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 | 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

Enable Like and Comment Medium | Discrete Maths |

At a party of N people, some have a symmetric friendship. Symmetric means that if A is friends with B, then B is in turn friends with A. Prove that there are at-least two people with same number of friends.

Solution

Source: Top Quant Interview

Enable Like and Comment Medium | Discrete Maths |

An 8x8 chessboard can be entirely covered by 32 dominoes of size 2x1. Suppose we cut off two opposite corners of chess (i.e. two white blocks or two black blocks). Prove that now it is impossible to cover the remaining chessboard with 31 dominoes.

Hint

The two diagonally opposite corners are of the same color.

Solution

The two diagonally opposite corners are of the same color. A domino covers adjacent faces & hence a domino always covers 1 black and 1 white square. The 31 dominoes will cover 31 blacks and 31 whites. The chess has 30 & 32 square instead. Hence this can't be done.

Source: Martin Gardner

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