In a bar, there are two strangers called A and B. Each of them knows the day of the week they were born on, but not the day of the week the other person was born on.

They are told that they were born one day apart, that is, on consecutive days. Furthermore, A is told that B was not born on a Monday. A and B alternately answer the question “Can you deduce the day of the week the other person was born on?”. A said “No”, then B said “No”, then A said “No”, then B said “No”, then A said “No” and then finally B said “Yes”.

Their answers are all based on logic. On which day of the week was A born?

If you don’t know how to go about solving this puzzle or you’re just wondering whether your thought process (or guess) is correct, then you should continue reading.

The days of the week are seven. For each possible day A was born on, there are two possible days B was born on because A and B were born one day apart. Hence there are 14 possible combinations of consecutive days on which A and B were born. For instance, (Thu, Fri) is a possible combination, standing for A being born on a Thursday and B being born on a Friday.

The fact that B was not born on a Monday implies that the pairs (Sun, Mon) and (Tue, Mon) are impossible and thus the number of possible combinations reduces to 12. The answers of the two strangers exclude other possible combinations, leaving two of them out. In fact, A’s first answer excludes the combinations (Tue, Wed) and (Sun, Sat).

This can be reasoned out as follows. Suppose A was born on a Tuesday. Then A would have concluded that B was born on a Wednesday because A knew that B was not born on a Monday. However, if this were the case, A would have said yes. Since A said no, one can deduce that (Tue, Wed) is impossible. Similarly, one can argue that (Sun, Sat) is impossible. To summarise, there are 10 possible combinations left after A’s first no.

In light of B’s first no, one can rule out the combinations (Thu, Wed) and (Fri, Sat), so the number of possible combinations drops to eight. By excluding two combinations after every answer (see flow chart), one ends up with the combinations (Mon, Tue) and (Mon, Sun). That’s why B said yes eventually! Eureka – A was born on a Monday.

Alberto Cassar is a mathematics PhD graduate of the University of Warwick.

Sound Bites

•        In February 2023, two computer scientists, UCLA professor Raghu Meka and student Zander Kelley, stunned mathematicians with a breakthrough in an old combinatorics question: How many positive integers can you throw into a bucket while making sure that no three of them form an arithmetic progression (like 2, 5, 8 or 80, 130, 180)? They derived an upper bound on the number of positive integers smaller than some N that could be put in the bucket without creating such a pattern. The result is in the paper: Strong Bounds for 3-Progressions, 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS).

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DID YOU KNOW?

•        The word ‘logic’ derives from the Greek word logos which means reason, idea, word.

•        A person who is skilled in logic is called a logician. There have been various mathematicians who were logicians such as Bertrand Russell, Kurt Gödel and Ernest Zermelo.

•        An interesting problem to look into is the barber paradox: In a village, a barber shaves all those who do not shave themselves. Does the barber shave himself? Any answer leads to a contradiction.

•        The pattern of reasoning and the conclusion of the barber paradox is similar to Bertrand Russell’s paradox: Let R be the set of all sets which are not members of themselves. Is R a member of itself?  There’s no such barber and no such set R. This paradox exposed a huge problem in naïve set theory and changed the direction of 20th-century mathematics.

For more trivia, see: www.um.edu.mt/think.