Tumblers on a Turn Table Puzzle

You are blindfolded. There is a square turntable in front of you with four tumblers at four corners. Each might be oriented "up" or "down". You make a sequence of "moves". Each move consists of five phases:
(1) A referee turns the table an unknown number of quarter-turns(90,180, 270 or 360 degrees);
(2) You select "adjacent" or "opposite", and then take a pair of tumblers which are either "adjacent" (90 degrees apart) or "opposite" (180), accordingly.
(3) You observe (by touch) the current orientation of these two tumblers.
(4) You turn over one or the other tumbler, or neither, or both.
(5) The referee tells you whether all four tumblers are in the same orientation; if so, you win and the game is over.

After how many moves can you guarantee that you will win no matter whats the starting position or whatever way referee plays?
What is your strategy?

Solution: tumblerPuzzleSolution.pdf

Why Euler Constant ‘e’ is used as preferred logarithm base?

Euler Constant

When I read about logarithm for first time in High school I understood as it as “The logarithm of a number is the exponent to which a base(usually 10) must be raised to produce that number”

For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: 1000 = 10 ×10 ×10 = 10^3. More generally, if x = b^y, then y is the logarithm of x to base b, and is written y = log_b(x), so log₁₀(1000) = 3. This was quite reasonable to understand.

Handshakes in a Party Puzzle

You went to a get together with your wife where three of your friends arrived with their spouses. One couple among your friends were "Raj" and "Kiran". When every one arrived, people shook hands with one another but nobody shook hand with oneself(naturally) or with one's own spouse.

After the party you spoke to each person(except yourself) and asked them how many people they had shaken hands with. Each person replied with a different number.

You shook hands with "Raj". The question is:
1. Did your wife also shake hands with Raj?
2. Did you or your wife shake hands with Kiran?

Solution: HandshakePuzzleSolution

Bowling Pin Arrangement Puzzle

Once you went to bowling with your friend. All the bowling pins were labelled with different integer from 0,1,2 .. to 9. The arrangement of pins (in a triangle form) was very interesting. If two pins are side-by-side, the sum of their labels (reduced modulo 10) is equal to the label of the pin in front and between them. For Example

1   2   4   5

  3   6   9

   9   5

   4

BFS, DFS and Connect Graph Algorithms

 

In mathematics, and more specifically in graph theory, a graph is a representation of a set of objects where some pairs of objects are connected by links. Graphs are incredibly important part of Modern day life. We come across them almost daily. For example we can reach our office via separate ways possible but which one to take(Considering distance, traffic). Whole of social media like Facebook, twitter, linkedin is nothing but a huge graph with people acting as node and their relationship(friendship) acting like links or edges. Computer networks, electricity distribution network, city pipeline network, road network, airline network, Internet are nothing but a form of a Graph. There are several types of Graphs such as Directed, Undirected, Tree, Weighted, Cyclic, Acyclic etc to name a few of them. And we find examples of almost all of them in real life.