Quantitative Research

‘N’ people are standing in a circle numbered from ‘1’ to ‘N’ in clockwise order. First, the person numbered 1 will proceed in a clockwise direction and will skip K-1 persons including itself and will kill a Kth person. Now (K+1)th person from 1 will start and will kill the Kth person from itself.

You have to find the position of the last person surviving with respect to initial numbering.

Note:

A person can also kill himself.

Ninja and his friend are playing a game called flip game. They are given a string ‘STR’ containing only two characters, ‘$’ and ‘*.’

In this game, Ninja and his friend take turns to flip two consecutive “$$” to “**”. The flip game ends when Ninja or his friend can no longer make a move, i.e., there is no consecutive “$$” present in the ‘STR’ and, therefore, the other person will be the winner of the game.

Both the players play the game optimally in alternate turns. Given that Ninja starts the game i.e. takes the first turn, your task is to find out if he wins the game.

For example:

Let 'STR' = "$$**".

There are consecutive "$$" in 'STR' and it is Ninja's turn to begin the game. So, Ninja changes the consecutive "$$" to "**". This changes 'STR' to "****". 

Now, as it is his friend's turn and there are no more consecutive "$$", Ninja wins the game.

Let’s say you have a biased coin, but you don’t know the probability p of it landing heads. How would you estimate the value of ppp? I was expected to discuss confidence intervals and variance.

The expected number of throws required to get two consecutive sixes when using a fair six-sided die.

This was a game puzzle.

There are two players on the same team; let’s call them A and B. The host has an 8×8 chessboard with a coin placed on each of the 64 cells. Each coin can show either heads or tails, as decided by the host.

The host shows the chessboard to A and whispers a secret number xxx (from 0 to 63) into A’s ear. Now, A has to help B guess this secret number. The only action A can take is to flip exactly one coin on the board (or choose not to flip any) before showing the board to B.

B must then guess the number by observing the final state of the chessboard. If B guesses correctly, A and B win; otherwise, they lose. They are allowed to strategize before the game begins.

Can they win the game?

Yes, they can.

The solution uses the XOR operation. Number each cell on the chessboard from 0 to 63. A computes the XOR of all indices where the coins show heads.

• If this XOR value is equal to the secret number xxx, A does nothing and passes the board to B.
• Otherwise, let the XOR value be ccc. Compute y=c⊕x. A flips the coin at position yyy.

After this operation, when B computes the XOR of all indices showing heads, the result becomes c⊕y=x, allowing B to correctly determine the secret number.

This is the expected solution.