Finding a princess

The prince of a small kingdom travels to a neighboring land to meet with their king. After the meeting, he agrees to take one of the king's daughters' hand in marriage. The prince is told:

Princess Anne always tells the truth.
Princess Beatrice always lies.
Princess Catherine speaks randomly.

The prince gets to meet the 3 princesses at the same time, but unfortunately they are identical triplets and he cannot tell them apart. He may ask one of them a yes or no question. If he asks a question that she cannot answer, then he will be executed for rudeness. After that, he must choose a bride.

The prince is happy to marry Anne or Beatrice, but does not want to marry Catherine since he will never know what she is really thinking. How should he proceed?

#Logic

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

For any X, Y, and Z, ask X, "if I asked you if Y is the random one, would you say yes?" If X is either the truth-teller or the liar she will answer yes iff Y is the random one. If X says yes, then, either X or Y is the random one, so choose Z. If X says no, then either X or Z is the random one, so choose Y.

Mark C - 5 years, 1 month ago

yes well given that if you ask are you catherine you will know who is not catherine, which depending a the randomness you will have a non catherine answer :) so if catherine says no than you will marry beatrice if she says yes than you will marry anne :)

Trent Smith - 4 years, 10 months ago

Do you know the solution? I think that it is impossible to know who of them is Catherine with just one yes or no question... give me a light.

Mateo Matijasevick - 5 years, 1 month ago

There is a solution that guarantees that you do not pick Catherine, but you won't know whether you have chosen Anne or Beatrice and you won't know which of the other two is Catherine. Remember that you may ask one princess the question but choose a different princess. :)

Maggie Miller - 5 years, 1 month ago

Well , but I don't find that if it is what I am thinking you say is the solution then you are guaranteed not to choose Catherine. After a little bit of meta-logical , I would say , thinking I wanted to ask the same question as Mateo cause it didn't seem to be any solution. Since there needs to be only such questions then it is impossible to say by one question if the person which is asked is either Catherine or someone else and that because even if I would knew the truth value of the questions asked and could make the difference between who is Beatrice and Anne I couldn't say if it is also Catherine since she "speaks randomly". By this it means that the only way to get rid of Catherine would be to ask a question by which I can say if it is Anne or Beatrice and consider for each of these cases the option that that person might also be Catherine and chose other princess by which I would also found which one of the other two will be either Anne or Beatrice. This because by putting that question i will find that one of them is either Anne or Catherine or Beatrice or Catherine but since I would choose one of the other two and for each of the two cases it remains opened the possibility that one of the other 2 is Catherine I should know which might be Catherine from them. But by following this strategy which would be necessary a question which has this characteristics doesn't seem to be possible.

A A - 5 years, 1 month ago

O my gosh. What’s with the princesses 👸?

Annie Li - 3 years, 5 months ago

The solution:

Say the princess are P1, P2, P3. The prince asks P1, "Does P2 tell the truth more often than P3?" If the answer is yes, he chooses P3. If the answer is no, he chooses P2.

Then if the prince spoke to Anne, he has chosen Beatrice. If he spoke to Beatrice, he has chosen Anne. If he spoke to Catherine, then he has chosen one of the other two.

Maggie Miller - 3 years, 5 months ago

just ask are you catherine the only one that is different will be either anne or beatrice

Trent Smith - 2 years, 11 months ago

My solution is as follows. You can ask the princesses, " Are you the King's Daughter?" since we know that all three sisters are in fact the King's daughters,we will be able to accurately guess who is lying and who is telling the truth.In this instance you have two scenarios: one of which Catherine Answers "Yes" and the other of which Catherine answers "No". You ask the Question,"Are you the King's Daughter?" The truth teller will say yes, the liar will say no, and for this situation the random one will say yes.(Assuming she is random in that she can say yes or no) Since the triplets are identical and we can not distinguish between the two sisters who replied yes, we select the sister who said no because we can conclude that we have selected Beatrice (The one who lies). In the alternate scenario, the truth teller will say yes, the liar will say no, and the random one will say no. In that instance since we still cannot distinguish between the two sisters who say no, we select the one princess who answered yes. From that we can conclude that we have selected Anne. "The princess who always tells the truth." This solution is based on the assumption that we can ask the same question three different times. If that assumption proved illogical, then this whole conclusion is rendered invalid.

James Wyche - 4 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$