Riddles, brain puzzles and mathematical problems

[Froge]

A teacher gives his students 10 problems and tells them that 5 of those 10 problems will be on the exam, and they will be the only questions. A student memorizes the solutions for 7 of the 10 problems. He does not know how to do any of the other 3. What is the probability the student gets at least 4 questions correct on the exam?

[BAndrew]

@MyRedNeptune

Your answer is correct! You continue to impress me
My solution was completely different, but that's the awesomeness; that there are many different solutions.

[Froge]

anyone?

[MyRedNeptune]

@BAndrew

Thank you! So you're not fazed by the last step of my solution? That's mainly what I was referring to when I mentioned lousy technique.

I'm quite curious what your solution is, too.

EDIT: Hm, today I found another solution to the same problem. This one takes like 5 minutes, and it's more elegant. Also, this problem is awesome.

Spoiler below!

This is totally easy. To start, write down your digit of choice. This will be the "moving digit" and the first digit of the number m. Let's call it a. Multiply a by 2 and write the first digit of the result in the space to the left of it, adding any applicable 1's in the process. Repeat the process with the resulting new digit.

Keep doing this until two conditions are met:

-the leftmost digit is 1/2*a (if a is even) or 1/2*(a-1) (if a is odd)
-if a is even, the second leftmost digit times two gives a single-digit number; if a is odd, the second leftmost digit times two gives a double-digit number

This will lend us the smallest possible m for any a. Multiply by 2 to get n.

@Chrono Sorry, probability is... I just forgot everything. :S

[BAndrew]

@MyRedNeptune

So you're not fazed by the last step of my solution?

No.

Hm, today I found another solution to the same problem. This one takes like 5 minutes, and it's more elegant.

Well done again!

I'm quite curious what your solution is, too.

My solution is more similar to your second solution, but instead of multiplying I use division. It is a bit hard to describe though. So I will use some pictures to illustrate the steps.

Pick up a digit (I picked up 9 because why not) and then you devide it with 2 using long division.

We know that the number n must be divisible by 2 exactly. We start to do division normally. So in my case the first step is this:



Because we put the first digit (the 9 in this case) in the end we know that 4 is the second digit of the number n.



Then 7 must be the third digit for the same reason.



Continue to do this until both of the following conditions are met:

• The last digit in the quotient is a 9 (or the digit you began with)
  
• The remainder is 0
  

Then you have your number!

I am actually too lazy to continue the process but you can see that it works. The number I found was 18 digits long or something like that, but I lost the paper I have written it down. It doesn't take more than 5 minutes to find it though.

@Froze
OK Froze we got it, probabilities are your Jesus. Every problem you've posted so far has probabilities in it. I don't have anything against probabilities, but I know only the basics or even less about the subject.

EDIT:

I'll post a new problem either today 21:00-22:00 GMT or tomorrow the same time.

There is also Froge's riddle if anyone wants to try.

[MyRedNeptune]

@Bandrew

That's quite clever! I had a feeling my second solution would be closer to yours.

[BAndrew]

@MyRedNeptune
Thanks


Next riddle:

You are suddenly abducted by aliens while sleeping. You wake up in a room and you see an indescribable creature which you soon realise, that somehow, it knows and can speak English. Before you ever have the chance to react the alien makes things clear:
" We were looking for other intelligent life in the universe and we finally found Earth. Humans have confused us in a great extent. We want to measure human intelligent and we abducted a human being randomly (that happened to be you). You see if a civilisation is not smart enough then it must be destroyed. If you manage to solve the intelligence test we have prepared for you then we will let you free and leave your planet in peace. However if you fail, your civilization probably isn't smart enough."

"This has to be a nightmare" you think, but then the alien continues:

"Next to you there is another room which you can't see or interact in any way. In this room there is a square table with 1 card on it in each corner (total of 4 cards). You don't know which cards are face up and which are face down. Your goal is to turn all the cards face up by following the following rules:
At each phase you tell me which card or cards you want me to turn. Then I go to the room and turn the cards you told me. However in the end of the phase I am going to rotate the table as many times as I want to make it more difficult. If all the cards after turning them as you told me are face up then you win and you will be set free. If not, then you try again with the current position of the cards. You have 3 hours to solve the test."

How are you going to solve the problem and save the Earth?


PS. If you have difficulty understanding something in the riddle then let me know.

NOTE: The riddle has not time limit. It's just for the sake of the plot.

[Bridge]

A teacher gives his students 10 problems and tells them that 5 of those 10 problems will be on the exam, and they will be the only questions. A student memorizes the solutions for 7 of the 10 problems. He does not know how to do any of the other 3. What is the probability the student gets at least 4 questions correct on the exam? What about scoring perfect?

Had to brush up on probability a bit before I could do this but I think I have the solution:

10C7 / 10C5 = 120 / 252 = 0.476 = 47.6%

This is the probability of the answers he memorized appearing on the exam and consequently the probability of him scoring perfect. To find out what the probability of him getting at least 4 correct on the exam is, you do the same thing but limiting the amount of questions that are chosen to 4. So, 10C7 / 10C4 = 120 / 210 = 0.571 = 57.1%

[MyRedNeptune]

You are suddenly abducted by aliens while sleeping. You wake up in a room and you see an indescribable creature which you soon realise, that somehow, it knows and can speak English. Before you ever have the chance to react the alien makes things clear:
" We were looking for other intelligent life in the universe and we finally found Earth. Humans have confused us in a great extent. We want to measure human intelligent and we abducted a human being randomly (that happened to be you). You see if a civilisation is not smart enough then it must be destroyed. If you manage to solve the intelligence test we have prepared for you then we will let you free and leave your planet in peace. However if you fail, your civilization probably isn't smart enough."

"This has to be a nightmare" you think, but then the alien continues:

"Next to you there is another room which you can't see or interact in any way. In this room there is a square table with 1 card on it in each corner (total of 4 cards). You don't know which cards are face up and which are face down. Your goal is to turn all the cards face up by following the following rules:
At each phase you tell me which card or cards you want me to turn. Then I go to the room and turn the cards you told me. However in the end of the phase I am going to rotate the table as many times as I want to make it more difficult. If all the cards after turning them as you told me are face up then you win and you will be set free. If not, then you try again with the current position of the cards. You have 3 hours to solve the test."

How are you going to solve the problem and save the Earth?

Spoiler below!

Because the table is rotated an unknown number of times after each turn, the absolute position of each card is impossible to predict. Therefore, for the sake of simplicity, we can view our table as stationary and our choices of flipping cards as falling randomly on the table. It is the position of the cards relative to each other that is important. There are 5 such patterns possible, and they are each assigned a number in the chart below (the dark circles represent cards that are face up and the win state is unlisted):



There are also 5 possible moves we can make, and they coincide with the states above. For example, 1 is an inversion - all cards are flipped, 2 is flipping two cards on opposite corners, etc. Now I will make a list of ways in which each move can affect each state.


state/move + move/state = possible state(s)

1 + 1 = win
1 + 2 = 2
1 + 3 = 3
1 + 4 = 4
1 + 5 = 5
2 + 2 = win, 1
2 + 3 = 3
2 + 4 = 4, 5
2 + 5 = 4, 5
3 + 3 = win, 2, 1
3 + 4 = 4, 5
3 + 5 = 4, 5
4 + 4 = 1, 3, 2
4 + 5 = win, 3, 2
5 + 5 = 1, 3, 2


Using this list, we can now create a strategy for solving the puzzle. It will be written in the following format:

(possible states);
move (possible states);
move (possible states);
...etc

So, the semicolon marks the beginning of a new turn, the first number of each line is the move we make in the current turn, and the numbers in the parentheses indicate the possible states after the move. Each new turn is given with the assumption that the previous turn did not result in a win state.

(1, 2, 3, 4, 5);
1 (win, 2, 3, 4, 5);
2 (win, 1, 3, 4, 5);
1 (win, 3, 4, 5);
3 (win, 1, 2, 4, 5);
1 (win, 2, 4, 5);
2 (win, 1, 4, 5);
1 (win, 4, 5);
5 (win, 1, 2, 3);
1 (win, 2, 3);
2 (win, 1, 3);
1 (win, 3);
3 (win, 1, 2);
1 (win, 2);
2 (win, 1);
1 (win)

I should be given the title of Earth's Champion because I solved it in 30 minutes.

[BAndrew]

@MyRedNeptune

Congratulations! You saved the Earth! You keep doing an excellent job.




Funny coincidence: It took me 30 minutes to solve it, too.