Medium | Discrete Maths |

A rabbit sits at the bottom of a staircase with n stairs. The rabbit can hop up only one or two stairs at a time. What kind of sequence is depicted by the different ways possible for the rabbit to ascend to the top of the stairs of length n=1,2,3...?

Hint

Recursion

Answer

Fibonacci Sequence.

Solution

Suppose f(n) are the number of ways to reach nth stair. Notice that the final hop is either a single jump or double jump, i.e. its from (n-1)th stair or (n-2)th. Thus f(n) = f(n-1) + f(n-2), where f(0)=f(1)=1. This is Fibonacci sequence.

Medium | Discrete Maths |

A. B & C live together and share everything equally. One day A brings home 5 logs of wood, B brings 3 logs and C brings none. Then they use the wood to cook together and share the food. Since C did not bring any wood, he gives $8 instead. How much to A and how much to B?

Hint

Its not 5 & 3

Solution

Since each person consumed 8/3 woods. A gave 5-8/3 = 7/3 woods to C and B gave 3-8/3 = 1/3 woods to C.

So, Out of the 8 dollars, A gets 7 and B gets 1

So, Out of the 8 dollars, A gets 7 and B gets 1

Source: CSEblog

Enable Like and Comment Medium | Strategy |

Suppose you have a hotel which has one floor with infinite number of rooms in a row and all of them are occupied.

1) A new customer wants to check in, how will you accommodate her?

2) What if infinite number of people want to check in, how will you accommodate them?

3) Suppose infinite number of buses arrive at the hotel, each having infinite number of people, how will you accommodate them?

1) A new customer wants to check in, how will you accommodate her?

2) What if infinite number of people want to check in, how will you accommodate them?

3) Suppose infinite number of buses arrive at the hotel, each having infinite number of people, how will you accommodate them?

Hint

Define Infinity ;)

Solution

1) Since there are infinite number of rooms and infinite+1= infinite

Just ask person in room k to move to k+1, thus making the first room vacant. :)

2) In the other case, since infinite+infinite = infinite

asking person in room k to move to 2k solves the problem.

3) Since NxN is countable set. We can get a 1-1 mapping from N to NxN

Hence, we can accommodate (infinite people X infinite buses) in the hotel.

Relevant article:

http://en.wikipedia.org/wiki/Cantor_pairing_function

Just ask person in room k to move to k+1, thus making the first room vacant. :)

2) In the other case, since infinite+infinite = infinite

asking person in room k to move to 2k solves the problem.

3) Since NxN is countable set. We can get a 1-1 mapping from N to NxN

Hence, we can accommodate (infinite people X infinite buses) in the hotel.

Relevant article:

http://en.wikipedia.org/wiki/Cantor_pairing_function

Source: CSEblog

Enable Like and Comment Medium | Probability |

In this gambling game, a player can buy a ticket for Rs 1 on any number from 1 to 6. Three identical and unfair dice are rolled. If the booked number appears on 0, 1, 2 or 3 dice, player wins Rs 0, 1, 2 or 3 respectively, without returning the original Rs 1. What is expected money you can win after buying a ticket for Rs 1?

Hint

Book the full house

Answer

1/2

Solution

We bet Rs 1 on each number 1-6. In any case, we get Rs 3 back. That is 1/2 per ticket. Hence the expected amount of money we can win is 1/2. A tedious way to arrive at this answer is to calculate the probability of getting 1, 2 or 3 faces common to our booking.

Source: 50 puzzles in prob

Enable Like and Comment Medium | Discrete Maths |

A group has 70 members. For any two members X and Y there is a language that X speaks but Y does not, and there is a language that Y speaks but X does not. At least how many different languages are spoken by the members of this group?

Solution

8 choose 4 is 70

Source: Quantnet

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