months there is no unique date.*

This is perfect reasoning that Albert and you also carry out. Bernard's statement just after this will let you and Albert know what Bernard really knows.

The status of logic table at this stage is shown below.

#### Result after analyzing Bernard's first statement: Stage 2.

By the first part of his statement Bernard confirms second part of Albert's previous statement. But, **significant is Bernard's confirmation that now he knows the birthday.**

It means,

The

date can only be one of15, 16 or 17 andmonth July or August.

The date cannot be 14th July, **as it appears in both July and August**, and *as Bernard doesn't know the month*, he won't have known the birthday if the date given to him by Cheryl were 14.

At this point Bernard knows the birthday, you don't know it and also don't know what Albert will know after this statement. Fun, isn't it?

The status of logic table is as below.

#### Result after analyzing Albert's second statement: Final stage 3.

The implication to you that the date must be one of 15, 16 or 17 also was clear to Albert. Only difference was in Albert knowing the month precisely that you didn't know for sure.

If month were August, there would have been two possible dates 15 and 17. Knowing the date, Bernard would have known the birthday also, but not Albert or least of all you.

So *when Albert confirmed that he also knows the birthday*, **it must be 16 and July**. For certain.

To Bernard first, then Albert and finally to you the knowledge became clear.

The final logic table is shown below.

That's long. isn't it. But it has its own values.

### End note

First the math puzzle Cheryl's birthday problem is solved by informal commonsense reasoning that makes sense to common folks like us.

In contrast, the second **methodical step by step analysis **is a more general approach and suited to solve logic puzzles in general. This approach lays bare all the mysteries of the puzzle and apparently removes the fun in searching for the solution blindfolded in the dark.

Which solution do you like? Or better still, do you have a different solution approach altogether?

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