Prime Numbers

How many prime numbers exist between 2 consecutive prime numbers?

Taxi Service

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?

Four Fruits

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

How many times one can subtract 2 from 32?

Descending down a building

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?

Color matching in concentric circles

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.

Dominoes on a square board

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

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)

Fox in a Hole

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?

Pieces of a Stone

Consider a stone of 'n' kg. What is the minimum number of pieces should the stone be broken so that you can measure all the weights from 1 to 'n' kg in a weighing balance (taraju)? What should be the weight of the each individual piece?

Guess the polynomial

Consider a polynomial p(x) of any degree such that all its coefficients are positive integer valued. You can ask the value of p(x) for any two values of x. What values of x will you ask and can you find the exact polynomial using these values?

Sunday Born Boy

Given are 2 children. What is the probability that both of them are boys given that one of them is a boy and is born on Sunday?

Journey to home

A student leaves the coaching everyday at 11. His brother leaves the house some time before 11 to reach coaching at 11 to pick the student. One day, the coaching got over at 10 and the student started walking towards home at 8km/hr. In the way, he found his brother coming and then they went together reaching home 20 mins earlier than usual time. Find the speed of his brother?

Can Prisoners survive?

There were 100 prisoners which were asked to stand in a line such that the 100th one can see all, 99th one can see the other 98 and so on. There are two sets of cap Black and White. The king decides to put these caps on each of the prisoners(the prisoner can't see his cap but can see all people standing before him caps) randomly and then the king asked each of them to tell their cap color starting from the 100th prisoner, then 99th and so on. If the prisoner answers correctly, the king leaves him otherwise the prisoner is shot dead. If the prisoners are allowed to formulate a strategy, how many prisoners can be saved.

NOTE: Every prisoner can hear every priosoner's answer

Balls and Weighing balance

There are some 'n' identical balls of which 1 is defective( don't know heavy or light). If you are allowed to use weighing balance 'k' times what can be the maximum value of 'n'?