## Tuesday, July 31, 2007

### Brain Teasers

I like brain teasers. I discovered a site with a few fun questions yesterday, and here are some of the questions:

A train leaves New York for Boston. Five minutes later another train leaves Boston for New York, at double the speed. Which train will be closer to New York when they encounter?
If I go halfway to the town (which is 60 km away) at the speed of 30 km/hour, how fast do I have to go for the rest of the way to have the average speed of the entire way 60 km/hour?
These are the conditions in a town:
1. No two inhabitants have the same number of hairs on their head.
2. No inhabitant has exactly 518 hairs.
3. There are more inhabitants than any inhabitant's hair in the town.
What is the highest possible number of inhabitants?
An Arab sheikh tells his two sons to race their camels to a distant city to see who will inherit his fortune. The one whose camel is slower will win. The brothers, after wandering aimlessly for days, ask a wise man for advice. After hearing the advice they jump on the camels and race as fast as they can to the city.

What does the wise man say?
Three masters of logic wanted to find out who was the wisest one. So they invited the grand master, who took them into a dark room and said: "I will paint each one of you a red or a blue dot on your forehead. When you walk out and you see at least one red point, raise your hands. The one who says what colour is the dot on his own forehead first, wins." Then he painted only red dots on every one. When they went out everybody had their hands up and after a while one of them said: "I have a red dot on my head."

How could he be so sure?
Imagine you are in a room with 3 switches. In an adjacent room there are 3 bulbs (all are off at the moment), each switch belongs to one bulb. It is impossible to see from one room to another. How can you find out which switch belongs to which bulb, if you may enter the room with the bulbs only once?
Check out the solutions in the puzzle page! :) Sharpening our mind sometimes helps it works better.

day-dreamer said...

Before I start clicking on any links, I can only think of a solution for the first two questions only, which I'm not even sure is correct.

1. Both trains are equally near to NY when they encounter?

2. 60 km/hour?

Okay, time to check answers.

ShouFarn said...

1. both equally near
2. umm, you can't, can you? i mean, if you reached halfway (30km) with 30km/hour, then you've already used your 1 hour up.
3. 518 inhabitants, accounting for the bald man.
4. Wise men told them to use each other's camel
5. can't figure this one out arghh
6. Heh, assuming they are edison bulbs, and you may touch them, you switch one on for 5 mins, turn it off. then switch on another one. Walk into the room. Feel for the one that is hot but off, look at the one that is on, and feel for the one that is cool. :p

If not, then... i dunno haha.

Now to check the answers...

bluez_aspic said...

Here's some more from the new Paradox magazine =)

1. I am thinking of one of three numbers: 1, 2, or 3. You may ask me exactly one yes-no question to find out what number it is, and I will answer truthfully (yes, no, or I don't know). What do you ask?

2. You are blindfolded before table, and on the table there are some coins - exactly 247 of which are heads up. How can you divide all of the coins into 2 piles, such that each pile has the same number of heads facing up?

3. Two strings each burn for exactly 1 hour at uneven rates. How would you measure 45 minutes with the candles and some matches?

youngyew said...

Hey bluez_aspic, your questions are interesting! :D

To clarify, what does it really mean by "burn at uneven rates"? Does it mean different parts of the same string will have different rate of burning?

I haven't done number 2, by the way. But no.1 piqued my interest, and I constructed a few answers for that (and grrh, spent some of my time which should have been spent doing work :P)

My first answer was this: "I took some 1s, 2s and 3s, and by doing some multiplications using only these numbers, I got a product of 18. Now, have I used your number twice in the multiplication process?"

Yes: 3
No: 2
Don't now: 1

The answer above took me a good half 20 minutes or so. But once I got it, some other answers dawned upon me.. For example:

"I am an eccentric drinker, whenever I have a number of glasses of beer, I will drink at least one glass but I will also drink at most two glasses, sometimes when I feel like it. Now I began with N glasses of beer (N being the number you thought of), would you still have beer for yourself when I am done?"

Yes: 3
No: 1
Don't know: 2

There's also another variation:

"There are three items on the floor: an empty matchbox (with the abrasive surface at the side), a blue match and a red match. Now, if I take up N items randomly from the floor (N is the number you thought of), would I be able to light up a match?"

Yes: 3
No: 1
I don't know: 2

The last one, is a cheat and might not be accepted if you are strict about the definition of a yes/no question:

"Now I define the number's names as follows: 1 is called yes, 2 is called no, 3 is called 'I don't know'. What is the number you are thinking?"

:P

bluez_aspic said...

I love your first answer! (multiplication of 1s, 2s and 3s yielding 18) It's by far the most simple and elegant one I've heard.

I came up with two answers - the first of which is a variation of yours ("I am thinking of one of two numbers: 2 or 3. Is my number greater than yours?"), and the second one unfortunately can only be posed to mathematicians ("Is the Ramsey R(N+2, N+2) - where N is the number you chose - divisible by 9?").

For the second question on dividing heads into two piles, you're allowed to do *anything* with the coins (except destroying them). Uneven rate of burning means that, say, burning the first half a string might take longer than burning the second half - so you can't measure off 30 minutes by burning half of the string.

And quick, submit your solutions - there're cash prizes to be won! (http://www.ms.unimelb.edu.au/~paradox/archive/issues/p07-2.pdf - scroll down to the last few pages)

I won't be jeopardize your chances (i.e. not submitting), so you're in good stead to win (armed with the wonderful '18' answer) :)

youngyew said...

For the string question, I assume it's a candle string?

Let's say the strings are called string A and B..

1. Burn TWO ends of string A and one end of string B. By the time string A is finished, half an hour must have transpired and we are left with half an hour of string B.

2. Now burn the other end of string B. Since string B has only half-an-hour's worth of string remaining, burning two sides of string B would make it finish in 15 minutes.

Viola! We have 30 minutes from the first part and 15 minutes from the second part. :)

Anyway I can't get question 2. Unless you can differentiate the head and the tail with your fingers, or ask your friends for help, I haven't really figured out how to do it.

The only manoeuvres I thought possible were flipping the coin, putting one coin on top of the other, or stacking towers of coins. Are those manoeuvres acceptable, and are there other manoeuvres that I haven't thought of?

bluez_aspic said...

If you had, say, 9 coins which were all heads, the only way you could divide them into two equal piles is to flip one (or some) of them.

Notice too that the total no. of coins in the question is not specified.

youngyew said...

I understand that the total number is unspecified. However, I tried to do a simplified version of 3 head 1 tail, but I can't even do it on this simplified version.

Say you flip one of them, so it would become either 2h2t or 4h. It's fine if it's 4h, but if it becomes 2h2t you still struggle to find out which is which.

Say you flip two of them, it would become either 1h3t or 3h1t. Doesn't really help.Flipping three of them would produce 4t or 2h2t. Flipping all four would produce 1h3t.

It seems that only 4t or 2h2t are the desirable results; but even with this result you still can't do much, can you?

Say we have 2h2t, then flipping 1 gives 1h3t or 3h1t. Flipping 2 gives 4t, 2h2t or 4t.

Etc. Etc.

Can you give me a hint where I went wrong in this thought process?

ShouFarn said...

He told me the answer, so i'll give the hint:

Think sets.

I love the string idea btw. I was working on that last night (well, while reading my notes, didn't go over too well for either), but yeah. I never considered the possibility of burning both ends. (mainly because i read the original article and it says "candles" not strings, so i didn't think of burning candles at both ends.)

But for string it works beautifully. *praise praise*

youngyew said...

Ah damn I got lazy and asked Shou Farn about the answer. Ingenious! I thought of dividing it to two sets, and also thought of doing something on the sets; but I didn't arrive at flipping the whole set consisting of 247 coins.

The coin problem has been more challenging to me than the others. Funny it's rated as easier than the string problem.