Brainteaser/Problem Solving Thread

General Discussion
1

So this idea came from a post I made in SRK. I personally love doing this kind of stuff, it’s one of my favorite pastimes (and I’m hoping I’m not the only one). I figured we could post some problems/brainteasers and see what sort of solutions the people from SRK come up with. I was also thinking people could even maybe post some schoolwork problems within reason (if you are doing a thesis on Galois Theory then maybe this isn’t the best place to come for assistance). If this thread ends up being a bust then so be it, but it’s worth a shot as this stuff can be really fun. One thing I would ask is that if you post a problem/solution that you didn’t write, then mention that you didn’t write it (don’t plagiarize). So to kick it off I’ll post two problems, one easy and one difficult, neither of which were written by me, although I’ll post my solutions in time:

The easy:
If exactly one of these statements is false, which statement is false?
A) Statement D is true
B ) Statement A is false
C-x) Statement B is false
D) Statement C-x is true

Apologies for the somewhat weird notation; I had to do that in order to avoid getting emoji. Problem written by Sandeep Bhardwaj

The hard:
A toilet paper roll of mass m = 150g with inner radius r = 1.5cm and outer radius R = 5cm is put on a small cylindrical bar with coefficient of friction k = 2 . If we pull the paper straight down very slowly and increase the force very gradually, the inner part of the roll will first move without slipping on the bar. Then at some critical force F_c , it will start to slip. Assume that the pulled out piece of paper is always vertical. Find F_c in Newtons. Assume that ** g = -9.8 m/s^2** Problem written by David Mattingly.

2

B is false?

I don’t understand the second problem; what does the small cylindrical bar look like? Why would the inner part move without slipping; how is it even connected to the bar? Can the bar rotate or something like that?

Have you tried XKCD’s blue eyes puzzle? Spoilered for length:

[details=Spoiler]A group of people with assorted eye colors live on an island. They are all perfect logicians – if a conclusion can be logically deduced, they will do it instantly. No one knows the color of their eyes. Every night at midnight, a ferry stops at the island. Any islanders who have figured out the color of their own eyes then leave the island, and the rest stay. Everyone can see everyone else at all times and keeps a count of the number of people they see with each eye color (excluding themselves), but they cannot otherwise communicate. Everyone on the island knows all the rules in this paragraph.

On this island there are 100 blue-eyed people, 100 brown-eyed people, and the Guru (she happens to have green eyes). So any given blue-eyed person can see 100 people with brown eyes and 99 people with blue eyes (and one with green), but that does not tell him his own eye color; as far as he knows the totals could be 101 brown and 99 blue. Or 100 brown, 99 blue, and he could have red eyes.

The Guru is allowed to speak once (let’s say at noon), on one day in all their endless years on the island. Standing before the islanders, she says the following:

“I can see someone who has blue eyes.”

Who leaves the island, and on what night?

There are no mirrors or reflecting surfaces, nothing dumb. It is not a trick question, and the answer is logical. It doesn’t depend on tricky wording or anyone lying or guessing, and it doesn’t involve people doing something silly like creating a sign language or doing genetics. The Guru is not making eye contact with anyone in particular; she’s simply saying “I count at least one blue-eyed person on this island who isn’t me.”

And lastly, the answer is not “no one leaves.”

I’ve done my best to make the wording as precise and unambiguious as possible (after working through the explanation with many people), but if you’re confused about anything, please let me know. A word of warning: The answer is not simple. This is an exercise in serious logic, not a lateral thinking riddle. There is not a quick-and-easy answer, and really understanding it takes some effort.[/details]

3

@forte95
B is false and can be found using a simple truth table (if anyone wants I’ll include one, but it can also be found just by sort of reasoning through it).

For the second problem, I will attach a diagram to help you better visualize it:

http://i.imgur.com/eUUmFz8.png

This problem is legit pretty hard and requires you make some assumptions in order to solve it. Since few people seem to be interested at this point, I’ll include the assumptions and hopefully that will help you find a solution, the main one being that since the paper is being pulled very slowly we can assume the roll is always in equilibrium. The “no-slip” part helps us solve the problem because it gives us a certain inequality that can be solved too. The bar isn’t rotating, maybe it would help to think of it as a “pivot” point.

I’ve never tried the puzzle you posted, but I will be sure to take a look at it this weekend and post up any thoughts I have about it here.

4

I have a bunch of logic puzzle books and other stuff around the house. Use them for weird health reasons though.

Should rewrite some of them for SRK.

5

brain teaser is what your mom called it when she deep throated me

or was it brain tickler, i cant remember i was too busy pumping her mouth full of babies

6

what would you call something that is beautiful and angelic in 5 letters?

7

Kromo

8

Lolis
Death
Poops
But he probably means Neesa

Not really a brain teaser, but a question I’ve been thinking about:
Why do games always have you move to the right? Ie, side scrollers, beat em ups, platformers. You start on the left side of the screen and move to the right.
I’ve heard “you read left to right” but a lot of these games (for example Mario) are Japanese and the Japanese read right to left.
I’ve seen a few answers of “subconscious bias” but that isn’t very satisfying.

9

@drunkards_walk If you assume the roll is always in equilibrium and the bar doesn’t rotate, the inner part wouldn’t move first without slipping, then.

In that case, F_c would just be whatever the friction needs to be to roll the tissue roll: normal force (weight of the roll) x coefficient of friction.

I thought about the case where it would move first without slipping. The roll would tilt off-center like what it looks like in your diagram. I realized the normal force would tilt away from the vertical, so fuck that shit, I’m not in the mood for 1st yr physics all over again, so I looked up the solution. It’s something new I’ve learned, thanks.

Here are some physics-related ones I know off the top of my head. You don’t need a calculator for any of these (excuse my Paint skills):

  1. There are three glasses: one with a solid ice cube, one with an ice cube with an air pocket inside, and one with an ice cube with a nail inside. Each is filled to the brim with water. When the ice cubes melt, what will happen to the water level in each of the glasses: overflow, stay the same level, or drop? Problem given to me by my friend.
  1. The air inside a car tire follows Pascal’s principle: any pressure the air exerts is transmitted equally throughout. That means all the air above the tire shaft exerts the same pressure as all the air below the tire shaft. Therefore, no net force should be exerted on the shaft. How is it that the shaft floats above the ground and supports the car’s weight anyway? Problem given to me by my dad.

@drunkards_walk If you’re interested in talking higher-level physics brainteasers, I’m game. We could bring it to the science thread too, if you want.

10

The next Mario level I make will have Mario move from right to left. And it will have elements in it that force you to go fast.

11

@forte95
Yeah the problem is talking about the case where the paper moves first without slipping. Glad you liked the problem; I thought it was a good one as well. I’d definitely be interested in doing some high-level physics brainteasers. I also have a book of some math ones as well, some of which I’ve not yet worked through, so if you’re interested I could post some of those as well

12

You could take a crack at the ones I posted first; I don’t have higher-level ones right now. You could post more math ones, yes.

13

@forte95
Attempt for ice cube problem

[details=Spoiler] Contrary to what someone would intuitively think, the water level in the first glass would decrease because ice has more volume than liquid water.As would be the case with glass 2 and 3. Sounds easy unless there is something weird at work.

EDIT: Oddly enough I wonder if some type of thermal expansion of the gas in the ice cube would cause anything to happen. Too bad most common gases aren’t very water soluble. I’ll stick with my answer.[details=Spoiler]

14

Cool, I’ll take a look at the ones you posted and post some math problems tomorrow when I finally have a chance to relax; work has been a bit crazy this past week

15

@BullDancer
Solution for ice cube problem (SPOILERS!)
First glass: solid ice cube

Spoiler

You’re on the right train of thought! The water won’t overflow, but actually as it turns out, the water from the molten ice cube will exactly match the decrease in water level; the water level will remain the same.

Here’s why:
The buoyant force relies on the Archimedes principle (that’s what it’s called IIRC): the upward buoyant force on a waterborne object is equal in magnitude to the weight of the water displaced by the object. Now, the ice cube was floating, so its downward weight (W_i) was being matched by the upward buoyant force (F_b). By the Archimedes principle, F_b should be equal to the weight of the water displaced, whose volume should be equal to the volume of the submerged part of the ice cube (V).

Thus (!), the ice cube has the same weight as the volume of water V, which incidentally (seen in the figure), is the volume of water needed to keep the water level at the brim! Since the ice cube has the same weight (even if it has different density at first), once it melts into water, it will be the same amount of water needed to fill up the volume V!

https://s16.postimg.org/yww5v8m2t/first_glass_solution.png

Second glass: ice cube with air pocket

Spoiler

This is similar to the first scenario, yes. However, the air pocket makes the ice cube float more than it would if it were solid, and as it turns out, the water will overflow out of the glass.

Here’s why:
The air pocket makes the ice cube a tad less dense in the middle, so it floats more. However, once the ice melts enough such that the air pocket breaks free, there is nothing keeping the ice cube floating more than it has to. The ice cube will sink a little bit, and that will push a little bit of water out of the glass!

https://s14.postimg.org/cgrjdp2k1/second_glass_solution.png

From that point on, the problem will carry on in the same way as the first scenario.

Third glass: ice cube with nail

Spoiler

This is actually just the opposite of the second scenario. The nail makes the ice cube sink more than it would if it were allowed to float alone, so if you’ve read the solution for the second scenario, you know the water level will decrease instead of overflowing.

Here’s why:
The nail weighs down the ice cube a tad, so it sinks more. However, once the ice melts enough such that the nail breaks free, there is nothing keeping the ice cube floating less than it has to. The ice cube will float a little bit, and that will free up some room inside the glass, causing the water level to drop!

If you read the solution for the second scenario, you might be thinking that the sinking nail would cause the water level to rise. Here’s the trick: the nail has always been too dense to float. However, in this case, it floated because it was stuck to much less dense material (the ice cube). That means the water had to put up more buoyant force to support the weight of the nail, and by the Archimedes principle, that means more water had to be displaced.

Once the nail breaks free from the ice, it starts to sink. This means (!) the water isn’t putting up enough buoyant force to support its weight anymore! Which means, some amount of displaced water must have been “replaced” (or stopped being displaced lol, for lack of a better term), which means the water level had dropped!

https://s4.postimg.org/lpp1hhn1p/third_glass_solution.png

From that point on, the problem will carry on in the same way as the first scenario.

16

Mario was one of the first platforming games. I think the reason why most platforms that follow the “rush to the end” gimmick are just copying the games before it, as it’s what gamers are use to.

So why does Mario always run to the right?

Some actually think that mario is themed as a play, as he always has to “exit stage right”. The levels are even called “stages.” There are other hints, like how Mario 3 starts with a curtain being drawn up. Mario 64 has a creature with a camera on a stick following Mario around, which seems to hint that mario hit the big time and actually became a tv/movie actor.

17

@forte95 Dammit if I put my thinking cap on the second one should’ve been obvious.

Idk if I would’ve gotten the third one because for some dumb reason I wasn’t considering this as a buoyancy problem. That’s an issue with education, you learn math and then you get unrealistic problems to solve so it’s alien to things that it should be obviously applied to.

#salt I’m incredibly mad about losing.

18

@forte95 I’ll take a crack at the second problem since the first has been answered already. Also, I am still interested in some problem solving in the Physics thread if you are still interested. I’ll try to respond whenever I can but I’ve been quite busy lately. Anyway, the shaft floats above the ground because the pressure below the shaft is greater than the pressure above the shaft. This is because below the shaft there is pressure due to air in addition to pressure due to the weight of the vehicle. If you could give me a link to the Physics thread we can move the disucssion there; I also have a few problems I’d like to post as well

19

@drunkards_walk Link’s in my sig.

Second one, good try, but remember that pressure is transmitted equally throughout. If the shaft pushes down on the air because of the weight of the vehicle, the air isn’t going to push up on you with the same pressure; it’s going to push all around you from all directions.

Think about it: if you lift a shaft into midair, then try to ride it, the air doesn’t bear your weight at all; you promptly fall down. What is it about the tire that makes it different from open air?

Hint: The pressure above is the same as the pressure below. There is nothing wrong with that statement. The force below, however…

EDIT: Also no worries, I’ve been very busy myself.

20

@forte95
Ok round two: the weight of the car act will act downward and so the normal force exerted by the road would be equal and opposite in magnitude/direction; this would give a net force of zero, so it is balancing the shaft to float and the weight is cancelled by the nornal force and so it supports the weight