Earlier today I set you the following two problems.
1. Puzzee Rascal always takes two sugars with his tea and cannot lie, while his brother takes none and cannot tell the truth. One day you meet a man who is either Puzzee or his brother. You are allowed to ask one yes-no question to establish his identity. What do you ask?
2. Puzzee Rascal always takes two sugars with his tea and cannot lie, while his brother takes none and cannot tell the truth.
When asked a question the brothers will only ever answer “Bonkers!” or “Dance wiv’ me!” These two phrases mean “yes” and “no”, although you don’t know which one is “yes” and which one is “no”.
One day you meet a man who is either Puzzee or his brother. You are allowed to ask one question to establish his identity. What do you ask?
Answers:
1. The simplest answer is that you can ask him anything to which you know the answer. For example, “Is the sky blue?” If the answer is yes, then you know it is Puzzee, who is the one who tells the truth. If the answer is no, it’s his brother.
But this doesn’t really help us build up the skills that we will need to solve the second question.
The way to determine who is who without recourse to a question that you know the answer to is to ask a question embedded in a question, such as:
“If I asked you whether you take two sugars would you say yes?”
Puzzee, who takes two sugars and tells the truth, will answer yes. If you asked his brother, who does not take two sugars and lies, he will say no.
It takes a moment to follow the logic of why he says no. If you asked the brother simply if he takes two sugars he will say ‘yes’, since he doesn’t take two sugars and always lies. But if you asked him if he would say yes when asked this question he must lie again and say no.
2. This is much more difficult. The challenge is to find a question that determines which brother is which, irrespective of the meanings of “Bonkers” and “Dance Wiv Me!”
This embedded question is a solution:
“If I asked you whether you take two sugars would you say ‘Bonkers!’?”
Puzzee will always reply “Bonkers!” and his brother will always reply “Dance Wiv’ Me!
Here’s why. Let’s call Puzzee T for truth-teller and his lying brother F for falsity-teller, lets abbreviate “Bonkers!” to BNK and “Dance Wiv Me!” to DWM. And let’s call the embedded question Q.
Option 1: BNK means yes/DWM means no
(This part is the same as the answer to the first problem above).
If you asked T ‘do you take two sugars?’, he would truthfully respond BNK. So he will answer BNK to the question.
If you asked F ‘do you take two sugars?’, he would lie and say BNK (yes), so he would respond to Q with a lie and say DWM
Option 2: BNK means no/DWM means yes
If you asked T ‘do you take two sugars?’, he would truthfully respond DWM. So he will answer Q with BNK.
If you asked F ‘do you take two sugars?’, he would lie and say he DWM. So he would respond to Q with a lie and say DWM.
Got it? “Bonkers?” “Dance Wiv’ Me?” Great, so we can move on.
Here’s The Hardest Logic Puzzle Ever.
Puzzee Rascal always speaks the truth and his brother Buzzee Rascal always lies. There is another brother Fuzzee. Whether he tells the truth or lies is completely random, as if he is flipping a fair coin in his brain, and if the coin says heads he tells the truth and if it says tails he lies.
When asked a question the Rascals will only ever answer “Bonkers!” or “Dance wiv’ me!” These two phrases mean “yes” and “no”, although you don’t know which one is “yes” and which one is “no”.
You have three questions, each of which must be asked to a single Rascal, although you can ask the same Rascal more than once. What questions do you ask to establish who is who?
The puzzle was nicknamed the hardest logic puzle ever by the logician George Boolos - who credited Raymond Smullyan as its originator.
I asked Chris Ovenden, who teaches Philosophy (and Logic!) at the University of Manchester, to explain this one:
Call the brothers A, B and C, in the order you find them, and lets say their names are just True, False and Random for the purpose of this explanation (to keep track of who is who regarding truth telling).
Ask A: (1) ‘If I asked you ‘is B Random?’ would you say ‘Bonkers’?’
If A says ‘Bonkers’, then either B is Random or A is Random, in which case C is not Random.
- now, ask C (who you know is not Random): ‘if I asked you ‘are you True?’ would you say ‘Bonkers’?’
- If C says Bonkers then they are True
- If C says Dance Wiv’ Me then they are False
- Now ask C: ‘if I asked you ‘is B Random’ would you say ‘Bonkers’?’
- If C says Bonkers then B is Random and A is the remaining brother (True or False depending on what C turned out to be)
- If C says Dance wiv’ me then A is random and B is the remaining brother (True or False depending on what C turned out to be).
If A says ‘Dance wiv me’ to question (1), then either C is Random or A is Random, but B is not Random
- So, ask analogous questions as set out above: determine whether B is true or false, then ask whether C is random to determine the final two identities.
Thanks Chris! Now I’m off for a cup of tea.
I post a puzzle here every second Monday. My most recent book is the mathematical adult colouring book Snowflake Seashell Star. (In the US its title is Patterns of the Universe.)
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