Brain Teasers

[»Blocks]

I think we have some people here (myself included) who would enjoy a good brain teaser or two. Post them here for people to work on. I'll keep track of the questions and solutions in this post.

 

Two Positive Integers (solved)

 

A man is thinking of two positive integers. He tells his son the product of the two numbers, and his daughter the sum.

 

He asks his son if he knows the two numbers. He says NO.

He asks his daughter if she knows the two numbers. She says NO.

He asks his son if he knows the two numbers. He says NO.

He asks his daughter if she knows the two numbers. She says NO.

He asks his son if he knows the two numbers. He says YES.

He asks his daughter if she knows the two numbers. She says YES.

 

What are the two numbers?

 

 

The numbers are

1 and 6

.

 

Son, first NO:

We know that the numbers aren't 1 and 1, or 1 and any prime number.

(1, 1), (1, 2), (1, 3), (1, 5)

...

 

Daughter, first NO:

We know that the numbers aren't 2 and 2, because if they were, the sum would be 4 and she would know (because she knows it can't be 1 and 3).

(2, 2)

 

Son, second NO:

We know the numbers aren't 1 and 4, because if they were, the product would be 4 and he would know (because he knows it can't be 2 and 2).

(1, 4)

 

Daughter, second NO:

We know the numbers aren't 2 and 3, because if they were, the sum would be 5 and she would know (because she knows it can't be 1 and 4).

(2, 3)

 

Son, YES:

We know the numbers are 1 and 6, because this is the only set whose product only has one set of factors available.

(1, 6)

 

Daughter, YES:

This is just a distraction.

Pills A and B (solved)

 

You have a life-threatening illness that you are treating with two prescription medications, A and B. You have to take one pill of each daily. The pills are indistinguishable from one another for the purpose of this riddle. One day, you accidentally pour two pills of B into your hand, after you already had one pill of A in the same hand. Taking a double dose of either pill is fatal, as is missing a day. The pills are prohibitively expensive, so throwing them away is not an option. Neither is counting how many pills are left in the bottle, as the bottles have different quantities and you started them at different times. What do you do?

 

 

Take out another pill of A and cut all the pills in half (being careful to separate the halves into two groups). Take one group today and one group tomorrow, guaranteeing you 1 pill of A and 1 pill of B each.

King and the Blind Mathematician (solved)

 

Once upon a time, a king wanted to hire the best mathematician in his kingdom to work in his palace. His servants brought to him the best two mathematicians, one of whom was blind. The king told both them that he couldn't hire two mathematicians, so he would ask a question, and whoever answered correctly would be hired.

 

The king said: "I have three sons. If you multiply their ages, the result will be 36. What are their ages?" Both mathematicians told the king that this information was not enough. The king then said: "The number of windows in the building across the street is equal to the sum of the ages of my sons." The first mathematician (who could see) counted the windows, and told the king that he still could not figure it out. The blind mathematician (who could not count the windows) told the king that he did not have an answer. The king then said: "My oldest son has red hair." Right away the blind mathematician gave the correct answer and got hired.

 

What are the sons' ages?

 

 

• List out the sets of three positive integers that multiply to 36.
  
  
• List out the sums of each set. There are two sets that have the same sum: (1, 6, 6) and (2, 2, 9). The others all have unique sums. The ages must be one of these sets since knowing the sum did not produce a definite answer.
  
  
• Knowing that there is an oldest son (age 9) means that it must be (2, 2, 9).

Ladybugs on the Complex Plane (solved)

 

Four ladybugs start out at the corners of a unit square on the complex plane, centered at the origin. Suddenly, each beetle starts moving along a smooth trajectory that is continuously adjusted to point toward the next beetle along the square in a counterclockwise direction. At any given time, all four beetles are moving at the same speed.

 

What is the length of each beetle's path at the time which they meet at the origin?

 

 

One.

Edited October 23, 2009 by Blocks

[Cheese]

not enough information Edited October 22, 2009 by Cheese

[»Blocks]

not enough information

There is no more information required.

[Arry]

not enough information

There is no more information required.

 

I'm gonna take a wild swing and say the numbers were 1 and 2

[CRe]

0 and 1?

[Dr Brain]

6 and 1. I brute forced it, though.

[Samapico]

I don't get it...

[»Blocks]

6 and 1. I brute forced it, though.

Winner! Though what do you mean by "brute forced it"?

 

Solution below:

 

The numbers are 1 and 6.

 

Son, first NO:

We know that the numbers aren't 1 and 1, or 1 and any prime number.

(1, 1), (1, 2), (1, 3), (1, 5)...

 

Daughter, first NO:

We know that the numbers aren't 2 and 2, because if they were, the sum would be 4 and she would know (because she knows it can't be 1 and 3).

(2, 2)

 

Son, second NO:

We know the numbers aren't 1 and 4, because if they were, the product would be 4 and he would know (because he knows it can't be 2 and 2).

(1, 4)

 

Daughter, second NO:

We know the numbers aren't 2 and 3, because if they were, the sum would be 5 and she would know (because she knows it can't be 1 and 4).

(2, 3)

 

Son, YES:

We know the numbers are 1 and 6, because this is the only set whose product only has one set of factors available.

(1, 6)

 

Daughter, YES:

This is just a distraction.

 

[Dr Brain]

I'm in the process of learning python, so I threw together a script.

 

The script generated a list of all possible sets of two integers X and Y (less than 50).

It then threw away all possibilities where there was just one item in the list to give X*Y

It then threw away all possibilities where there was just one item in the list to give X+Y

It then threw away all possibilities where there was just one item in the list to give X*Y

It then threw away all possibilities where there was just one item in the list to give X+Y

 

It then generated a new list of all items where there was just one item in the list to give X*Y

It then outputted all of the items in the new list which had just one possibility for X+Y. [6, 1] was the only output.

[»Blocks]

I'm in the process of learning python, so I threw together a script.

 

The script generated a list of all possible sets of two integers X and Y (less than 50).

It then threw away all possibilities where there was just one item in the list to give X*Y

It then threw away all possibilities where there was just one item in the list to give X+Y

It then threw away all possibilities where there was just one item in the list to give X*Y

It then threw away all possibilities where there was just one item in the list to give X+Y

 

It then generated a new list of all items where there was just one item in the list to give X*Y

It then outputted all of the items in the new list which had just one possibility for X+Y. [6, 1] was the only output.

Oh no ... I thought of that when you said you brute forced it, but then I thought nobody would actually do that. Though that's the same as doing it by hand, just automating the grunt work.

Edited October 22, 2009 by Blocks

[»Blocks]

Here's a new one. The premise is slightly ridiculous, but you have to go with it.

 

You have a life-threatening illness that you are treating with two prescription medications, A and B. You have to take one pill of each daily. The pills are indistinguishable from one another for the purpose of this riddle. One day, you accidentally pour two pills of B into your hand, after you already had one pill of A in the same hand. Taking a double dose of either pill is fatal, as is missing a day. The pills are prohibitively expensive, so throwing them away is not an option. Neither is counting how many pills are left in the bottle, as the bottles have different quantities and you started them at different times. What do you do?

Edited October 22, 2009 by Blocks

[AL3X]

i dunno Edited October 22, 2009 by AL3X

[p man]

Here's a new one. The premise is slightly ridiculous, but you have to go with it.

 

You have a life-threatening illness that you are treating with two prescription medications, A and B. You have to take one pill of each daily. The pills are indistinguishable from one another for the purpose of this riddle. One day, you accidentally pour two pills of B into your hand, after you already had one pill of A in the same hand. Taking a double dose of either pill is fatal, as is missing a day. The pills are prohibitively expensive, so throwing them away is not an option. Neither is counting how many pills are left in the bottle, as the bottles have different quantities and you started them at different times. What do you do?

Add another pill a, crush into powder, thoroughly mix, take half today/half tomorrow

 

win

[Hakaku]

I still don't understand the first one...

 

But here's what I would do for the second one:

 

Add another pill A to the collection. Now you should have two As, and two Bs. Cut them one by one in half, putting one part in one pile, the other in another pile. You should then obtain two piles of the four pills in half. Take one pile today, keep the other one the next day.

 

ABB + A

 

aabb = AB

----

aabb = AB

 

 

[Emit]

take another b cut all in half eat all halves. do so again tmr

[»Blocks]

Add another pill a, crush into powder, thoroughly mix, take half today/half tomorrow

 

win

Fair enough.

 

 

Take out another pill of A and cut all the pills in half (being careful to separate the halves into two groups). Take one group today and one group tomorrow, guaranteeing you 1 pill of A and 1 pill of B each.

 

[Kai Ossa]

Add another Pill A. Line them all up. Cut in half, take left half. Save right half for tomorrow. PROFIT.

 

edit: Oh, didn't see all the those posts above, was trying to log in. =/

Edited October 22, 2009 by Kai

[»Blocks]

I still don't understand the first one...

It's an information problem. You have less information than the son and daughter, but you can successively eliminate possible sets of numbers based on the responses of the people who do have more information, as in Dr Brain's script.

 

Here's a new one, similar to the first.

 

Once upon a time, a king wanted to hire the best mathematician in his kingdom to work in his palace. His servants brought to him the best two mathematicians, one of whom was blind. The king told both them that he couldn't hire two mathematicians, so he would ask a question, and whoever answered correctly would be hired.

 

The king said: "I have three sons. If you multiply their ages, the result will be 36. What are their ages?" Both mathematicians told the king that this information was not enough. The king then said: "The number of windows in the building across the street is equal to the sum of the ages of my sons." The first mathematician (who could see) counted the windows, and told the king that he still could not figure it out. The blind mathematician (who could not count the windows) told the king that he did not have an answer. The king then said: "My oldest son has red hair." Right away the blind mathematician gave the correct answer and got hired.

 

What are the sons' ages?

[Emit]

2 2 9

[»Blocks]

2 2 9

Too fast!

 

 

I don't have the full solution to this one written up, but it goes like this:

 

• List out the sets of three positive integers that multiply to 36.
  
• List out the sums of each set. There are two sets that have the same sum: (1, 6, 6) and (2, 2, 9). The others all have unique sums. The ages must be one of these sets since knowing the sum did not produce a definite answer.
  
• Knowing that there is an oldest son (age 9) means that it must be (2, 2, 9).

 

[»Blocks]

This is the last one I have written up, but it should be around for longer than the others. It's not for the faint of heart, and only fun if you're into this sort of thing. Feel free to brute force it as that would be a fun exercise (post your code if you do).

 

Four ladybugs start out at the corners of a unit square on the complex plane, centered at the origin. Suddenly, each beetle starts moving along a smooth trajectory that is continuously adjusted to point toward the next beetle along the square in a counterclockwise direction. At any given time, all four beetles are moving at the same speed.

 

What is the length of each beetle's path at the time which they meet at the origin?

[p man]

This is the last one I have written up, but it should be around for longer than the others. It's not for the faint of heart, and only fun if you're into this sort of thing. Feel free to brute force it as that would be a fun exercise (post your code if you do).

 

Four ladybugs start out at the corners of a unit square on the complex plane, centered at the origin. Suddenly, each beetle starts moving along a smooth trajectory that is continuously adjusted to point toward the next beetle along the square in a counterclockwise direction. At any given time, all four beetles are moving at the same speed.

 

What is the length of each beetle's path at the time which they meet at the origin?

Over 9000.

[rootbear75]

aww hell, i know this one... they start to make a fractal type of image...

i dont know the exact math behind it though...

 

http://www.tkheaven.com/answer.jpg

 

yes i realize its sloppy

 

 

EDIT: I also have some teasers i was going to throw up too if you dont mind

Edited October 23, 2009 by rootbear75

[»Blocks]

aww hell, i know this one... they start to make a fractal type of image...

i dont know the exact math behind it though...

 

http://www.tkheaven.com/answer.jpg

 

yes i realize its sloppy

On the right track -- but since their direction is continuously adjusted, it ends up being a spiral.

 

EDIT: I also have some teasers i was going to throw up too if you dont mind

Go for it, I'd like to hear new ones. I'm kind of anal about spelling, grammar, punctuation, and sentence structure, so don't be offended if I edit it for the main post.

[Dr Brain]

It's equal to the length of a side of the square (one, in this case), and it's easy to explain.

 

The ladybugs are always crawling at right angles to each other. This means that their motion is always completely toward each other (no wasted motion), and they must travel the original distance, and no more.

 

Yes, I believe it's a logarithmic spiral.