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Logical Puzzles
These logical puzzles will test your brain-power! Common sense probability and elementary math are required on a few of them.

Clicking on 'Solution' for a puzzle pops up a solution window like this. All solutions will appear in that window. You can then re-arrange the main window and the solutions window to appear on your desktop separately. This way, you can have a permanent solutions window that is refreshed when you view a new solution without disturbing the main window.

The Puzzles (Difficulty rating 1 Easy - 4 Difficult indicated by star)
 

I'm sure you are familiar with the probabilities associated with rolling two dice. You know that there is a 1/36 chance of getting a 2 or a 12, a 2/36 chance of a 3 or an 11, a 3/36 chance of a 4 or a 10, a 4/36 chance of a 5 or 9, a 5/36 chance of a 6 or 8, and a 6/36 chance of a 7. 

All this assumes, of course, that each die contains the numbers 1,2,3,4,5,6. There is, however, a way of renumbering the die with non-zero integers such that the probabilities of rolling any number from 2 to 12 remains the same. Can you find what numbers would be on each of the renumbered dice? You are allowed to use the same number more than once, even multiple times on the same die, if necessary.

What is the 5-digit number in which the sum of the first two digits is one less than the third, the third is double the fourth, the fourth is double the last, the third is the product of the fourth and fifth, and the second is five more than the first.

A man has two children. When asked the sex of his children, he replies "At least one of my children is a girl". What is the probability that his other child is a girl ?

Someone has prepared two envelopes containing money. One contains twice as much money as the other. You pick an envelope. 

Then the following argument occurs to you: Suppose my chosen envelope contains £X, then the other envelope either contains £X/2 or £2X. Both cases are equally likely, so my expectation if I take the other envelope is .5 * £X/2 + .5 * £2X = £1.25X, which is higher than my current £X, so I should change my mind and take the other envelope. 

But then you can keep applying this argument over and over so you never settle on one envelope. Where is the flaw in the logic, and what should you actually do ?

Someone has prepared two envelopes containing money. One contains twice as much money as the other. You have decided to pick one envelope. This time you peek inside and see £100.

As in Envelopes I, the following argument occurs to you: Suppose my chosen envelope contains £X, then the other envelope either contains £X/2 or £2X. Both cases are equally likely, so my expectation if I take the other envelope is .5 * £X/2 + .5 * £2X = £1.25X, which is higher than my current £X, so I should change my mind and take the other envelope. 

But then you can keep applying this argument over and over so you never settle on one envelope. Where is the flaw in the logic, and what should you actually do ?

This fascinating problem turns out to be quite a mind-blower.

(Thanks to Steve Schultz for providing this puzzle)

An explorer named Marco wants to cross an Australian desert on foot. The track will begin in the outback at the aboriginal village of Kwazi and takes six days. However, one man can only take enough food to last him four days.

Fortunately, Marco can hire porters from Kwazi to assist him. Each porter cost $100 per day that he is away from the village. What is the least possible cost to Marco of hiring enough porters so that he can cross the desert and the porters can return home safely?

You have two glasses filled with the exact same amount of liquid. One contains wine, the other water. First, take one teaspoon of water from the water glass and pour it into the wine glass. After stirring the wine glass, take a teaspoon of liquid and pour into the water glass.

Is there now more wine in the water glass than water in the wine glass, or is there less ?

A Western rancher, finding himself well advanced in years, called his boys together and told them that he wished to divide his herds between them while he still lived. 

"Now, John," he said to the eldest, "you may take as many cows as you think you could conveniently care for, and your wife Nancy may have one ninth of all the cows left."

To the second son he said, "Sam, you may take the same number of cows that John took, plus one extra cow because John had the first pick. To your good wife, Sally, I will give one ninth of what will be left."

To the third son he made a similar statement. he was to take one cow more than the second son, and his wife was to have one ninth of those left. The same applied to the other sons. Each took one cow more than his next oldest brother, and each son's wife took one ninth of the remainder.

After the youngest son had taken his cows, there were none left for his wife. Then the rancher said: "Since horses are worth twice as much as cows, we will divide up my seven horses so that each family will own livestock of equal value."

How many cows did the rancher own, and how many sons did he have ?

You have two pieces of wire and a lighter. Each piece of wire takes 40 minutes to burn from one end to the other, but burns unevenly. For example, it may only take 7 minutes to burn half the wire, and 33 minutes for the remaining half. How do you time 30 minutes ?

Oasis are late for a concert. On their way, they come across a bridge. The bridge is only strong enough to hold two of them, and they only have one torch between them. Liam can get across the bridge in 1 minute, Noel takes 2, Bonehead takes 5, but Guisy takes 10 minutes. Two band members together can only move as fast as the slowest member of the band, and one band member must be carrying the torch during a crossing. What is the shortest time it takes for the band to cross the bridge ? Hint : It can be done in less than 19 minutes.

You have twelve balls that weigh the same, except for one. This ball may be lighter or it may be heavier than the other eleven. You are given some scales to balance balls against each other. How do you discover the rouge ball, and whether it is heavier or lighter, with only three weighings ?

You are a contestant on a quiz show. In front of you are three closed doors. The game show host informs you that behind one of these doors is the motor car of your dreams, but behind the other two doors lies a peanut (which you're allergic to anyway!).

The quiz-master asks you to select a door. After you have selected, he then opens one of the other two doors that does not contain the car. He does this every week to build up the suspense for the watching millions. He asks if you would like to open the door you originally selected and take home that prize, or switch to the remaining door and go home with that prize. 

Is it in your best interests to switch, or to remain with your original selection ?

After gathering 770 chestnuts, three little girls divided them up so that their amounts were in the same proportion as their ages. As often as Mary took four chestnuts, Nellie took three, and for every six that Mary received, Susie took seven.

How many chestnuts did each girl get?

Five clever newsboys formed a partnership and disposed of their papers in the following manner. Tom Smith sold one paper more than one quarter of the whole lot. Billy Jones disposed of one paper more than a quarter of the remainder. Ned Smith sold one paper more than a quarter of what was left, and Charley Jones disposed of one paper more than a quarter of the remainder. At this stage the Smith boys had together sold just one hundred papers more than the Jones boys had sold. Little Jimmy Jones, the youngest kid in the bunch, now sold all the papers that were left. The three Jones boys sold more papers than the two Smith boys, but how many more?

A prisoner waits on death row. The night before his execution, he is offered a possible reprieve. Before him are two large urns. One urn contains fifty black balls, the other fifty white balls. Tomorrow, the executioner will, while blindfolded, draw a ball randomly from one of the two urns. If it's black, it's curtains for the prisoner. If it's white his sentence will be commuted to life.

The prisoner wants very much to live, and is pleased that with the current state of affairs his chances of living are fifty-fifty. He is then presented with an option. He may change the contents of the urns. He can swap white balls for black, move balls from urn to urn, etc. There is a stipulation that when he is done, there must be fifty white and fifty black balls total between the two urns. He can't eat some of the black balls or paint them or anything.

It occurs to the prisoner he might be able to help his situation by moving the balls so that there were twenty-five of each color in each urn, then making sure the white balls were on top. But the executioner might have guessed this, and may shake up the urns. Worse yet, he might deliberately reach to the bottom of the urn he chooses.

Is there another way the prisoner can help himself?

You are standing next to a well, and you have two jugs. One jug has a content of 3 litres, and the other one has a content of 5 litres. How do you measure out 4 litres ?

You have to boil a single egg for exactly nine minutes.

All you have are two hourglasses, able to precisely measure in sand seven and four minutes, respectively.

How quickly using only these two hourglasses can you boil with his egg?

After years of debate, the hare and the tortoise decide to renew their rivalry. The race is to be run over 100metres and the hare to give the tortoise a 16metre head start. When the pistol fires to signify the beginning of the race, the hare soon makes up the 16metres head start that the tortoise started with. But in the time it took the hare to cover this distance, the tortoise has moved on a little bit. In the time it takes the hare to cover this distance, the tortoise has moved on a little further. In the time it takes the hare to cover this distance, the tortoise has moved on a little further still. Therefore, surely it reasons that the hare can never catch the tortoise.  However, the hare won the race. How ?

(Assume the hare is at least twice as fast as the tortoise) Puzzle by Ming Song

In a shop there are sufficient numbers of two kinds boxes of candies, which contain A kilograms and B kilograms of candies respectively. There may exist the smallest number of kilograms of candies such that if a client buys a number of kilograms of candies more than or equal to that smallest number, the shop clerk doesn't need to open any boxes of candies.

Part A

When doesn't the smallest number exist?

Part B

If the smallest number does exist, what is it ?

Example

If there are boxes of 3 kilograms and 5 kilograms, the smallest number is 8.

8  = 1*3 + 1*5
9  = 3*3
10 =       2*5
11 = 2*3 + 1*5
12 = 4*3
13 = 1*3 + 2*5
14 = 3*3 + 1*5
15 =     + 3*5
 ..etc..

Because 7 cannot be expressed this way, the smallest number is 8.

A B C D want to get across a lake 312 miles across.

Swimming speeds in mph: A=4  B=8  C=12  D=16

They have a motorboat that goes at 48mph but this boat can only take 2 persons.

D leaves first; 2 hours later the other 3 leave.

During the last hour, a strong salmon decides to push the boat. This causes the boat to go 72mph during the last hour.

All 4 arrive on other side at same time, 16 hours after D left.

Part A

How many miles did the boat travel?

Part B

For how many hours did A swim?

Part C

How many miles did C swim?

Part D

How many hours was B in the boat?

NOTE: Everything is in complete miles and hours (no fractions of nothing), all speeds are constant, no interfering currents, no lost of time getting in/out of boat, no loss of time doing boat u-turns, no hanging on to the back of the boat or other such dishonest stuff; in other words, no dirty tricks to make your life miserable.

 Puzzle by Denis Borris

Canadian Wild Strawberries (CWS) are tiny but tasty.

A and B each have a jar containing 400 CWS; they decide to have a CWS eating race; A wins, swallowing his last CWS when B still has 23 left. Took A 13.2 seconds.

Next, B takes on C, each with a jar containing 261 CWS

B wins, C left with 117 CWS

Next A took on C, and each had a jar containing N CWS's :

You are then shown a number that equals the sum of digits of the number of CWS's that C had left when A finished .

What is the value of N?

 Puzzle by Denis Borris

Picture the scene ....

I'm sitting outside, so is Andy; Andy's right vision is 10% better than mine.

Andy yells to me: "Hey Tim, I can see Sandy sunning herself on her balcony, in her famous one-piece bikini; lucky me: happens to be the last house I can see!"

What is the number of houses I can see on my left, the number I can see on my right, and Sandy's address assuming she lives at the maximum address possible ?

 Puzzle by Denis Borris

John: So Jack, how many kids you got now?
Jack: Less than 5
John: Well...how many?
Jack: Figure it out; the product of their ages equals twice the sum.
John: That's not enough information!
Jack: My wife gave birth a year ago.
John: Still not enough info!
Jack: The age of my oldest is the same as the number on this door.
John: Hmmm...that helps...still not enough though
Jack: The two in the middle...
John: Stop there! I know how many kids you have, and also their ages...

Do you ?

You have £100 to spend on farm animals. You must buy exactly 100 animals and spend exactly £100. You must have a combination of the following animals; sheep, goats and cows. Sheep are £10. Goats are £1. You get 8 cows for £1. You must have at least one of each  

How many of each animal needs to be purchased ?

A 40 mile-long army is marching at a constant speed through the desert. A horseman at the back of the army rides at a constant speed towards the front bearing a message for the general who is at the head of the army. When the horseman arrives at the front of the army, he immediately delivers the message and reverses direction, riding back towards the rear (still at a constant speed). By the time he arrives at the back of the army, the army has marched 40 miles. How far has the horseman rode?

i.e. If the horseman had an odometer on the horse, how many miles would it say ? (Hint : It is NOT 80 or 120 miles)

Sally and Sue have a strong desire to date Sam. They all live on the same street yet neither Sally or Sue know where Sam lives. The houses on this street are numbered 1 to 99. Sally asks Sam "Is your house number a perfect square?". He answers. Then Sally asks "Is is greater than 50?". He answers again. Sally thinks she now knows the address of Sam's house and decides to visit. When she gets there, she finds out she is wrong. This is not surprising, considering Sam answered only the second question truthfully. Sue, unaware of Sally's conversation, asks Sam two questions. Sue asks "Is your house number a perfect cube?". He answers. She then asks "Is it greater than 25?". He answers again. Sue thinks she knows where Sam lives and decides to pay him a visit. She too is mistaken as Sam once again answered only the second question truthfully. If I tell you that Sam's number is less than Sue's or Sally's, and that the sum of their numbers is a perfect square multiplied by two, you should be able to figure out where all three of them live.

Three prisoners on death row are told that one of them has been chosen at random for execution the next day, but the other two are to be freed. One privately begs the warden to at least tell him the name of one other prisoner who will be freed. The warden relents: 'Susie will go free.' Horrified, the first prisoner says that because he is now one of only two remaining prisoners at risk, his chances of execution have risen from one-third to one-half! Should the warden have kept his mouth shut?

A man walks into a bar, orders a drink, and starts chatting with the bartender. After a while, he learns that the bartender has three children. "How old are your children?" he asks. "Well," replies the bartender, "the product of their ages is 72." The man thinks for a moment and then says, "that's not enough information." "All right," continues the bartender, "if you go outside and look at the building number posted over the door to the bar, you'll see the sum of the ages." The man steps outside, and after a few moments he reenters and declares, "Still not enough!" The bartender smiles and says, "My youngest just loves strawberry ice cream." How old are the children? A variant of the problem is for the sum of the ages to be 13 and the product of the ages to be the number posted over the door. In this case, it is the oldest that loves ice cream. Then how old are they?

 

You are playing the numbers game on the TV program Countdown. The numbers are 1,2,25,50,75 and 100. How do you make 383 ?

A store has announced that it will give £1000 to the first person in a queue outside the store whose birthday is the same as someone in front of them. The person at the head of the queue is asked first, followed by the person behind them etc... you can get in the line at any time at any place. What position in line gives you the best chance of winning the cash ?

Assumptions :

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