JCC The quarantine puzzles/enigmas thread

Here everybody can post a puzzle/enigma and the rest of the JCC can try to answer.

Just few rules

-There are no constraints on the kind of enigma that someone can propose, but whoever asks is supposed to know the solution and possibly to add a warning if it requires extra-ordinary knowledge on some topic.

-Questions should be numbered and with a title in bold.

-Answers always under spoiler tag. The answer must contain the number of the question outside the spoiler tag.

-You can post as many questions as you want, no matter whether the previous ones already have a solution (and this is why numbering the questions is useful).


I will start with a relatively famous one

1) Monty Hall problem
Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. (Suppose you really prefer the car over a goat). You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

So, if possible, in this case the answer should have the form

1)

Spoiler

**whatever the answer is **

^ This problem usually stimulates quite heated discussions. At a family dinner we discussed for hours, and at the end I think that I failed to convince my father of the correct solution.

 

1)

Spoiler

Because Monty has perfect knowledge and will never open a non-dud door, the probability that the remaining door is not the dud is disproportionately higher, therefore it is mathematically prudent to switch. This bears out in long-term simulations and actual data from the show's run showing that roughly 66% of participants who switched won, versus roughly 33% who didn't. It is perhaps best understood by the analogous 100 door variant - if Monty has perfect knowledge, and opens 98 doors, all duds, it becomes easier for people to accept that they might well have picked a dud and should switch. Again, modeling indeed bears this conclusion out.

This is not to suggest the answer is intuitive: in a famous anecdote, the great combinatorialist Paul Erdös could not accept the result until he saw the simulation data.

PICK SOMETHING THAT ISN'T A MATH PROBLEM YOU COWARDS.

 

One night aliens land in your back yard. They produce for you 9 metal spheres and a double pan balance scale. They inform you that you must determine which of the 9 metal spheres is slightly lighter than the others (8 of them are exactly the same, 1 is slightly lighter, and the difference cannot be determined by just holding them or looking at them) and that you may use the balance scale ONLY TWICE to figure it out or they'll blow up the world. How do you do it?

 

The key is to remember that in a situation where only one person on brown eyed island has blue eyes, they will immediately notice no one else has blue eyes and will isolate themselves.

 

@Mortimer Snerd please number the question and write a little bold title.
Also. Spoiler tag the answers!

3) Four litres of water
I have a bottle that can contain 5 L and a bottle that can contain 3 L . I have unlimited water at my disposal. How can I get 4 L of water as precisely as possible? I cannot use any weird measuring instruments.

2) Find the fake ball
EDIT. this is what Mortimer Snerd wrote. When I read it it was too late. Apologies!

5) (this requires good mathematical knowledge) Fibonacci and Phi
@Ramza
Demonstrate that the ratio between a Fibonacci number F(n) and the previous one F(n-1) converges to the golden ratio 1.6180... as you increase the two numbers (that is, as you increase n).

 

Spoiler

Fill both bottles to half-full, then pour the contents of the second bottle into the first bottle. 2.5 L + 1.5 L = 4 L

 

@Iron_lord
The shape of the bottles makes it impossible to accurately fill them precisely by half.

 

This is immediate from Binet's formula.

 

Spoiler

Put three spheres on one pan and three on the other. In this way you will be left with three spheres as possible candidates (in case those you tried weigh the same, the three spheres are the ones you left out).
Weigh one of the three spheres on one pan of the scale and another one on the other.
In this way you will find the sphere you are looking for (again, if the two you chose weigh the same, the correct one is the one that was left out).

 

6) The Wise Dwarfs
You are in front of two boxes and you must pick the one that contains a treasure. You can ask one question to one of two wise dwarfs. They only answer either Yes or No. They know everything. One of them always tells the truth and the other one always lies. You don't know who is the one who tells the truth.
How can you find the correct box by asking a single question?

7) The Wise Dwarfs, but more difficult
What do you need to ask in case you only know that each dwarf either always lies or tells the truth, but you do not know how many of the two dwarfs tell the truth (so, in case it's also possible that they both lie or both tell the truth)?

 

6)

Spoiler

Shoot one in the leg, or break their leg if a gun is not available. Then ask if it hurt. if they say no while screaming then that's the liar.

 

Spoiler

Ask one "If I were to ask the other dwarf which was the right box, what would he say?"

The correct box is the other one.

 

Sorry 'bout that. Edit time passed.

You got it though @3sm1r .

 

Then, you would have used your only question and still not know where the treasure is.

 

8) The wolf, the sheep and the cabbage
I must take my boat and bring on the other side of a river a wolf, a sheep and a cabbage. There is space on the boat for only one thing for every travel. When I am absent, there is the risk that the sheep eats the cabbage and that the wolf kills the sheep. What strategy should I use to bring all of them on the other side safe?

 

Spoiler

Cross with the sheep, come back. Cross with the wolf and return with the sheep. Cross with the cabbage and come back. Cross with the sheep.

 

From The Guardian

9) The coloured socks
Ten red socks and ten blue socks are all mixed up in a dresser drawer. The 20 socks are exactly alike except for their colour. The room is in pitch darkness and you want two matching socks. What is the smallest number of socks you must take out of the drawer in order to be certain that you have a pair that match?

10) Cutting the pie
With one straight cut you can slice a pie into two pieces. A second cut that crosses the first one will produce four pieces, and a third cut can produce as many as seven pieces. What is the largest number of pieces that you can get with six straight cuts?

 

Spoiler

Three. This is one of those misdirection problems where you’re meant to say “Ah, one more than half.” But they’re socks. It doesn’t ****ing matter which one you grabbed, so you just need three to get two with the same color.

 

@Ramza
Yes, correct. Number 10) seems a bit less straightforward. I broke my own rule and I posted it before checking the solution, since it was published in The Guardian so I guess it can't be excessively difficult. But I'm not sure how to do it, on the top of my head.

 

My guess is you want to maximize the number of intersections with each cut - hence two intersections with the third cut, which gets you seven pieces (seven irregular, worthless pieces, but pieces). So next would be three, then four, up to five intersections giving however the hell many actual chunks.

 

11) Father and son
A father is twice as old as his son. 15 years ago he was three times older. How old are they?

 

Spoiler

x=2y, x-15 = 3(y-15) -> x = 3y - 30 -> 2y = 3y - 30 -> y = 30. So 60 and 30. There’s probably some trick about 15 but screw that, man, two equations and two unknowns!

 

12) The two fuses
I have two fuses without any bomb attached, one longer than the other. If I light one end of the first fuse, it will take 1 hour for it to burn completely. If I do the same for the other fuse, it only takes 40 minutes, since it is shorter. How can I use the two fuses to measure 50 minutes as accurately as possible?
I do not have any instrument to precisely cut either fuse in any way, and I cannot light them up in any place other than their ends.

 

13) The three switches
You are in a room that has three switches and a closed door. The switches control three light bulbs on the other side of the door. Once you open the door, you may never touch the switches again. How can you definitively tell which switch is connected to each of the light bulbs?