Which number survives at the last?

100 people are standing in a circle in an order 1 to 100 and 100 adjacent to 1. No.1 has a sword. He kills the next person (i.e. no. 2) and gives the sword to the next (i.e  no.3). This continues until only one person remains. Which number survives at the last? Can you generalize your solution to N people? Hint: First solve the problem for powers of 2.

Birthday on a Sunday!!

It seems to me that people do not value how often their birthday falls on a Sunday, or for that matter a weekend - when many friends and relatives get a chance to wish them. So supposing your birthday falls on a Sunday this year, value it and count the number of years you need to wait until your birthday falls on a Sunday again. Do let me know your answers. Mind it, the wait can be long, but it is so worth it.

Now that you would have thought about this problem in your head, here's something that will make you grab a sheet of paper and a pen. If your birthday falls on a weekend, how long should you wait for a birthday on a weekend again? Do post your answer, for this is bound to get interesting!! Count the number of birthdays in a lifetime :)

Fair experiment with unfair coin!

The puzzle is simple to state: You are given a biased coin with 60% chance for getting heads and 40% chance of getting tails while tossing. Design an experiment with two equally likely outcomes using the coin.

Think out of the box!

2 rooms, 1 light bulb in one room and the other room has 3 switches out of which only one of the light bulb switches on/off. The rooms are not viewable from each other. You as a single person has just one chance to find which switch operates the bulb. How can this be done?

An Interval Graph

Six professors had been to the library on the day when the rare tractate was stolen. Each had entered once, stayed for some time and then left. If two were in the library at the same time, then at least one of them saw the other. Detectives questioned the professors and gathered the following testimony:


A said he saw B and E
B reported he saw A and I
C claimed he saw D and I
D said he saw A and I
E testified to seeing B and C
I said that she saw C and E

One of the professors lied. Who was it?

Unit distance coloring

Imagine that each point on the plane takes a color. What is the minimum number of colors needed to color all the points so that any two points which are at distance exactly 1 unit get two different colors?

The Plane in the Wind

An airplane flies in a straight line from airport A to airport B, then back in a straight line from B to A. It traverses with a constant engine speed and there is no wind. Will its travel time for the same round trip be greater, less, or the same if, throughout both flights, at the same engine speed, a costant wind blows from A to B?

Back with a bang :)

Manupulative codewords - Hofstadter's MIU system

This system of codewords have only stings of letters formed from the alphabets, M, I, U with MI as one of the codewords. You can form other codewords in the system by the following four rules :

x in the below lines can be any string of letters from M, I, U that makes a codeword in each of the rules.

1. If xI is a codeword, then xIU is also a codeword.
2. If Mx is a codeword, Mxx is also a codeword.
3. In any codeword, III can be replaced by U.
4. In any codeword, UU can be omitted.

Questions : Show that MUII is a codeword. Is MU a codeword?

Students in the Library

Five students visited the library on a certain day. The library is open from 8 AM - 8 PM on all days. Each went for a lunch at some time and returned after it. Any two of them met in the library. Prove that at some moment of time three of them were in the library. Show also that this is not always true for four students.

Hint (posted on July 27, 2005): Draw interval lines for each student for the period during which he was present.

Girlie company

25 boys and 25 girls are to be seated in a round table with 50 seats. Prove that no matter what the arrangement is, there would be a person who is seated in-between two girls.

The hole principle

Let A = (a_1, a_2,...,a_n) be a sequence on real numbers with n > k m (k and m - integers). Prove that there is a subsequence of length k +1 of A of which is increasing OR a subsequence of length m +1 of A of which is decreasing.

A non-rational path

Consider R^2 - the Euclidean plane consisting of 2-tuples of real numbers. Remove all the points that have rational co-ordinates from it. For example (1/4, 5/100) should be removed, (1/2, sqrt(7)) should NOT be removed. We are left with what is denoted R^2 - Q^2; Q being the rational points. Pick any two points a and b in R^2 - Q^2. Is it possible to find a path (a line - not nessesarily straight) from a and b in R^2 - Q^2?

If the answer is yes, give a precise path and if the answer is no, why?

Count the sum of the determinants

To each positive integer with n^2 number of digits, we can associate a corresponding determinant by writing the digits in order across the rows.

That is for n = 2, we have n^2 = 4. Thus take all 4-digit positive integers. For 7615, we have the determinant
| 7 6 |
| 1 5 |
which is equal to 35 - 6 = 29.

Can you find the sum of all determinants associated with n^2-digit integers as a function of n?

Just as a motivating example - for n = 1, we have only 9 determinants and their sum is 45. For n = 2, we have 9000 determinants.

A computational Problem

Let x = .1234567891011121314...998999, where the digits are obtained by writing the integers 1 through 999 in order. Find the 1983rd digit to the right of the decimal point.

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!!