The most intelligent prince!!

Read the problem carefully. Its a very interesting problem

A king wants his daughter to marry the smartest of 3 extremely intelligent young princes, and so the king's wise men devised an intelligence test.

The princes are gathered into a room and seated, facing one another, and are shown 2 black hats and 3 white hats. They are blindfolded, and 1 hat is placed on each of their heads, with the remaining hats hidden in a different room.

The king tells them that the first prince to deduce the color of his hat without removing it or looking at it will marry his daughter. A wrong guess will mean death. The blindfolds are then removed.

You are one of the princes. You see 2 white hats on the other prince's heads. After some time you realize that the other prince's are unable to deduce the color of their hat, or are unwilling to guess. What color is your hat?

Note: You know that your competitors are very intelligent and want nothing more than to marry the princess. You also know that the king is a man of his word, and he has said that the test is a fair test of intelligence and bravery.

Give ur answer as black or white and then give the explanation of why is it so.

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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}

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\sum_{i=1}^3

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Easy Math Editor

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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

i dont see how someone could determine. explanation please.

john smith - 6 years, 9 months ago

This problem is solved easily as follows. You are Prince A. Let's see whether your hat is black. Although A can see that B and C have white hats, B and C don't know what their hat color is yet. If B sees black on A and C doesn't immediately claim that their hat is white, B must know that their own hat is white since, if B was wearing black, C would have immediately claimed to be wearing white. But C is silent. However, B is smart too so, why are they silent too? The only possible reason is, A must be wearing white. The only way any of the princes could quickly and easily determine their own hat color would be the presence of at least one black hat. Given that no fast answer was forthcoming, all three hats must be white. Presumably one of the three princes eventually figures this out and claims to be wearing white to claim the princesses hand ... afterall, they are all exceedingly smart.

Doug Johnston - 4 months ago

White hat!

Daniya Quadros - 6 years, 7 months ago

Correct me if I'm wrong:

Let the other two princes be B and C.

We are given B and C are white.

Case 1: you are Black.

Case 2: you are white

Then B sees both white and black, hence he is unable to answer with certainty.

This is also the case for C. But, w.l.o.g C sees that B can't answer. Hence he deduces he is white.

The question says that neither B nor C can answer with certainty.

So we reject case 1. Therefore you must be white.

However, we see a paradox here. If you are white, then we have a cyclic symmetry, where all 3 princes are in the same situation. In that case, both B and C should be able to deduce that they are white. Unless this just means you were able to deduce you are white before the other two, which means you win and get to marry the princess?

Jihoon Kang - 5 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}`	$\boxed{123}$