Can the mouse eat the cube?
A mouse eats its way through a 3*3*3 cube of cheese by tunnelling through all of the 27 1*1*1 subcubes. If he starts at one corner and always moves on to an adjacent uneaten subcube, can he finish at the center of the cube?

More on Primes

Prove that there is atleast one prime number between n and 2n.

Consecutive no-prime numbers
Tell me 1000 consecutive non-prime numbers.

I am not naming this! Well, I name it
Handshakes
At a recent convention of Scientists, there was a lot of handshakes between the scientists. But the number of Mathematicians in attendence who shook hands an odd number of times was even, why?! Remember that an handshake requires exactly two different people!

The Frog and the Wall
Suppose there is a frog at a certain distance from a wall, say, ten feet. Imagine that this frog jumps halfway to the wall, so that he is five feet away. Suppose he continues to jump toward the wall, each time getting halfway there. Will the frog reach the wall?

Another Question will follow in this context!

The Structure of the Universe
We say that space is three-dimensional. But in the world of Mathematics, though, we are not limited by physical constraints!! We might have four, five, or six geometric dimensions, or even infinitely many!!!

Let's go back to two-space. Thos might be a plane or the surface of a smooth objects. Suppose we have little cretures on this two-dimensional surface. Their movements are limited compared to what we are used to.

Imagine one of these two-dimensional creatures, imprisoned inside a square. He cannot get out. But it is easy for us to think of a way to get him out: reach in, grab him, yank him off the two-dimensional surface that is his universe, and plop him out down outside the square! So easy for us, but impossible for him, or anyone else in his universe.

We can extend this idea to our own universe. Suppose somebody is locked in a cubicle that cannot be broken open. There is no escape unless the walls are breached. But a four-dimensional being, looking at this poor three-space person, could quite easily grab him and pull him ouot without the walls of the cubicle even being touched. We cannot envision how this could be done, but mathematically it could be, if we allow four dimensions to exist. More on this in Flatland.

There is another way a person could escape from the cubicle, and the method is easy for us to imagine. Figure it out!!

Tricky Track
Three High schools-Washington, Lincoln and Roosevelt-competed in a track meet. Each school entered one man, and one only, in each event. Susan, a student of Lincoln High, sat in the bleachers to cheer her boyfriend, the school's shot-put champion.

When Susan returned home later in the day, her father asked how her school had done.

"We won the shot-put all right," she said,"but Washington High won the track meet. They had a final score of 22. We finished with 9. So did Roosevelt High."

"How were the events scored?" her father asked.

"I don't remember exactly," Susan replied, "but there was a certain number of points for the winner of each event, a smaller number for the second place and still a smaller number for third place. The numbers were the same for all events." (By "number" Susan of course meant a positive integer.)

"How many events were there all together?"

"Gosh, I don't know, Dad. All I watched was the shot-put."

"Was there a high jump?" asked Susan's brother.

Susan nodded.

"Who won it?"

Susan didn't know.

Incredible as it may seem, this last question can be answered with only the information given. Which school won the high jump?

The Twin Paradox

This blog is as a result of my own attempt to understand a difficult-to-understand paradox, Twin Paradox which is an outcome of of Einstein's relativity (Any physicians around?!). Please tell me if you see anything insightful!(I have question-marked whatever I did not understand!)

Suppose we have two twin brothers, in spaceships somewhere in the intergalactic void where there is no reference point that can be called "stationary." The special theory of relativity asserts that the motion can be ascertained only when there is a reference frame (this one is okay for me!). Suppose these ships are initially hovering close to each other and that there is no relative motion. The twins-call them Jim and Joe-syncronize their watches so that they read the same date and time. What will happen if Joe speeds off to some distance place, at some high speed, and returns to Jim, who sits patiently waiting for him without using his engines during that time? According to the special theory of relativity, Jim will see Joe's time flow slow down while Joe is moving(???). Joe may not have to travel very far away from Jim at all; he might go around Jim in circles for a while, or zig-zag in Jim's vicinity, periodically checking the time. When Joe is through, we can conjecture that Joe's clock will be behind Jim's clock by a certain amount that depends on Joe's average speed and duration of this trip(is there a formula for this???). Joe will be younger than Jim by that amount when the journey is over. (This is okay for me, provided I understand the previous ?-marks!).

But who says Joe is moving and Jim is stationary. If Joe pulls away from Jim and goes a few light years into space and then comes back, is this different than Jim going away from Joe ? Relatively speaking, aren't these two models, one from Jim's point of view and the other Joe's point of view, iddentical? Then we should be able to say that Joe will se Jim's time frame as moving slower than his own, and should find that Jim's clock is behind his own when the trip is over. Now we have a paradox. Because both points are equally valid according to special theory of relativity, it follows that Jim is younger than Joe and Joe is younger than Jim! It is enough to make us want to trash the whole theory of relativity and start over.

We cannot trash the whole theory, however. Its conclusions have been proven to be true! An experiment was conducted not long ago in which an airplane was taken aloft having a payload that consisted of an atomic clock, among other things. The time according to the clock was compared with the time according to another, synchronized, atomic clock on the ground. When the plane returned, there was a discrepancy, though small, that exactly matched the discrepancy according the special theory of relativity.

There is only one way to resolve the paradox,find out how?. I have the answer, but the theory of relativity is still puzzling to me. If someone could explain that to me,....!

I should quote Einstein! "God may be sophisticated but he is not malicious" meaning that although the universe is elegant it is not designed with the intention of being too complicated!

Yesterday, we went out of the city and saw the meterio showers on the sky, saw the milky way. It is wonderful. Thanks to Kasthuri and two other friends. Ok I digressed a lot, I will stop.

A Band Around The World
Suppose the earth were a smooth, perfectly round sphere, with no hills or mountains. Imagine a rope around the earth's equator, strung so that it is snug and does not strech. If we were to add, say, ten feet to this rope, and then prop it up all the way around the planet so that it stood out equally far everywhere, how far above the surface would it stand? We can assume the earth is 8,000 miles diameter, and one mile is 5,280 feet. Do the simple calculation. The answer may surprise you!

A Fixed-Point Theorem One Morning, exactly at sunrise, a Buddhist monk began to climb a tall mountain. The narrow path, no more than a foot or two wide, spiraled around the mountain to a glittering temple at the summit.
The monk ascended the path at varying rates of speed, stopping many times along the way to rest and to eat the dried fruit he carried with him.He reached the temple shortly before sunset. After several days of fasting and meditation he began his journey back along the same path, starting at sunrise and again walking at variable speeds with many pauses along the way. His average speed descending was, of course, greater than his average climbing speed.
Prove that there is a spot along the path that the monk will occupy on both trips at presicely the same time of day.

The Car-and-Goats II
This is a continuation of the previous question... If the car is actually at Door I (probability 1/3), then when you switch you lose; but if it is at Door 2 or Door 3(probability 2/3) then the host's revelation of a goat shows you how to switch and win. Therefore the chance you win by switching is 2/3. Elegant. But in this argument, we are still considering the possibility that the car is behind Door 3; so the host cannot have already opened the door (much less to reveal a goat). In this game, you have to announce before a door has been opened whether you intend to switch.

The Car-and-Goats
A number of mathematicians were thrown into a tizzy by the following problem, which appeared in Marilyn vos Savant's column, "Ask Marilyn", in Parade(a Sunday newspaper supplement):

One of the three doors hides a car (all three equally likely) and other two hide goats. You choose Door 1. The host, who knows where the car is, then opens one of the other two doors to reveal a goat, and asks whether you wish to switch your choice. Say he opens Door 3, should you switch to Door 2?

Random Hats Three people are given hats. Each hat is either red or blue, chosen at random. Each person can see the other 2 hats, but not their own. They each must simultaneously either guess their own hat's color, or pass. No communication is allowed, although they can agree on a strategy ahead of time. What strategy will give them the best chances of at least one person guessing right, and nobody guessing wrong?

Probability While your back is turned, a friend places a penny, nickel, and dime on the table. He arranges them in a pattern of heads and tails provided that the three coins are not all heads or all tails.

Your object is to give instructions, without seeing the coins, that will cause all heads to be the same (all heads or all tails). For example, you may ask your friend to reverse the dime. He must then tell you whether you have succeeded in getting all the coins alike. If you have not, you again name a coin for him to turn. This procedure continues until he tells you that the three coins are the same.

Your probability of success on the first move is 1/3. If you adopt the best strategy, what is the probability of success in two moves are fewer? What is the smallest number of moves that guarantees success on or before the final move.


If you feel the above question easy, here is your question!
The situation is the same as before, except this time your intent is to make all the coins show heads. Any initial pattern except all heads is permitted. As before, you are told after each move whether or not you have succeeded. Assuming that you use the best strategy, what is the smallest number of moves that guarantees success? What is your probability of success in two moves or fewer, in three moves or fewer, and so on up to the final move at which the probability reaches 1 (certainly)?

Guess the numbers
Two mathematicians are placed in two rooms and one is given the sum of two
distinct numbers between 2 and 99 (2 and 99 excluded) and the other is given
the product of same two numbers. They both come out of the room and the
following conversation takes place.

Mathematician with product : "I don't know the two numbers".
Mathematician with sum : "I know, you won't know the two numbers".
Mathematician with product : "Now, I know the numbers".
Mathematician with Sum : "Now, I too know the numbers".

What are the numbers and how did they find it out ?

Hi there, I installed YACCS, so you can comment!

Mirror, Mirror...
I found this problem in "No Ordinary Genius" by Christopher Sykes. It's pretty easy. But I do not have an answer!
You look in a mirror, and let's say you part your hair on the right side. You look in the mirror, and your image has its hair parted on the left side, so the image is left-to-right mixed up. But it's not top-to-bottom mixed up, because the top of the head of the image is there at the top, and the feet are down at the bottom. The question is: how does the mirror know to get the left and right mixed up, but not the up and down?