IIT JEE , AIEEE Forum - Mathematics , Algebra

sravan

1. Find a 5 digit number, as big as possible, that when you multiply it by a single digit number, you get a six digit number, in which all digits are identical.

Create the number 24 using (all of) 1, 3, 4, and 6. You may add, subtract, multiply, and divide. Parentheses are free. You can (and must) use each digit only once. Note that you may not "glue" digits together. (14 - 6) * 3 is not a solution. 1³ * 4 * 6 is not a solution either (powers not allowed).

2. There are 100 prisoners locked up in solitary cells. There is a central living room with one light bulb; the light bulb is initially off. No prisoner can see the light bulb from his cell. Every day, the warden picks a prisoner at random (i.e., the warden may pick one prisoner multiple days in a row), and that prisoner is taken to the living room. In the room, the prisoner can do one of three things: 1) toggle the light switch, 2) do nothing, or, 3) assert that all 100 prisoners have been in the living room at least once. If this assertion is false, all 100 prisoners are shot. If it is true, they go free. Before they were locked up, they got an hour to discuss a plan. What should be their strategy? [I know of at least three solutions, only one of which will get them out within their lifetime. Try to estimate the runtime of your solution.]

3. A person cashes a cheque in the bank, and after he leaves the bank he buys chewing gum for 50 cents. Now, he didn't have any money when he entered the bank and when he counts the money in his pocket now he sees, to his utmost surprize, that the bank teller made a mistake: instead of the dollars he gave him the same number of cents, and instead of cents, he gave him the same number of dollars. The most bizarre situation was, that after buying the chewing gum, he now has twice the amount of money on the original cheque. What was the amount on the cheque? [Try not to write a program to solve this. There is an analytical solution.]

4. You and an opponent are playing a game where each person, in turn, gets to place one coin on the surface of a round table. The only rules are that the coins have to be placed flat on the table, and you are not allowed to place a coin on top of another one. The last person to successfully place a coin on the table wins. If you get to play first, what is your winning strategy?

5. You have 11 coins, one of which is slightly heavier or lighter. Using a balance, can you find the coin within 3 tries and say whether it's lighter or heavier? Now do the same with 12.

6.You have 100 bags with coins. All coins are equal, but one bag contains coins that are all 1 gram heavier than the coins in all the other bags. If you have a scale (not a balance) that you can use only once, can you find the bag with the bad coins?

7. A Classic: you come to a fork in the road. One road leads to Hell, the other to Heaven. On both sides of the road there is a person. One is a pathological liar (everything he says is false), the other a pathalogical truth teller (everything he says is true). If you get to ask a single question to one of them, what would you ask to choose which road you will continue your journey on?

8. You have a number that consists of 6 different digits. This number multiplied by 2, 3, 4, 5, and 6 yields, in all cases, a new 6-digit number, which, in all cases, is a permutation of the original 6 different digits. What's the number?

I go to a party with my wife. When we get there, four other couples arrive at the same time. We all know each other, so we all greet each other. A greeting can be a handshake, a kiss, a hug, or whatever. When everyone is done I ask everyone how many times they shook another person's hand. All answers I get are different. Given that nobody greets their own spouse, how many hands did my wife shake?

9. Every day, the prison warden takes out his 100 prisoners to play an evil game. He places them in a circle where everyone can see each other. He then places a hat on each prisoner's head. The hat is either red or white and he always gives the same hat to the same person. The prisoner cannot see the color of his own hat. Then, on his command, all prisoners with a white hat have to step forward. If one too few or one too many steps forward, they will all be executed. If they do it right, they all go free. If they all do nothing, they all go back to their cells and the game continues the next day. If all prisoners know there is at least one person with a white hat, how many days did it take for the prisoners to get out, and how did they do it? (Note: they are not allowed to communicate in any way about the color of their hats. For fear of execution, they don't.)

10. You have two wicks and a box of matches. Both wicks burn for 30 minutes each, but they burn irregularly, i.e., any partial amount of wick that burned says nothing about the amount of time that passed. How do you measure 45 minutes?

11.You have 55 matches arranged in some number of piles of different sizes. You now do the following operation: pick one match from each pile, and form a new pile. You repeat this ad infinitum. What is the steady state? Is it unique? [Once you solve this puzzle, you can play around with this little Ruby program to visualize the process.]

12. Another Classic: there are three desk lamps standing on a desk in room A. You, however, are in room B where you cannot see the lamps. There are three light switches in room B that control the three desk lamps in room A. All light switches are in the off position. You get fifteen minutes to think and act and then you will be brought to room A, where you have to state which switch corresponds to which lamp. What do you do?

You throw three darts onto the surface of a globe, each from a randomly chosen direction. What is the probability that all three darts are in one hemisphere?

13.What's the next number in the following sequence?

     1      11      21    1211  111221  312211

14. A riddle for computer scientists: if you have an array of positive numbers, zeroes, and negative numbers, how do you determine the highest sum you can make using a subarray (i.e., consecutive numbers within the array)? Can you do this in linear time?

15. Another riddle for computer scientists: if you have an array with random odd and even numbers, what is the most efficient algorithm you can think of to put all even numbers on one side- and all odd numbers on the other side in this array?

sravan

Click on some questions for the answers.

This problem is actually damn hard, I don't know why I put it first.
You are given a set of scales and 12 marbles. The scales are of the old balance variety. That is, a small dish hangs from each end of a rod that is balanced in the middle. The device enables you to conclude either that the contents of the dishes weigh the same or that the dish that falls lower has heavier contents than the other.

The 12 marbles appear to be identical. In fact, 11 of them are identical, and one is of a different weight. Your task is to identify the unusual marble and discard it. You are allowed to use the scales three times if you wish, but no more.

Note that the unusual marble may be heavier or lighter than the others. You are asked to both identify it and determine whether it is heavy or light.

You are given a set of scales and 90 coins. The scales are of the same type as above. You must pay $100 every time you use the scales.

The 90 coins appear to be identical. In fact, 89 of them are identical, and one is of a different weight. Your task is to identify the unusual coin and to discard it while minimizing the maximum possible cost of weighing (another task might be to minimizing the expected cost of weighing). What is your algorithm to complete this task? What is the most it can cost to identify the unusual coin?

You are a bug sitting in one corner of a cubic room. You wish to walk (no flying) to the extreme opposite corner (the one farthest from you). Describe the shortest path that you can walk.

A mythical city contains 100,000 married couples but no children. Each family wishes to ?continue the male line?, but they do not wish to over-populate. So, each family has one baby per annum until the arrival of the first boy. For example, if (at some future date) a family has five children, then it must be either that they are all girls, and another child is planned, or that there are four girls and one boy, and no more children are planned. Assume that children are equally likely to be born male or female.

Let p(t) be the percentage of children that are male at the end of year t. How is this percentage expected to evolve through time?

How many degrees (if any) are there in the angle between the hour and minute hands of a clock when the time is a quarter past three?

There are 100 light bulbs lined up in a row in a long room. Each bulb has its own switch and is currently switched off. The room has an entry door and an exit door. There are 100 people lined up outside the entry door. Each bulb is numbered consecutively from 1 to 100. So is each person.

Person No. 1 enters the room, switches on every bulb, and exits. Person No. 2 enters and flips the switch on every second bulb (turning off bulbs 2, 4, 6, ?). Person No. 3 enters and flips the switch on every third bulb (changing the state on bulbs 3, 6, 9, ?). This continues until all 100 people have passed through the room.

What is the final state of bulb No. 64? And how many of the light bulbs are illuminated after the 100^th person has passed through the room?

A windowless room contains three identical light fixtures, each containing an identical light bulb. Each light is connected to one of three switches outside of the room. Each bulb is switched off at present. You are outside the room, and the door is closed. You have one , and only one, opportunity to flip any of the external switches. After this, you can go into the room and look at the lights, but you may not touch the switches again. How can you tell which switch goes to which light?

What is the smallest positive integer that leaves a remainder of 1 when divided by 2, remainder of 2 when divided by 3, a remainder of 3 when divided by 4, ? and a remainder of 9 when divided by 10?

In a certain matriarchal town, the women all believe in an old prophecy that says there will come a time when a stranger will visit the town and announce whether any of the men folks are cheating on their wives. The stranger will simply say ?yes? or ?no?, without announcing the number of men implicated or their identities. If the stranger arrives and makes his announcement, the women know that they must follow a particular rule: If on any day following the stranger?s announcement a woman deduces that her husband is not faithful to her, she must kick him out into the street at 10 A.M. the next day. This action is immediately observable by every resident in the town. It is well known that each wife is already observant enough to know whether any man (except her own husband) is cheating on his wife. However, no woman can reveal that information to any other. A cheating husband is also assumed to remain silent about his infidelity.

The time comes, and a stranger arrives. He announces that there are cheating men in the town. On the morning of the 10^th day following the stranger?s arrival, some unfaithful men are kicked out into the street for the first time. How many of them are there?

You and I are to play a competitive game. We shall take it in turns to call out integers. The first person to call out ?50? wins. The rules are as follows:
1. The player who starts must call out an integer between 1 and 10, inclusive;
2. A new number called out must exceed the most recent number called by at least one and by no more than 10.

Do you want to go first, and if so, what is your strategy?

You are to open a safe without knowing the combination. Beginning with the dial set at zero, the dial must be turned counter-clockwise to the first combination number, (then clockwise back to zero), and clockwise to the second combination number, (then counter-clockwise back to zero), and counter-clockwise again to the third and final number, where upon the door shall immediately spring open. There are 40 numbers on the dial, including the zero.

Without knowing the combination numbers, what is the maximum number of trials required to open the safe (one trial equals one attempt to dial a full three-number combination)?

Inside of a dark closet are five hats: three blue and two red. Knowing this, three smart men go into the closet, and each selects a hat in the dark and places it unseen upon his head.

Once outside the closet, no man can see his own hat. The first man looks at the other two, thinks, and says, ?I cannot tell what colour my hat is.? The second man hears this, looks at the other two, and says, ?I cannot tell what colour my hat is either.? The third man is blind. The blind man says, ?Well, I know what colour my hat is.? What colour is his hat?

You are standing at the centre of a circular field of radius R. The field has a low wire fence around it. Attached to the wire fence (and restricted to running around the perimeter) is a large, sharp-fanged, hungry dog. You can run at speed v, while the dog can run four times as fast. What is your running strategy to escape the field?

You have 52 playing cards (26 red, 26 black). You draw cards one by one. A red card pays you a dollar. A black one fines you a dollar. You can stop any time you want. Cards are not returned to the deck after being drawn. What is the optimal stopping rule in terms of maximizing expected payoff? Also, what is the expected payoff following this optimal rule?

Why is that if p is a prime number bigger than 3, then p²-1 is always divisible by 24 with no remainder?

You have a chessboard (8×8) plus a big box of dominoes (each 2×1). I use a marker pen to put an ?X? in the squares at coordinates (1, 1) and (8, 8) - a pair of diagonally opposing corners. Is it possible to cover the remaining 62 squares using the dominoes without any of them sticking out over the edge of the board and without any of them overlapping? You cannot let the dominoes stand on their ends.

You have a string-like fuse that burns in exactly one minute. The fuse is inhomogeneous, and it may burn slowly at first, then quickly, then slowly, and so on. You have a match, and no watch. How do you measure exactly 30 seconds?

Can the mean of any two consecutive prime numbers ever be prime?

How many consecutive zeros are there at the end of 100! (100 factorial). How would your solution change if there problem were in base 5? How about in Binary???

How can this be true???? Have a look at the picture (click to enlarge.) All the lines are straight, the shapes that make up the top picture are the same as the ones in the bottom picture so where does the gap come from????

A man is in a rowing boat floating on a lake, in the boat he has a brick. He throws the brick over the side of the boat so as it lands in the water. The brick sinks quickly. The question is, as a result of this does the water level in the lake go up or down?

You have a 3 and a 5 litre water container, each container has no markings except for that which gives you it's total volume. You also have a running tap. You must use the containers and the tap in such away as to exactly measure out 4 litres of water. How is this done?

I have three envelopes, into one of them I put a £20 note. I lay the envelopes out on a table in front of me and allow you to pick one envelope. You hold but do not open this envelope. I then take one of the envelopes from the table, demonstrate to you that it was empty, screw it up and throw it away. The question is would you rather stick with the envelope you have selected or exchange it for the one on the table. Why? What would be the expected value to you of the exchange?

You're a farmer. You're going to a market to buy some animals. On the market there are 3 types of animals for sale. You can buy:

Horses for £10 each, goats for £1 each and ducks, you get 8 of these per bunch and each bunch costs £1.

The aim is to acquire 100 animals at the cost of £100, what is the combination of horses, goats and duck that allows you to do this? (You must buy at least one of each.)

Adam, Bob, Clair and Dave are out walking: They come to rickety old wooden bridge. The bridge is weak and only able to carry the weight of two of them at a time. Because they are in a rush and the light is fading they must cross in the minimum time possible and must carry a torch (flashlight,) on each crossing.

They only have one torch and it can't be thrown. Because of their different fitness levels and some minor injuries they can all cross at different speeds. Adam can cross in 1 minute, Bob in 2 minutes, Clair in 5 minutes and Dave in 10 minutes.

Adam, the brains of the group thinks for a moment and declares that the crossing can be completed in 17 minutes. There is no trick. How is this done????

A man has built three houses. Nearby there are gas water and electric plants. The man wishes to connect all three houses to each of the gas, water and electricity supplies.

Unfortunately the pipes and cables must not cross each other. How would you connect connect each of the 3 houses to each of the gas, water and electricityic supplies???

How many squares are there on a chessboard?? (the answer is not 64)

Can you extend your technique to calculate the number of rectangles on a chessboard.

sravan

i. The Bridge

Four men want to cross a bridge. They all begin on the same side. It is night, and they have only one flashlight with them. At most two men can cross the bridge at a time, and any party who crosses, either one or two people, must have the flashlight with them.
The flashlight must be walked back and forth: it cannot be thrown, etc. Each man walks at a different speed. A pair must walk together at the speed of the slower man. Man 1 needs 1 minute to cross the bridge, man 2 needs 2 minutes, man 3 needs 5 minutes, and man 4 needs 10 minutes. For example, if man 1 and man 3 walk across together, they need 5 minutes.

The Question: How can all four men cross the bridge in 17 minutes?
The Answer:

Ask & Discuss Questions with Community & Experts

i. The Bridge

ii. Alphabet Blocks

iii. To Know or not To Know

iv. A Quiz...

v. John & Julia

vi. Square Puzzle

vii. Coin Weighing

viii. Stacking Coins

ix. 3 Heads & 5 Hats

x. Zebra

xi. Cube Creatures

xii. Colourful Dwarfs

xiii. Pirate Treasure

xiv. Bizarre Boxes

xv. The Truel

xvi. Fourteen Fifteen

xvii. Coconut Chaos

xviii. Numbers and Dots