Puzzle 7 - Roads Again!!

The natives of Roadland are at it again. They have now upgraded their network so that their is exactly one path between any two cities in Roadland. They have moved onto the next level of urbanisation and now want X-Way outlets (these serve yummy yooters: very popular in Roadland). But as the are Roadlanders, they have their set of constraints. The X-Way mandarins would set up their supply stations in some of the Roadlandian cities, but they would oblige a city only if the maximum of its distance from all other Roadlandian cities is the least possible. At most how many supply centres can their be? Oh, I forgot to tell you that all Roadlandian roads mysteriously have the same length. (This problem is based on a very simple and well known graph theoretical result) - The problem was given by Piyush Srivastava (currently in University of California, Berkley) for NERD Volume 1 Number 2.

Puzzle 6 - The Number Game

This problem is simple, how would you find the last digit of the integer part of a^2000, where
a = 3+7^0.5? (Based on a problem from Google CodeJam) - The problem was given by Piyush Srivastava (currently in University of California, Berkley) for NERD Volume 1 Number 1

Extra Puzzle - How many roads?

In the state of Roadland (sorry for the terribly unimaginative name), there are 42 major cities. Some of these cities are connected by roads, where each road connects exactly two cities, and there is at the most one road between any two cities. Inhabitants of Roadland also try to remove obvious superfluity from there road network: so if any three cities A, B and C are such that there is a road from A to B, and one from B to C, then there would not be any road from A to C. How many roads can there be in Roadland? (Based on a well known but simple graph theoretical result)

Puzzle -5-Gluing Pyramid

This is an easy but beautiful puzzle of Geometry.
A Solid square based pyramid,with all edges of unit length,and a solid triangle-base pyramid(tetrahedron),also with all edges of unit length, are glued together by matching two triangular faces.
How many faces does the resulting solid have?

Puzzle 4 - Strange Multiples

Number theory has been beautiful and oldest subject of mathematics.It's simplicity create a intrinsic interest in people.This is the reason we have many simple and beautiful conjecture and at the same time very difficult to prove.So this time we have number puzzle for you.

Let n be a natural number.Prove that
a)n has a (non-zero)multiple whose representation(base 10) contains only zeros and ones
b)2^n has a multiple whose representation contains only ones and twos.

Puzzle 3 - Again, x,y

After playing with the chess board, lets have some some fun with cryptic and predict. Hoping your answers the hint of 'logic'. With the next puzzle, you can post the answer.

Which two numbers come at the end of this sequence?
2, 4, 6, 30, 32, 34, 36, 40, 42, 44, 46, 50, 52, 54, 56, 60, 62, 64, 66, x, y

Puzzle 2 - Play to win

Hello friends!! We are back with our second posting.

Before getting started with the main puzzle, lets build up some confidence with famous interview question.

"You have 100 coins labeled 1 to 100 on them. They are put up in line randomly, such that no order maintained. Its your turn first and you can pick up any coin which is either first or last. At the end of 5oth round, whoever will have higher the summation of coin-values will be the winner. If you are given first chance, find out winning strategy (or at least 'non loosing' strategy)."

Now lets move on to the main puzzle.

"Sid begins by marking a corner square of n by n chessboard. Sud marks an orthogonally adjacent square. Thereafter, Sid and Sud continue alternating each marking a square adjacent to the last one marked, until no unmarked adjacent square is available at which the player whose turn it is to play loses.

For which n does Sid have a winning strategy??"