## Logic 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. You have to find Cheryl's birthday.

#### The logic puzzle

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.

Knowing the clues, Albert and Bernard make three statements,

**Albert:** I don’t know when Cheryl’s birthday is, but I know 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.

#### Preamble to Cheryl’s birthday problem

As per Wikipedia:

Cheryl’s Birthday is the unofficial name given to a brain teaser asked in the Singapore and Asian Schools Math Olympiad. It 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 find the birthday of a girl named Cheryl using a handful of clues she had given to her friends Albert and Bernard.

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

#### Barebones elementary logic example

When you apply logic, you apply it based on your knowledge on relevant facts, not on my knowledge or, for that matter Rini’s knowledge. That is the usual logic analysis we do, by ourselves.

For example, when you know that ‘*Surya grew up in Kolkata*’, you say with confidence, “Surya is from West Bengal.”

This is elementary logic statement based on your knowledge that *‘Surya grew up in Kolkata*’ and also on the fact that *‘Kolkata is in West Bengal’.*

The first part is personal knowledge, and the second part public knowledge. Both are TRUE, at least to you. The important thing is: you have made a CONCLUSION from two pieces of FACTS. Any of the two facts might have been FALSE turning your CONCLUSION also FALSE.

#### Dependency logic: based on relative knowledge

In logic puzzles, many complications are introduced intentionally to make the puzzles interesting.

A special category of complexity is in DEPENDENCY LOGIC.

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

But in Cheryl’s puzzle,

You have TWO logical persons Albert and Bernard making THEIR statements based on THEIR knowledge. Put yourself in their shoes and think logically to decipher the IMPLICATION of THEIR statements.

* Your logic analysis depends on results of others’ logic analysis.* This is DEPENDENCY LOGIC ANALYSIS and so is discomforting and unnatural to us. The challenge is to analyze dependent logic statements.

Two approaches to the solution of the logic puzzle Cheryl’s birthday problem are presented,

- Informal Reasoning based solution.
- Formal logic analysis based Solution.

### Informal Reasoning based solution to logic puzzle Cheryl’s birthday problem

The ten dates given by Cheryl have *four unique months*.* Each month appears more than once on the list of ten dates.*

So knowing solely Cheryl’s birth-month, Albert couldn’t have known the birthday.

But, he persisted in his analysis. He continued to analyze Bernard’s awareness of the birthday. Knowing that Bernard was told only the date, Albert analyzes THE TEN DATES GIVEN BY CHERYL.

Month of May has three dates: 15, 16 and 19.

15 and 16 both appear more than once, but 19 appears ONCE only.

In June, out of two dates, 17 appears also in August, but 18 appears ONLY in June.

In July and August, each of the dates 14, 15, 16 and 17 appears more than once among the ten dates.

**Albert reasons:** If Bernard were told the date was 18 or 19, he would have known the birthday at once, as both dates appear only once in the ten dates.

So when he asserts, Bernard doesn’t also know the birthday, it can only mean (TO YOU and also to Albert),

Albert was told the month was July OR August. Not May or June.

* Probables are the five dates in July and August. *But you are not Bernard.

*He knows the date (but not the month). You don’t know it (date or month).*

When Bernard hears Albert confidently telling, “Bernard doesn’t also know the birthday”, he analyzes knowing,

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

Same as you, first he reasons **the month must be July or August.**

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

Bernard announces after completing his analysis, “Now I also know the birthday.”

Based on 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 (without knowing the month). THE DATE CAN ONLY BE ANY OF THE THREE 16th July, 15th August or 17th August, each unique.

Bernard was told the date. So he could be certain of the birthday choosing one out of the three.

The short-list for you is reduced to three dates, not a unique one.

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, Albert confirms, “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 probables.

He is sure only because the date is 16th of July.This is the single probable date in a month among the three short-listed probable dates.

You also are sure now that the **birthday told was 16th of July.**

### Formal logic analysis based Solution to the Logic Puzzle Cheryl’s birthday problem

To solve a difficult logic puzzle, use of logic tables helps. 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 information Cheryl had given. The three statements will form the remaining part of the puzzle.

When we analyze to make our conclusions, logic table will change, but the fact table and the statements will not change.

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

The 6 unique dates 14, 15, 16, 17, 18, 19 are columns and four unique months May, June, July and August are 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. This obviously will be the key to the solution.

#### The Statements: Logic statements

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

Analyze the three components—*fact table, statements and logic status table together to arrive at final implication at each stage.*

#### The Logic table

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

When a date is proved 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 analyzed will give Cheryl’s birthday.

#### Result after analyzing Albert’s first statement: Stage 1

First, Albert knows the month (but not you or Bernard). As he declares, “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. As Albert knew the month to be July or August with no unique date, he could confidently make the declaration.

This is perfect reasoning that Albert, Bernard and you carry out. Bernard’s next statement will let you and Albert know what Bernard knows.

The status of logic table at this stage is shown.

#### Result after analyzing Bernard’s first statement: Stage 2

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

It means,

The

date can only be one of 15, 16 or 17 and month July or August.All the three are unique dates in these two months.

**The date cannot be 14th of July, as it is 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 14th.

Now Bernard knows the birthday, but you don’t know it. You also don’t know what Albert will know after this statement. Fun, isn’t it?

The status of logic table is shown.

#### 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 is, Albert knows the month, but not you.

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

So when Albert confirmed that he also knows the birthday,** it must be 16th of July**,* the single date in a month out of three probable dates.*

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

The final logic table is shown.

Which solution do you like? Or do you have a different solution?

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