[OlymUSec_20150412BDP] Guessing Cheryl’s Birthday

Question

Introduction

     This “Primary 5 mathematics” (actually an upper secondary Olympiad) logic puzzle has gone viral.  It has been making its rounds in various forums in Singapore and overseas, stumping adults and children alike.  It is actually a parody of an old puzzle.  Can it even be solved?  It seems that there is no information given by each parties that we can exploit.  Actually there is!  In a subtle way ...

Solution

     In the beginning, everybody knows that Albert knows only the month and Bernard knows only the numerical day of the month.

     When Albert tells us “I don’t know when Cheryl’s birthday is, but I know that Bernard does not know too.” he is leaking out information (from his knowledge of the month) that the day of the month appears more than once and cannot be (June 18 or May 19).  Actually, the original phrasing is more like “If I don’t know when Cheryl’s birthday is, then Bernard does not know too.”.  The person who set this question merely changed the names of the people and the dates, without appreciating the subtle but crucial difference between a statement of fact and an implication (an “if ... then ... ” statement). 

     Ruling out June 18 and May 19, we also know that Albert knows that the birthday month is neither June nor May.  Otherwise, how would he have been so confident in saying that he knows Bernard would not know Cheryl’s exact birthday?  So we can eliminate those months.

     Bernard acknowledges the above state of affairs and the embedded hint.  With the choice narrowed down and with his knowledge of the numerical date, he now knows Cheryl’s birthday.  Since we know that Bernard knows Cheryl’s birthday, we know that it cannot be a numerical date that appears more than once (otherwise he would not have been able to know).  So we can cross out July 14 and August 14.

     Now Albert would telepathically thank Bernard for this helpful hint.  Because now he is able to deduce Cheryl’s birthday with his knowledge of the month.  That would mean that this cannot be a month with two candidate dates.  We blot out the August dates and see for ourselves the only remaining possibility.

Conclusion: Cheryl’s birthday is  July 16.

Remarks

     This puzzle was solved using the process of elimination and analysing our knowledge of what each party knows and can know.  Thus we successively narrow down the possibilities until the answer becomes obvious.  Here we learn that

     knowledge of other people’s knowledge can itself give us knowledge. 

This principle is actually employed in cryptology (the use of secret codes) which finds applications in fields like banking, the military (cf. interesting story of how the German Enigma code was broken in WorldWar II) and communications.  As an example, radio communication can tell the enemy of troop positions and warn of an impending attack, and that is why radio silence is imployed as a precaution.  Sensitive information in certain organisations is restricted on a “need to know” basis.

Suitable Levels
* Upper Secondary Olympiad
* other syllabuses that involve knowledge or epistemology
* application of mathematical principles in real life
* for all people interested in logic puzzles

[OlymPri20150412LGC] Your #logic a House of Cards?

Question

Introduction

     This is an olympiad type of question that tests one’s use of logic.  Sadly, logic is seldom taught in schools, except maybe in courses like knowledge inquiry, theory of knowledge, or in selected topics like geometry.  Mathematics is actually very much connected with logic, but it seems to be regarded as difficult to teach.  Being able to use logic and think critically is very important in our lives in various aspects.

     Let us study how we can analyse conditional statements using logic.

Analysing Conditional Statement with Logic

     A conditional statement is a statement of the form “If  H  then  C.”  An example would be: “If the blue litmus paper turns red, then the liquid is an acid”.  The front clause H (the part about the litmus paper turning red) is called the hypothesis (or premise).  The latter clause  C  (the part about the acid) is called the conclusion.  Every conditional statement has three other related forms: the contrapositive, the converse, and the inverse.

     For example, let say there is a NC16-rated movie and you can watch it only if you are at least 16 years old.  So  H := “watch NC16”  and  C := “16 years old and above”.  The conditional statement means “If you want to watch the movie, then you must be 16 years old and above”.  The contrapositive is “If you below 16 years old, then you cannot watch the movie”.  The contrapositive always has exactly the same meaning as the original conditional statement.  If one is true, so is the other.  If one is false, so is the other.  The contrapositive is merely phrased in a different way.

     The converse says “If you are 16 and above, then you will watch the movie.”  The inverse would say “If you don’t want to watch the movie, then you are not 16 and above.”  Note that the inverse is the contrapositive of the converse.  The inverse always has exactly the same meaning as the converse.  If one is true, so is the other.  If one is false, so is the other.

     The converse (and the inverse) has a different meaning from the original statement.  If the original statment is true, the converse may or may not be true.  In our example, “If you want to watch the movie, then you must be 16 years old and above” is true, but the converse “If you are 16 years old and above, then you want to watch the movie.”  may not be true, as some people who are 16 years old and above may want to do something else.  Like reading a book.  Or playing golf.  So you see, there are really two camps: the original conditional statement and its contrapositive in one camp, and the converse and the inverse in the other.

     Let us analyse the given problem using logic.

Analysis

     The statement  ‘any card with a letter A on one side always has the number “1” on the other side’ can be rephrased as ‘If letter A appears, then the other side is “1”’.  Let us analyse each position in order.  For (a), since the letter A appears, we definitely need to know what is behind the card to test whether the claim is true.  For (b), note that the claim does not say ‘If the letter is not A, then the other side is not “1”’ (inverse)  nor  ‘If the number is “1”, then the other side must be an A.’ (converse).  It is possible that A is on the other side and we are fine.  But it is also possible to have a non-A with the number  “1”.  The given condition does not prohibit that.  So, if the other side is not an A, it is still OK.  So the other side can be anything, and it does not matter.  For (c) we definitely have to flip to check the other side in case the other side is an A, then the claim would be proven false.  For (d), since the card is not an A, it is irrelevant.  The claim is talking about  A.  It is not talking about B.  We can summarise the analysis in the diagram below:-

Ans:  We need to flip (a)  and  (c)

Suitable Levels
* Primary School Olympiad

* syllabuses that involve logic and epistemology