Thread: How Good Are You At Reasoning?

Originally Posted by Xei

Sorry tommo, didn't see the posts. To be fair you did put them in a silly place.

Hehe. Probably didn't understand my post in the meta forum thread then about time-travel.

Originally Posted by Xei

If it were the highest number on the die that determined the winner then A would indeed always beat C. But as I imagine you guessed, it's not true. It seems very obvious, but if you relied on this intuition, and a reason that you 'know but can't put into words', you'd be incorrect. It is of course normally true. But there are various dice for which it is not true. For instance,

die A has sides: 2, 2, 4, 4, 9, 9
die B has sides: 1, 1, 6, 6, 8, 8
die C has sides: 3, 3, 5, 5, 7, 7.

You can work out the probabilities of A beating B and so on (any pair of numbers from the two dice is equally likely to come up, so just count the ones which win). You find that the probability that A rolls a higher number than B is 5/9 (55.55%), the probability that B rolls a higher number than C is 5/9, and the probability that C rolls a higher number than A is 5/9.

So.... yes my "intuition" or whatever, was correct.
(Not really intuition, I did think about it a bit, just not too much).

Originally Posted by Xei

Like all of these things, the answer is hard to get, but very easy to understand. It can be explained in two or three short sentences, I think.

Ok what about this....
There will be, at worst (because any more of each will create multiple circuits of one transport type), 7 buses and 8 trains.

Ala,


The one from top left to bottom right can be red, otherwise it creates 2 green circuits. But making it red creates a red circuit.



And breaking that circuit in any way will create a green circuit (light green and yellow are green ones breaking red circuit).



This I guess is still not proof, but you can't prove it wrong at least. So fuck it.

Originally Posted by PhilosopherStoned

Since people are playing again, I'll hold off on solutions. I will give a counter-example to the claim that one can circum-navigate the mountain by starting at the stop with the most fuel...

We'll have four depots. One has one third of the gas and each of the other three has two ninths. Starting from the stop with one third, one can traverse the circle in either direction and can hence potentially reach any point on two thirds of the circle. So one only needs to cluster the remaining three depots within the remaining one third of the circle to assure that starting at the point with the most gas will end in failure.

And fuck you.

Originally Posted by tommo

And fuck you.

You're welcome.

Originally Posted by PhilosopherStoned

You're welcome.

Feel free to ruin my theories any time.

And Xei, why won't you just let me give up!!?!?!????

I did realise all that, but I guess it will just take me some time to figure out how to go about doing it.

That's not bad at all tommo, you're definitely approaching the right direction to tackle it, in particular the bit about a single line not being able to be green or red.

Choosing a 'worst' case is also a good way to go about it, though you need to start simpler, and you need to think about some kind of 'bad' feature that will be present for every situation, so you get a nice general proof, rather than some specific bad cases where it's not clear how to extend them to every instance.

Edit: Oh, and looking at modified versions of the puzzle is also a common way to gain insight. In particular if you try the second part of the question first, about five towns, and then analyse the solution there.

I believe there is only one solution for 5. The solution is obviously special if you look at it / draw it right, and should give you quick insight into how to do 6.

By all means feel free to give up.

That looks solid to me. Definitely an interesting and elegant attack, why did you go about it like that? I did it by induction on the number of stops, though I did read a nicer answer than either of these (it was closer to your idea).

Thank you kindly. I saw the opportunity for an induction proof but figured that one based on contradiction would be good for non-mathematicians to read. In my experience, it's the easier method to grasp and apply.

Actually, the depot that I call T₂ is a depot from which one can complete the entire route, it just seemed easier to claim a contradiction than to prove it. Starting at T₂ one arrives at T₁ with enough fuel to cover precisely the distance which one would have to push the car if one had started at T₁ instead. That would probably be a nicer way to do it.

I think that was a fairly round about way of doing it, tbh.

It was quite literally a round-about way of doing it in that one went all the way around the route. But it was my first stab at the problem. Here's my second draft. This is probably closer to the nicer way that Xei alluded to.

Choose any depot to start at and procede, stopping to collect all fuel, until the car runs empty. Transport the car to the next fuel depot and repeat the process until arriving at the starting depot. At this point the car was transported without fuel for some distance D. Yet enough fuel was collected (by the terms of the problem) to go all the way around the route precisely once. Hence there is enough fuel to travel for D distance in the tank. If D is zero, then the problem is solved. Otherwise, there is some last stop into which the car needed to be tranported. Were one to start at this stop, then one would reach the original stop with enough fuel to drive a distance of D which is precisely the amount of fuel that is needed to traverse the route from that point.

I'm not sure if it's about social situations per se, but rather domains of human intuition in general... for instance, if I phrased the question as,

'A friend claims that bananas with green skins are sweeter with ones with yellow skins. I want to prove him wrong. I'm given two unpeeled bananas, yellow and green, and two peeled bananas, one of which tastes sweet and the other of which not. What information do I ask for to try to disprove my friend'?

is that any harder?

Originally Posted by tommo

I think that was a fairly round about way of doing it, tbh.

I don't understand what this means at all... it's a simple and precise answer, what did you want instead..?

Originally Posted by PhilosopherStoned

Choose any depot to start at and procede, stopping to collect all fuel, until the car runs empty. Transport the car to the next fuel depot and repeat the process until arriving at the starting depot. At this point the car was transported without fuel for some distance D. Yet enough fuel was collected (by the terms of the problem) to go all the way around the route precisely once. Hence there is enough fuel to travel for D distance in the tank. If D is zero, then the problem is solved. Otherwise, there is some last stop into which the car needed to be transported. Were one to start at this stop, then one would reach the original stop with enough fuel to drive a distance of D which is precisely the amount of fuel that is needed to traverse the route from that point.

Yeah that's pretty much the same. Certainly much closer than my brute forcing by induction.

Spoiler for Simple solution to 4:

Drive around the route in a different vehicle from some random position, with a large tank and enough fuel to make it without using the fuel stops. Fill up at each stop anyway, and record how much more or less fuel there is in the tank than your starting amount. To get all the way around using only the fuels in the stops, simply start from the stop where you recorded having least fuel arriving there. The amount of fuel in your car can never drop below this level, so you can make it all the way.

I thought that's what you were saying before.... Never mind my feeble mind then.

Originally Posted by tommo

I thought that's what you were saying before.... Never mind my feeble mind then.

It's basically the same. I just phrased it as a constructive proof rather than as a proof by contradiction. This is simpler because by the time one had done enough work to get the contradiction, one had already constructed a solution to the problem.

Originally Posted by PhilosopherStoned

It's basically the same. I just phrased it as a constructive proof rather than as a proof by contradiction. This is simpler because by the time one had done enough work to get the contradiction, one had already constructed a solution to the problem.

Ah! ok. Yes that's what I gathered, cool.

Originally Posted by Xei

I don't understand what this means at all... it's a simple and precise answer, what did you want instead..?

It was a joke. Because

This is simpler because by the time one had done enough work to get the contradiction, one had already constructed a solution to the problem.

And he had to go.... around.... and stuff

Originally Posted by Xei

Spoiler for Simple solution to 4:

Drive around the route in a different vehicle from some random position, with a large tank and enough fuel to make it without using the fuel stops. Fill up at each stop anyway, and record how much more or less fuel there is in the tank than your starting amount. To get all the way around using only the fuels in the stops, simply start from the stop where you recorded having least fuel arriving there. The amount of fuel in your car can never drop below this level, so you can make it all the way.

Is this even proof? Why can you get all the way around from the stop where you had the least fuel? Just because it cannot drop below that level, doesn't mean you can make it to the next stop.

I wanted to comment on problems 5 and 6. These same puzzles were just mentioned in the book I'm reading, The Tipping Point.

You are right, Xei, that it is the same problem framed two different ways and that most people get 6 right, but not 5. The reason for this is that humans are inherently social creatures. We understand and can solve problems involving people much easier than abstract problems.

The amount of fuel in your car will go up and down by the same amounts that it went up and down in the test car with the large tank with ample fuel. You start at the stop where the test car had least fuel. When you start there, you have 0 fuel. So the least fuel you will have on your journey will be this amount; 0. Hopefully this now makes it clear that you can make the journey. You collect the fuel at the stop, and the amount never falls lower than 0. If you always have fuel during the journey, there won't be any point at which you stop.

Late to the party, but I thought I'd take an honest stab at a few of these.

Originally Posted by Xei

1. You have a row of 1,000 coins, all of which are heads up. You flip the second, fourth, sixth, and all other even coins over so that they're tails. Then you flip over the third, sixth, and all other coins which are multiples of three. Now you do this for every fourth coin, every fifth coin, etcetera, all the way up to every thousandth coin (which means just flipping the last one).

The question is this: which coins are heads up, and why? There is a concise and surprising answer.

Given that flipping over that many coins is tedious (and that I can't trust the help staff not to steal any of them), I've hired a team of software engineers to write a script to simulate the sequence. In full 3D and everything!

I'll get back to you when they're done.

Originally Posted by Xei

2. What's the maximum number of pieces you can cut a pizza into using a cheese wire (i.e. straight cuts) if you can only cut six times? The pizza is to thin to cut horizontally, and too hot to pick up and move around.

Well, since the pizza would likely be a novelty appetizer, I would be content having it cut into 16 quaint squares. No need to maximize the piece yield per pizza, as the kitchen staff can always make more.

Originally Posted by Xei

3. There's a 1% chance that the average person has cancer. Somebody with cancer has an 80% chance of testing positive when they go for a scan. Somebody without cancer has a 9.6% chance of testing positive (i.e. getting a false positive). You go for a scan and receive a positive result. What is the probability that you have cancer?

Well, balls.

Either way, the statistics are unimportant. I will assume a 100% chance and solicit myself for pity sex and Reddit karma immediately.

I'll get a second opinion tomorrow.

Originally Posted by Xei

4. 100 people out of a group of 10,000 are tree-huggers. You ask all 10,000 people if they hug trees or not. 80 out of the 100 people who hug trees are honest about it and say yes. 950 out of 9,900 people who don't hug trees also pretend that they do. What fraction of people who claim to hug trees are genuine tree-huggers?

An important question, as we must quickly find and detain these genuine tree huggers indefinitely and without trial.

A cursory review shows 80 people out of the 1030 who claim to be tree huggers do, in fact, hug trees. However, this isn't good enough. We must find the 20 out of 8970 people who are covering up their true identity.

Furthermore, I propose action against these supposed tree hugging sympathizers who would lie to my face about their loyalties to industry, capitalism, and imperialism!

I will have my father send notice to his government contacts immediately.

Originally Posted by Xei

5. I have four cards with numbers on one side and colours on the other. I claim that cards with even numbers are red on the other side. I put the cards in front of you. They are

1, 2, red, blue.

Which cards do you need to turn over to check my claim?

While I only need to turn over 2 and blue to check your claim, I will instead flip them all over to ensure you aren't trying to conceal any tree hugger secrets from me.

I'm on to you lot.

Originally Posted by Xei

6. In the UK you can drink at 18. There are four people drinking at the bar. One is 21, one is 16, one is drinking coke, one is drinking beer. What do I have to check to make sure nobody is breaking the law?

Are you a police officer? If so, good. I need you to come with me immediately; the tree huggers are holding a peaceful protest--where the new mall is planned to go up--and they must all be arrested so they can face proper justice!

If not...why are you bothering these people? What gives you the right, huh? HUH?!

Originally Posted by Xei

7. In any group of (two or more) people, there are always at least two people with the same number of friends in that group. True or false?

N.B. you can only be friends with somebody who is friends with you!

Don't be silly. My father taught me at a very young age that there's no such thing as "friendship." Only money and what you can buy with it.

Originally Posted by Xei

8. On Hawaii, everybody has mobile phones, but naturally there are some people who don't know the phone numbers of other people (and if Alohi has Laka's number, Laka doesn't necessarily have Alohi's number).

We say that Alohi can communicate with Laka if Alohi knows the number of somebody who can then phone somebody else and so on until finally somebody phones Laka.

It can be shown that this splits the island into a distinct number of communicating groups, in which everybody can communicate with everybody else in the group.

Show that there is at least one communicating group which can't contact anybody outside the group.

[Part 2: show this isn't true for an infinite island].

Some lonely bastard either has no one's number, or no service plan with which to place a call, and thus can only communicate with himself, putting him outside of all other communication circles. Sad, really.

Also, an infinite island is impossible because with infinite land, you can have no surrounding water. Come on, man.

Originally Posted by Xei

1. I draw N circles on a piece of paper. Prove that they divide the paper into no more than N(N - 1) + 2 regions.

Hard to say unless I know how good you are at drawing circles. At how high an N do your muscles start to spasm?

It might be better to have a robot built to figure this out. Or better yet! I'll add this simulation to my software engineers' to-do list!

Originally Posted by Xei

2. x is any number on the number line such that x + 1/x is a whole number. Prove x^n + 1/x^n is also a whole number, for any positive whole number n.

Well, since x can only ever be 1 or -1, and 1 to the power of any whole number n is always one, you can simplify x^n + 1/x^n to the form x + 1/x and be done with the whole mess.

How do I know x can only ever be 1 or -1? I paid people to do that part for me; that's how.

Originally Posted by Xei

3. We have the set of consecutive numbers from 1 to some even number, for instance, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

Now we take one more than half of the numbers from this set, for instance, {2, 3, 4, 7, 9, 10}.

a) Prove that in this new set, there will be two numbers with no common factors (for instance, 4 and 9).

b) Prove that in this new set, one number will divide another (for instance, 3 and 9).

The guy I paid to work through this one said something about there always being at least one even or odd number in each of the 210 odd sets, but I was too busy talking to people in my communication group and exploiting my positive cancer results to pay attention.

Originally Posted by Xei

4. You wish to drive around a mountain. There are fuel stops at various places on the road, and the total amount of fuel in them is exactly enough to make the journey once. Prove that it's possible to make the journey in a car with an empty fuel tank from some fuel stop.

Why drive around the mountain when I can have a tunnel built through it and have fuel delivered to me via helicopter?

That seems much more fun.

Originally Posted by Xei

5. There are six towns. Between each pair of towns is either a bus service or a train service, but never both. Prove that you can travel through three towns in a loop (without visiting any others) with only one type of transport.

Does this have to be true for five towns?

Given that I have access to private helicopters, I could care less about the public transport options and instead fly wherever I'd like. Be it in a loop between three towns, or no.

But alas, the second and third tests I got all came back positive and the MRI scan have revealed a malignant tumor growing in my neck. There's little chance of recovery at this point, so screw these towns, I'm going somewhere more interesting with the little time I have left...

Originally Posted by Xei

The amount of fuel in your car will go up and down by the same amounts that it went up and down in the test car with the large tank with ample fuel. You start at the stop where the test car had least fuel. When you start there, you have 0 fuel. So the least fuel you will have on your journey will be this amount; 0. Hopefully this now makes it clear that you can make the journey. You collect the fuel at the stop, and the amount never falls lower than 0. If you always have fuel during the journey, there won't be any point at which you stop.

But how do you know that it only hits zero once

tommo think he funny bunny

Tommo, you need to get a better web-cam. You look like ace ventura.

Very nice once again.

Have you stumbled across any puzzles yourself for a lay audience?

Originally Posted by Xei

Have you stumbled across any puzzles yourself for a lay audience?

Haven't been looking. One interesting extension to (2) would be to determine for which integers, n, some real x exists with x + 1/x = n. One could then prove that as n -> infinity, x/n approaches 1. This is more amateur mathematician stuff than "lay audience" friendly though. If we let x(n) be such that x(n) + 1/x(n) = n, and let w_n(k) = x(n)^k + 1/x(n)^k, then it might be interesting to explore the sequences w_n(k) as k tends to infinity. Right now we just have the recurrence formula w_n(k) = w_n(k - 1)n - w_n(k - 2).

Here's one:

100 passengers are queuing to board a plane (which seats 100 passengers). The first passenger has lost his boarding pass and so sits in a random seat.

The remaining passengers board one at a time. If their seat is free, they take it. Otherwise, they sit in a randomly chosen seat.

What is the probability that the last passenger who boards will be able to sit in his allocated seat, and why?

Edit: Oh and also assume that the first passenger isn't thrown off for not having the pass, and that the remaining passengers don't refuse to fly for fear of the incompetence displayed by staff and airport security.