Puzzled

Coin Tossing Experiment

2013-10-10T00:17:00.001-07:00

Consider a coin tossing experiment. Let X be the number of heads if you toss a coin for 99 times. Let Y be the number of heads if you toss a coin for 100 times. What is the probability that X is less than Y?

Prime Numbers

2013-09-17T23:05:00.000-07:00

How many prime numbers exist between 2 consecutive prime numbers?

House with too many doors

2013-09-17T03:15:00.002-07:00

Have a look at the house plan at the right hand side, with five rooms and sixteen doors in total. The intention is to make a run through the rooms and pass each of the sixteen doors exactly once.

How many ways can it be done?

Color the Cube

2013-09-14T03:52:00.001-07:00

Given the cube and 2 colors such that each color is on exactly 3 faces, how many different coloring combinations are possible?

Taxi Service

2013-09-14T03:46:00.001-07:00

Mr. Kothari runs a taxi service in Ahmedabad. Every day, he travels from Ahmedabad to one nearby city and back. All the cities are less than 100 km away from Ahmedabad. Everyday morning, Mr. Kothari fills his car’s fuel tank to the brim. On a full tank, the car can travel maximum 250 km. Unfortunately, Mr. Kothari forgot to fill his tank today. What is the probability that he will run out of fuel during the day?

Maximize the product

2013-07-18T03:31:00.000-07:00

Using the digits 1 up to 9, two numbers must be made. The product of these two numbers should be as large as possible. All digits must be used exactly once.
What are these 2 numbers?

Hole and the Board - 3

2013-07-18T03:05:00.000-07:00

A carpet of size 10 by 10 meters should be placed in a room of size 12 by 9 meters. In the center of the room, there is an aquarium of size 8 by 1 meters (see the figure below). The carpet should be cut into no more than two pieces (i.e. one cut in total).

Four Fruits

2013-06-14T07:27:00.000-07:00

In a contest, four fruits (an apple, a banana, an orange, and a pear) have been placed in four closed boxes (one fruit per box). People may guess which fruit is in which box. 123 people participate in the contest. When the boxes are opened, it turns out that 43 people have guessed none of the fruits correctly, 39 people have guessed one fruit correctly, and 31 people have guessed two fruits correctly.

How many of them guessed three fruits correctly?

Subtraction

2013-06-14T02:46:00.000-07:00

How many times one can subtract 2 from 32?

Hole and the Board - 2

2013-03-07T03:08:00.000-08:00

Cut this board into only 2 pieces so that they fit into the given hole.

Series - I

2013-02-21T05:35:00.000-08:00

1, 4, 4, 8, 32, 40, 1280, ....

Theory of Countability

2013-02-21T05:33:00.000-08:00

Consider a uniform distribution between 0 and 1. Let us randomly pick up a number between 0 and 1, say X
a) What is the probability that X = 0.5?
b) What is the probability that X is rational?
c) What is the probability that X is irrational?

Descending down a building

2013-02-07T06:28:00.000-08:00

You need to descend down a 200m building but you only have a 150m long rope. There is a hinge present at the top and also at the 100m level of the building. How will you descend down?

6 Colored balls

2013-01-26T06:55:00.006-08:00

We have two red, two green and two yellow balls. For each color, one ball is heavy and the other is light. All heavy balls weigh the same. All light balls weigh the same. How many weighing on a beam balance are necessary to identify the three heavy balls?

Hole and the Board

2013-01-26T06:29:00.003-08:00

Cut this board into only 2 pieces so that they fit into the given hole.

2n digit number

2012-10-23T01:00:00.002-07:00

i) Consider a 4 digit number abcd. Will there exist a 'k' such that

abcd = k*ab*cd

If so, find the k and abcd?

ii) Now, consider a 6 digit number abcdef. Will there exist a 'k' such that

abcdef = k*abc*def

If so, find the k and abcdef?
iii) Using these 2 results, what can you conclude about a existence of a 'k' for a generic 2n digit number?

Maximum and Minimum of array

2012-10-23T00:56:00.003-07:00

Find the maximum and minimum of an array of 'n' numbers using only 3n/2 comparisons.

PS: This question was given to us in the DS class.

Color matching in concentric circles

2012-10-20T10:52:00.001-07:00

A circle is divided into 8 equal sectors. Half are coloured red and half are coloured
blue. A smaller circle is also divided into 8 equal sectors, half coloured red and half
coloured blue. The smaller circle is placed concentrically on the larger. Prove that
no matter how the red and blue sectors are chosen it is always possible to rotate the
smaller circle so that at least 4 colour matches are obtained.

Monte Hall Variant - 2

2012-10-13T08:01:00.001-07:00

Continuing to the previous problem, if each door contain a car with probability 'p'.
What will be your decision to the options listed in previous post?
Also, at what value of 'p' will you change your option?

Monty Hall Variant

2012-10-10T07:59:00.001-07:00

Consider 3 doors A,B,C and each of them can have a car worth Rs 3 lac with 0.5 probability. Now, you chose door A and the show host opened one of the door which contained a car. Now, he has given you 3 offers
Offer 1: You can stick to door A
Offer 2: You can change your door
Offer 3: Take 1,25,000 and leave the game

Which offer would you chose? What is the minimum amount the show host must offer you so that you can take his offer and leave the show?

Assumption: Show host knows everything behind the doors

Center of a circle

2012-09-16T06:04:00.001-07:00

Using only a compass, can you find the center of a circle?
NOTE: You can't use anything other than compass. Not even a ruler or a scale. This implies, you can't make a straight also.

Dominoes on a square board

2012-09-16T06:03:00.002-07:00

Suppose you have cut off the two diagonally opposite corner squares of a regular 8×8 square board, as shown above, 62 squares remain. Now you have a set of 2×1 dominoes, each of which covers two squares that are adjacent vertically or horizontally. Is it possible to use 31 of these dominoes to tile all 62 squares?

13 coins

2012-09-16T03:25:00.001-07:00

You are given 13 coins out of which one is defected. You don't know the defect 'heavy' or 'light'. Also you are given an authentic coin. You are allowed to use weighing balance 3 times. Give a strategy to find the defected coin and its defect.

NOTE: This is in continuation to the puzzle "Balls and Weighing Balance". If you are asked to only the defected coin, you can do so without the authentic coin also. But to find the defect, you need to figure out a new strategy making use of authentic coin.
Just, to avoid confusion, total coins would be 14(13 + 1authentic coin)

Grid Infection

2012-09-16T03:19:00.000-07:00

In an n by n grid of squares, two squares are neighbors if they share an edge. Initially, some squares are "infected". At successive clock ticks, an uninfected square gets infected if at least two of its neighbors are infected. How many squares must initially be infected so that all squares eventually get infected?

Fox in a Hole

2012-09-16T03:03:00.000-07:00

Consider 'n' holes in a line. One of them is occupied by a fox. Each night, the fox moves to a neighboring hole, either to the left or to the right. Each morning, you get to inspect a hole of your choice. What strategy would ensure that the fox is eventually caught?