1. You are given two candles of equal size, which can burn 1 hour each. You have to measure 90 minutes with these candles. (There is no scale or clock). Also u r given a lighter.

Ans: 1. First light up the two ends of the 1st candle. When it will burn out light up one end of the second candle. (30+60=90)

2. Try the similar problem to measure 45 minutes.

Ans: First light-up the two ends of the 1st candle and one end of the 2nd candle.

When the 1st candle will burn out ,then light up the both ends of the 2nd candle(15+30=45)

3. You r given a thermometer. What can u do by this without measuring the temperature?

Ans: if u put thermometer into a tree it won?t grow anymore, will just die off

4. How it is possible to place four points that are equidistance from each other?

ORU r a landscape designer and your boss asked u to design a landscape suchthat you should place 4 trees equidistance from each other. (Distance from each tree to the other must be same)

Ans: Only 3 points can be equidistant from each other. But if u place points in the shape of a pyramid then its possible

5. You are given a cake; one of its corner is broken. How will u cut the rest intoTwo equal parts?

Ans: Slice the cake

6. How will you recognize the magnet & magnetic material & non-magnetic material?

Ans: Drag one piece of material over another. There is no attractive force in the middle portion of the magnet.

ORGet a piece of thread and tie up with the one bar and check for poles. If it iron barthen it moves freely and if it is magnetic bar then it fix in one direction according to

poles.

7. If one tyre of a car suddenly gets stolen.... and after sometime u find the tyre without the screws how will u make ur journey complete?

Ans: Open 3 screws, 1 from each tyre and fix the tyre.

8. How can u measure a room height using a thermometer?

Ans: temp varies with height. but its dependent on various other factors like humidity, wind etc.

9. What is the height of room if after entering the room with a watch ur head strikes a hanging bulb?

Ans: Oscillate the hanging bulb. Calculate the time period for one complete oscillation by Simple Harmonic Motion (SHM) of the handing bulb. Put it in the formula T=2 * 3.14 * (L/G)^1/2 L will be the length of the hanging thread.Add the L with ur height to get the height of the room.

ORAns: Drop it from the room and find the time at which it strikes the floor. Usingphysics formula s = (at^2)/2 (IM NOT SURE ABOUT THIS ONE)

10. Color of bear.... if it falls from 1m height in 1s.

Ans: We get 'g' perfect 10 which is only in poles...hence polar bear...color White

11. How will you measure height of building when you are at the top of the building?And if you have stone with you.

Ans: Drop the stone and find the time taken for the stone to reach the ground. find height using the formula s = a + gt ( s = height, a= initial velocity=0, g=9.8m/s, t = time taken)

12. How wud u catch and receive a ball in same direction? (Dropping is from northAnd receiving from bottom not accepted, as it is 2 directions)

Ans: ?

13. 25 statements given. Some tell truth, some false and some alternators. Find outthe true statements.

Ans: ?

14. Can u make 120 with 5 zeros?

Ans: Factorial (factorial (0)+factorial (0)+factorial (0)+factorial (0)+factorial (0)) = 120

15.There are three people A, B, C. Liars are of same type and Truth speaking peopleare of same type. Find out who is speaking truth and who is speaking false from the following statements:

a) A says: B is a liar.b) B says: A and C are of same type.

Ans: lets assume A is speaking truth. It means B is a liar then it means A and C arenot of same type.

16.5 swimmers A, B, C, E, F and many conditions of their positions like there areTwo b/w A & F, B doesn't win etc the question was to find who was b/w like E & D?

Ans: ?

17. in a race u drove 1st lap with 40kmph and in the second lap at what speed u must drive so that ur average speed must be 80kmph.

Ans: its impossible! if u drove the first lap in 40 kmph, its impossible that the average speed of both the laps is 80kmph.

for eg. consider one lap distance = 80km.time req. to cover 1 lap = 80km/40kmph = 2 hrs.

if the avg. speed is 80kmph, then the total time would have taken =

160kms/80kmph = 2 hrs.

same is the case with any other distance u consider. so the avg to be 80kmph is impossible

18. You have to draw 3 concentric circles with a line passing thru their center

without lifting hand.

Ans: Start the line complete one circle move inside circles along the line and thendraw second circle. Like wise rest.

19. A rectangular paper is there. At a corner a rectangular size paper is taken fromit. Now you have to cut the remaining paper into two equal halves.

Ans: try it on the paper. You must fold the part that has complete paper and selectHalf of it and then fold the part that cut and selects half of it and then cut along the folding. (I DONT UNDERSTAND THIS ONE!!)

20. Value of (x-a)(x-b)???..(x-z)

Ans: 0 as there?s X-X term

21. There are 9 coins. 8 are of 1 gm and 1 is of 2 grams. How will you find out theheavier coin in minimum number of weighing and how many weighing it will need?

Ans: 2 weighing ( Divide the number of coins into 3 parts at each weighing)

2. Google Interview Puzzle : 2 Egg Problem

3.My intention here is not to trouble Google interviewers. I was just collecting some classic puzzles and found this one and a small Google search showed me that this is a Google interview puzzle to my pleasant surprise. But many of the answers I found were either wrong or totally twisted. I am making no surety of the answer I give and I am open to your remarks or suggestion or corrections.

4. The Standard Problem in simple writing goes like this:5.

* You are given 2 eggs.* You have access to a 100-storey building.* Eggs can be very hard or very fragile means it may break if dropped from the first floor

or may not even break if dropped from 100 th floor.Both eggs are identical.* You need to figure out the highest floor of a 100-storey building an egg can be dropped without breaking.* Now the question is how many drops you need to make. You are allowed to break 2 eggs in the process

If you are one of the people who likes to solve a puzzle before seeing the answer you must quit the blog now and come back later for checking the answer.

Now that this is a Google interview question I am taking the normal "Interview-Style" of solving a problem. Simply saying thinking aloud through the solution from worst to the best correcting the flows optimizing the solution or taking the 5-minute hard thinking acting pause to a problem, which you know already and just want to make your interviewer think that you are a challenge lover.

6. Solution7.

Drop the first egg from 50.If it breaks you can try the same approach for a 50-storey building (1 to 49) and try it from 25th floor. If it did not break try at 75th floor. And use linear search with the remaining portion of storey we need to test. For example if the first egg breaks at 50 we need to try all possibilities from 1 to 49.

Now this looks a feasible solution. In computer student's jargon do a binary search with first egg and linear search with the second one. Best case is log (100) and worst is 50.

Now the optimal solution for the problem is that you figure out that you will eventually end up with a linear search because you have no way of deciding the highest floor with only one egg (If you broke one egg and you have to find the answer among 10 all you can do is start from the lowest to the highest and the worst is the total number of floors). So the whole question grinds up to how to make use of the first egg to reduce the linear testing of the egg.

(For strict computer science students, well this problem can be solved using binary search on the number of drops needed to find the highest floor.)

Now let x be the answer we want, the number of drops required.

So if the first egg breaks maximum we can have x-1 drops and so we must always put the first egg from height x. So we have determined that for a given x we must drop the first ball from x height. And now if the first drop of the first egg doesn’t breaks we can have x-2 drops for the second egg if the first egg breaks in the second drop.

Taking an example, lets say 16 is my answer. That I need 16 drops to find out the answer. Lets see whether we can find out the height in 16 drops. First we drop from height 16,and if it breaks we try all floors from 1 to 15.If the egg don’t break then we have left 15 drops, so we will drop it from 16+15+1 =32nd floor. The reason being if it breaks at 32nd floor we can try all the floors from 17 to 31 in 14 drops (total of 16 drops). Now if it did not break then we have left 13 drops. and we can figure out whether we can find out whether we can figure out the floor in 16 drops.

Lets take the case with 16 as the answer

1 + 15 16 if breaks at 16 checks from 1 to 15 in 15 drops1 + 14 31 if breaks at 31 checks from 17 to 30 in 14 drops1 + 13 45 .....1 + 12 581 + 11 701 + 10 811 + 9 911 + 8 100 We can easily do in the end as we have enough drops to accomplish the task

Now finding out the optimal one we can see that we could have done it in either 15 or 14 drops only but how can we find the optimal one. From the above table we can see that the optimal one will be needing 0 linear trials in the last step.

So we could write it as

(1+p) + (1+(p-1))+ (1+(p-2)) + .........+ (1+0) >= 100.

Let 1+p=q which is the answer we are looking for

q (q+1)/2 >=100

Solving for 100 you get q=14.So the answer is: 14Drop first orb from floors 14, 27, 39, 50, 60, 69, 77, 84, 90, 95, 99, 100... (i.e. move up 14 then 13, then 12 floors, etc) until it breaks (or doesn't at 100).

8. Classic Puzzles 9. Friday, December 15, 200610.Job Interview Puzzle: 3 Classic Weighing puzzles :Simple Medium and

Hard

11.

In this post I want to describe about a series of puzzles called weighing puzzles. These puzzle vary in hardness from simple to extremely mathematical involving either expertise in some fields or extreme ingenuity. Either you know it or you figure out it using extreme intelligence. So here is the chance for some people to burn your grey cells.

I am putting forward three puzzles with varying range of hardness. Sometimes you may feel the hardest one is very easy for you have already come across the theory, but I still made it the hardest for the people who will be solving it with out knowing the theory behind it, giving them an option to figure out a small part in evolution of computational history.

There is a shopkeeper who wants to weigh things who has a common balance. He must be in a position to weigh things of all possible integral weighing units from 1 to a given maximum sum. The question will be either about how many weights you will need or how will you weigh.

12. Problem 1: One Side Only (Simple Interview Question for Phone Screening usually)

13.In this version of the problem shopkeeper can only place the weights in one side of the common balance. For example if shopkeeper has weights 1 and 3 then he can measure 1,3 and 4 only. Now the question is how many minimum weights and name the weights you will need to measure all weights from 1 to 1000. This is a fairly simple problem and very easy to prove also.

14. Problem 2: Both Sides (Medium:5 to 10 mins onsite interview question)

15.This is same as the first problem with the condition of placing weights on only side of the common balance being removed. You can place weights on both side and you need to measure all weights between 1 and 1000.For example if you have weights 1 and 3,now you can measure 1,3 and 4 like earlier case, and also you can measure 2,by placing 3 on one side and 1 on the side which contain the substance to be weighed. So question again

is how many minimum weights and of what denominations you need to measure all weights from 1kg to 1000kg.

16. Problem 3: Incremental (Hard)17. This is an altogether different one in the same scenario. You have to make 125 packets of

sugar with first one weighing 1 kg, second 2 kg, third 3 kg etc ...and 125th one weighing 125kg.You can only use one pan of the common balance for measurement for weighing sugar, the other pan had to be used for weights i.e. weights should be used for each weighing.It has come into notice that moving weights into and out of the pan of the balance takes time and this time depends on the number on the number of weights that are moved. For example - If we need to measure 4 kg using weights 1 and 3 only, it will take twice as much time needed to measure 1 kg. Lets say we want to make sugar packets of weights 1,3,4 using weights 1 and 3 only. For this first we measure 1 kg, with 1 unit of time, we place 3 kg along with 1 kg and measure 4kg with again 1 unit of time, and finally we move 1kg out of pan to measure 3kg in 1 unit of time. So in 3 units of time we could measure 1,3 and 4kg using weights 1 and 3 only.

Now you have to make sugar packets of all weights from 1 to 125 in minimum time, in other words in minimum movement of weights. The question here is to find out the minimum number of weighs needed and the weight of each the weights used and the strategy to be followed for the creation of 125 packets of sugar.

Question

If given a rectangular cake with a rectangular piece removed (any size or orientation), how would you cut the remainder of the cake into two equal halves with one straight cut of a knife?

Solution

Find the centers of both the original cake and the removed piece. Cut the cake along the line connecting these two centers. As this line cuts the original cake and the removed piece in half, the remainder is two equal halves. 

OR

Cut the entire cake in half horizontally (i.e. parallel to the table). This will get you two even halves.

Question

      There are four people who need to cross a bridge at night. The bridge is only wide enough for two people to cross at once. There is only one flashlight for the entire group. When two people cross, they must cross at the slower member's speed. All four people must cross the bridge in 17 minutes, since the bridge will collapse in exactly that amount of time. Here are the times each member takes to cross the bridge:

Person A: 1 minute

Person B: 2 minutes Person C: 5 minutes Person D: 10 minutes If Person A and C crossed the bridge initially, 5 minutes would elapse, because Person C

takes 5 minutes to cross. Then Person A would have to come back to the other side of the bridge, taking another minute, or six minutes total. Now, if Person A and D crossed the bridge next, it would take them 10 minutes, totaling to 16 minutes. If A came back, it would take yet another minute, totally 17. The bridge would collapse with A and B on the wrong side. How can all four people get across the bridge within 17 minutes? Note: there is no trick-answer to this problem. You cannot do tricky stuff like throwing the flashlight etc.

Solution

1. A and B cross together. Total Time: 2 Minutes 2. B comes back. Total Time: 4 Minutes 3. C and D cross together. Total Time: 14 Minutes 4. A comes back. Total Time: 15 Minutes 5. A and B cross together. Total Time: 17 Minutes

Question

You have someone working for you for seven days and you have one gold bar to pay them. The gold bar is segmented into seven connected pieces. You must give them a piece of gold at the end of every day. If you are only allowed to make two breaks in the gold bar, how do you pay your worker?

Solution

As 7 = 1 + 2 + 4, this implies there are only 2 divisions (cuts). Day 1: Give [1] and you have [2+4] Day 2: Give [2] and take back [1]. You have [1+4] Day 3: Give [1+2] and you have [4] Day 4: Give [4] and take back [1+2]. You have [1+2]. Day 5: Give [1+4] and you have [2] Day 6: Give [2+4] and take back [1]. You have [1] Day 7: Give [1+2+4] and you have nothing! The same trick can be applied to: 15 Days = 1 + 2 + 4 + 8 31 Days = 1 + 2 + 4 + 8 + 16 And so on. The main point of division is: 1 = 2^0 2 = 2^1 4 = 2^2 8 = 2^3 16= 2^4 And so on...

Question

You have 5 jars of pills. Each pill weighs 10 grams, except for contaminated pills contained in one jar, where each pill weighs 9 grams. Given a scale, how could you tell which jar had the contaminated pills in just one measurement?

Solution

Follow the steps outlined below: Step-1: Mark the jars with numbers 1, 2, 3, 4, and 5. Step-2: Take 1 pill from jar 1, take 2 pills from jar 2, take 3 pills from jar 3, take 4 pills from

jar 4 and take 5 pills from jar 5. Step-3: Put all of the jars on the scale at once and take the measurement. Step-4: Now, subtract the measurement from 150 (1*10 + 2*10 + 3*10 + 4*10 + 5*10) Step-5: The result will give you the jar number which has the contaminated pills. This puzzle was apparently written by Einstein in the last century. He said that 98% of the

people in the world cannot solve the quiz. See if you can... Facts: 1: There are 5 houses in 5 different colors 2: In each house lives a person with a different

nationality. 3: These 5 owners drink a certain beverage, smoke a certain brand of cigar and keep a certain pet. 4: No owner has the same pet, smoke the same brand of cigar or drink the same drink.

Hints: 1: The British lives in a red house. 

2: The Swede keeps dogs as pets 3: The Dane drinks tea 4: The green house is on the left of the white house (it also means they are next door to each other) 5: The green house owner drinks coffee 6: The person who smokes Pall Mall rears birds 7: The owner of the yellow house smokes Dunhill 8: The man living in the house right in the center drinks milk 9: The Norwegian lives in the first house 10: The man who smokes Blend lives next to the one who keeps cats 11: The man who keeps horses lives next to the man who smokes Dunhill 12: The owner who smokes Blue Master drinks beer 13: The German smokes Prince 14: The Norwegian lives next to the blue house 15: The man who smokes Blend has a neighbor who drinks water.

The question is: who keeps the fish?

Solution

The fish is in the fourth house. Its green, the owner drinks coffee, smokes prince and he's German!

For those who want to know the approach for solving such problems: First figure out is the order of the houses. We can use the following facts: • The green house is on the left of the white house • Blue is the second house.

• The center house is important because we know anything about it   So the possibilities could be: ____, Blue, ____, Green, White or ____, Blue, Green, White,____ There are 2 blanks and two colors. Red can't be the first house because the owner of the first

house should be from Norway , and the owner of the red house is British. That leaves us with: 1) Yellow, Blue, ____, Green, White – OR- 2) Yellow, Blue, Green, White, ____ The blank spot is the red house. The 2nd option can't be true, as we have clues that the center house owner drinks milk, but

the green house owner drinks coffee. So we get the order as: Yellow, Blue, Red, Green, and White. Then it is only the matter

of reading the clues to figure out the answer.

Question

There are three wise men in a room: A, B and C. You decide to give them a challenge. Suspecting that the thing they care about most is money, you give them $100 and tell them they are to divide this money observing the following rule: they are to discuss offers and counter-offers from each other and then take a vote. The majority vote wins. Sounds easy enough... now the question is, assuming each person is motivated to take the largest amount possible, what will the outcome be?

Solution

It is unlikely that one wise man would be voted out because counter-offers are allowed. Consider: A and B decide to leave C out and split halfway. C offers B to leave A out instead, and as incentive offers B $60 as compared to $50 (Note: for C, $40 is better than nothing at all).

Now A will try to raise that figure, but keep no less than $33 for himself. It would also happen that as B starts to get more and more of the share, A and C would decide to keep B out instead and split amongst each other. This would go on and on until they reach equilibrium with each agreeing to take their rightful share of 33 dollars.

  You are given two 60 minute long fuse ropes (i.e. the kind that you would find on the end of a

bomb) and a lighter. The fuses do not necessarily burn at a fixed rate. For example, given an 8 foot rope, it may take 5 minutes for the first 4 feet of the fuse to burn, while the last 4 feet could take 55 minutes to burn (a much slower rate) (5+55=60 minutes). Using these two fuses and the lighter, how can you determine 45 minutes? HINT: You can use the lighter any number of times

Solution

Light the first fuse at both ends and the other fuse at one end simultaneously. When the first fuse is burned out, you know that exactly half an hour has passes. You also know that the second fuse still has exactly half an hour to go before it will be burned completely, but we won't wait for that. Now also light the other end of the second fuse. This means that the second fuse will be burned completely after another quarter of an hour, which adds up to exactly 45 minutes since we started lighting the first fuse.

Question

There was a sheriff in a town that caught three outlaws. He said he was going to give them all a chance to go free. All they had to do is figure out what color hat they were wearing. The sheriff had 5 hats, 3 black and 2 white. Each outlaw can see the color of the other outlaw’s hats, but cannot see his own. The first outlaw guessed and was wrong so he was put in jail. The second outlaw also guessed and was also put in jail. Finally the third blind outlaw guessed and he guessed correctly. How did he know?

Solution

Let us look at it this way. Here are our possibilities: 1) BBB 2) BBW 3) BWB 4) WBB 5) WWB 6) BWW 7) WBW Now we can eliminate # 6 because in this case the first outlaw would be sure to know that he

had on a black hat. # 7 can be eliminated for the same reason for the second outlaw’s guess. In # 2, the first outlaw has to see at least 1 black hat (if he saw two while hats he wouldn't

have guessed wrong). From this we know that outlaw 2 or outlaw 3 has a black hat (possibly both). Now outlaw 2 has the same dilemma, but he knows that one or both of outlaw 2 and 3 has a black hat. He can see that outlaw 3 has a white hat so in that case he would guess black and be correct, but he didn't (since we know he guessed wrong). Given this, we can remove option 2 from consideration.

Options 1,3,4,5 all have outlaw 3 wearing a black hat. Thus, assuming that convicts 1 and 2 are as logical as possible, the only options left all have outlaw 3 wearing a black hat.

Question

 One train leaves Los Angeles at 15 MPH heading for New York . Another train leaves from New York at 20mph heading for Los Angeles on the same track. If a bird, flying at 25mph, leaves from Los Angeles at the same time as the train and flies back and forth between the two trains until they collide, how far will the bird have traveled?  

Solution

The distance traveled by the bird can be calculated as:

1. Let's say the distance between LA and NY is d miles.

2. The time before which the trains will collide: d / (15+20) hours.

3. The distance traveled by the bird in that time: (d / 35) * 25 = 5*d/7 miles.

Assumptions made: The bird must follow the line of the track, remain at the same altitude, and the speed must be relative to the ground and not air speed.

Logic Puzzle Interview Question at MicrosoftPosted on April 5, 2006 by jamesdmccaffrey

For years, Microsoft has been famous (or infamous depending on your point of view) for asking logic and puzzle questions during hiring interviews.  Although that practice is not as common now as it once was, you still see them every now and then.  Here’s one I heard recently that I liked because seeing how candidates approach the problem gives some insights into their personality.  The problem goes like this: Two friends, a programmer and a mathematician, get together for drinks after work one day at the programmer’s house.  The mathematician asks the programmer how his three children are doing.  The programmer replies that one of his three children just had a birthday.  The mathematician asks, "How old are your children now?"  The programmer replies, "The product of their ages is 36."  The mathematician thinks for a moment and says, "That’s not enough information."  The programmer says, "OK, the sum of their ages equals my house address."  The mathematician steps outside to check the address number, comes back inside, and says, "That’s still not enough information."  The programmer then says, "Well my oldest child has red hair."  The mathematician immediately responded, "Oh, now I know the ages."  What are the ages of the programmer’s three children?

 

The children are aged 2, 2, and 9 years old.  The mathematician’s logic was that since there are three children whose ages multiply to 36, the possible combinations are:

 

1, 1, 36

1, 2, 18

1, 3, 12

1, 4, 9

1, 6, 6

2, 2, 9

2, 3, 6

3, 3, 4

 

Initially, any of these could be the correct ages.  After the programmer says that the sum of the ages is the same as the house address, the mathematician mentally computed the sum of each possible combination:

 

1, 1, 36 -> 38

1, 2, 18 -> 21

1, 3, 12 -> 16

1, 4, 9  -> 14

1, 6, 6  -> 13

2, 2, 9  -> 13

2, 3, 6  -> 11

3, 3, 4  -> 10

 

Notice that all the sums are different except for (1, 6, 6) and (2, 2, 9).  If the programmer’s address was anything except for 13, then the mathematician would know the ages, so the three ages must be one of those two combinations.  But after the programmer said that the oldest child has red hair, the mathematician knew that there was a single oldest child which eliminates the (1, 6, 6) combination which has oldest twins.

 

A Collection of Quant Riddles With (some) Answers

 

The quant riddles or logic or lateral puzzles 1-19 appear in the book 'Heard on The Street: Quantitative Questions from Wall Street Job Interviews' by Timothy Falcon Crack PhD available on his website the rest have been accumulated from the internet and emails that I receive. They are designed to help training for job or university interviews or just training your brain. The internet is littered with this kind of thing but the answers can be a little harder to find so I've thought about all of them and the ones that I know the answer to can be clicked on and have little

at the end. Questions 3 & 5 are probably the easiest and a good place to start. I've coloured them Red, Amber and Green to indicate Very Hard, Quite Hard and Not so Hard. So that's it good luck....

1. 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.  

2. 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?

3. 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.

4. 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?  

5. 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?

6. 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?  

7. 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?

8. 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?

9. 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?

10. 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:

a. The player who starts must call out an integer between 1 and 10, inclusive;

b. 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?

11. 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)?

12. 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?  

13. 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?

14. 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?

15. Why is that if p is a prime number bigger than 3, then p 2 -1 is always divisible by 24 with no remainder?

16. 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.

17. 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?

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

19. 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???

20. 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????

 

21. 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?

 

22. 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?

 

23. 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?

24. 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.)

25. 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????

26. 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???

27. 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.

28. 3 men go into a hotel. The man behind the desk said the room is $30 so each man paid $10 and went to the room.A while later the man behind the desk realized the room was only $25 so he sent the bellboy to the 3 guys' room with $5.On the way the bellboy couldn't figure out how to split $5 evenly between 3 men, so he gave each man a $1 and kept the other $2 for himself.This meant that the 3 men each paid $9 for the room, which is a total of $27 add the $2 that the bellboy kept = $29. Where is the other dollar?

29. There were two men having a meal. The first man brought 5 loaves of bread, and the second brought 3. A third man, Ali, came and joined them. They together ate the whole 8 loaves. As he left Ali gave the men 8 coins as a thank you. The first man said that he would take 5 of the coins and give his partner 3, but the second man refused and asked for the half of the sum (i.e. 4 coins) as an equal division. The first one refused. They went to Ali and asked for the fair solution. Ali told the second man, "I think it is better for you to accept your partner's offer." But the man refused and asked for justice. So Ali said, "then I say that who offered 5 loaves takes 7 coins, and who offered 3 loaves takes 1 coin."

Can you explain why this was actually fair???

30. A drinks machine offers three selections - Tea, Coffee or Random but the machine has been wired up wrongly so that each button does not give what it claims. If each drink costs 50p, how much minimum money do you have to put into the machine to work out which button gives which selection ? .

31. Consider a standard chess board. What is the diameter of the largest circle that can be drawn on the board whilst only drawing on the black squares. Hint: This problem is theoretical, so a line passing from one black square directly into it's diagonal neighbour through the intersecting corners is considered to have stayed within the black squares, the lines are infinitely thin.

32. Two creepers, one jasmin and other rose, are both climbing up and round a cylindrical tree trunk. jasmine twists clockwise and rose anticlockwise, both start at the same point on the ground. before they reach the first branch of the tree the jasmine had made 5 complete twists and the rose 3 twists. not counting the bottom and the top, how many times do they cross?

33. Assuming i wish to stay dry. Should i walk or run in the rain?