Showing posts with label

**logic**. Show all posts## Saturday, September 5, 2015

## Island X

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There are three categories of tribes in Island X; a Truther, who
always speaks truthfully; a Liar, who always speaks falsely; and an
Altetnator, who makes statements that are alternatively truthful and
false, albeit not necessarily in that order.

A visitor approaches three inhabitants and asks who is a Truther. They answer as follows:

A says:

1. I am a Truther

2. B is a Liar

B says:

1. I am a Alternator

2. C is a Liar

C says:

1. I am a Truther

2. A is a Liar

A visitor approaches three inhabitants and asks who is a Truther. They answer as follows:

A says:

1. I am a Truther

2. B is a Liar

B says:

1. I am a Alternator

2. C is a Liar

C says:

1. I am a Truther

2. A is a Liar

*Determine the identity of each of the three inhabitants from the information provided in the above statements.*### Island X Puzzle Solution

Assume that A is the Truther. If so, then B is the Liar as A's statement asserts. If so, B's second statement is false, so C is the Alternator. This implies that C's first statement is false as also his second statement that A is the Liar, so that C is the Liar which is a contradiction, so that A cannot be the Truther.Assume that B is the Truther. If so, his first statement is a direct contradiction, implying amongst other things, that B cannot be the Truther.

Assume that C is the Truther. If so, then A is the Liar in conformity with his second statement, so the remaining member B must be the Alternator. This checks out since as a Alternator, B's first statement is true while in his second statement he falsely identified C as the Liar. Both of A's statement are then clearly false, so this establishes the veracity of both the statements of C.

Consequently, (A, B, C) = (Liar, Alternator, Truther)

## Monday, August 31, 2015

## A Farmer's Good Fortune

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A farmer from a small community is out of money. After a mysterious
desease spread among and killed his lifestock, he now needs to quickly
make up for the lost animals. He needs a whole grand.

Knowing that the bank won't lend him any money, he pays a visit to the local loan shark. The outlaw, who's known to have a bit of an obsession with puzzles, proposes a deal.

With the $1,000 he gets, the farmer has to be able to buy a combination of cows, pigs, and sheep, to total exactly 100 heads of lifestock. The combination has to include at least one cow ($100 each), one pig ($30 each), and one sheep ($5 each). The total amount of money spent for the 100 animals has to equal exactly $1,000.

If the farmer manages to accomplish the task, he'll have to return the money with a "friendly" interest rate. Otherwise, he'll get the normal rate, and the threat of a broken pinkie...

Knowing that the bank won't lend him any money, he pays a visit to the local loan shark. The outlaw, who's known to have a bit of an obsession with puzzles, proposes a deal.

With the $1,000 he gets, the farmer has to be able to buy a combination of cows, pigs, and sheep, to total exactly 100 heads of lifestock. The combination has to include at least one cow ($100 each), one pig ($30 each), and one sheep ($5 each). The total amount of money spent for the 100 animals has to equal exactly $1,000.

If the farmer manages to accomplish the task, he'll have to return the money with a "friendly" interest rate. Otherwise, he'll get the normal rate, and the threat of a broken pinkie...

*How many of each kind of livestock did the farmer buy?*### A Farmer's Good Fortune Puzzle Solution

Here's one combination:94 sheep = $470 1 pig = $30 5 cows = $500 ----------------- 100 heads = $1000

Are there anymore?

## Wednesday, August 26, 2015

## A Special Old Guy

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"Buzz, I've finished tracing my family tree back from the year 500 AD, and I found one quite special guy".

"What's so special about him?" asked Buzz.

"Well, he was x years old in the year x^2 (x squared) and he had a son who was y years old in the year y^3 (y cubed)".

Buzz looked perplexed "Sorry Alvin, but I can't solve for x or y".

"Well, he was your age when his son was born." said Alvin.

"You're right" said Buzz "He was a special old guy! But I still can't solve for x or y".

*How old was the old guy when his son was born?*

**Notes:**

Assume that the nerds have the conversation this year, ie 2004 AD.

### A Special Old Guy Puzzle Solution

Firstly, we can say that the date of birth (DOB) for the special old guy (SOG) is:Equation {a}:

DOB(SOG) = x^2 – x

Similarly, for the son of the special old guy (or SOSOG for short):

Equation {b}:

DOB(SOSOG) = y^3 – y

SOG's age when SOSOG was born was:

Equation {c}:

Age(SOG) = DOB(SOSOG) – DOB(SOG) = (y^3 – y) – (x^2 – x)

There are unlimited solutions to equation {c} so we need some assumptions and limits.

We know that SOG must have been between, say 10 years old and 100 years old when SOSOG was born:

Equation {d}:

10 < Age(SOG) < 100

We also know that both SOG and SOSOG were born some time since the year 500 AD:

Equation {e}:

500 < DOB(SOSOG) < 2004.

Based on equation {b} and {e} we can clearly see that there are only 5 solutions for y. They are y = 8, 9, 10, 11 or 12. Any other solutions for y are in breach of equation {e}.

For each of these possible solutions for y there is only a limited number of solutions for x that comply with {d}. They are:

x y -------- 21 8 22 8 26 9 27 9 31 10 36 11 41 12

Any other solutions for x and y are in breach of equation {d}.

SOG's age when his son was born can be calculated for each of these possible solutions by using equation {c} as follows:

x y Age(SOG) ---------------------- 21 8 84 22 8 42 26 9 70 27 9 18 31 10 60 36 11 60 41 12 76

So there are 7 different solutions, which is why Buzz said he couldn't solve for x and y.

Now comes the lateral part of the puzzle: Alvin informs Buzz that "...he was your age when his son was born". Of course, Buzz knows his own age. He should therefore be able to pick the correct solution from the list of 7 possible solutions shown above. However, he can't. That means that Buzz (and the special old guy) must be 60 because there are multiple solutions for an age of 60, whereas any other age would yield a unique solution. Any other age and Buzz would be able to solve.

A 60 year old father – quite a special old guy.

## Friday, August 21, 2015

## Switching Logic

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You are given plenty of time to play around with the switches in the basement, where you can put each individual switch in either its ON or OFF state.

However, you can only go upstairs once to check on the state of the bulbs!

Once you've gone upstairs and checked on the bulbs, you must be able to tell with 100% certainty which bulb is connected to which switch, without having to go down again.

*How can you tell which switch is connected to each bulb?*

**Notes:**

- You can't put a switch halfway between ON and OFF, hoping that this would make the bulb flicker like a bad neon light...
- You can't control the switches from a distance, eg with a string or whatever other form of remote control.
- You can't have anyone cooperating with you on the other floor, and that includes your dog who knows how to bark once for a 'yes' and twice for a 'no'.

### Switching Logic Puzzle Solution

Turn switch #1 ON. After about five minutes or so, turn switch #1 OFF and turn switch #2 ON.Then go upstairs and check on the bulbs.

The one ON is obviously #2. The other ones are OFF, but one of them should be very hot by having been ON for five minutes. That's #1, and the remaining bulb is #3.

## Sunday, August 16, 2015

## Boxes, Beads, and a Blindfold

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You have been named as a traitor by the King, the punishment for this
crime is death. Although he is a cruel tyrant he gives you one last
chance at freedom. When you are finally brought before him he has this
to say to you:

"There are 100 beads, 50 black and 50 white. You will be allowed to draw one bead, whilst blindfolded of course. If it is black you will be condemned to death, if it is white you will be set free".

So far so good you think to yourself, at least I have a 50/50 chance.

"The beads will be distributed amongst four boxes by me," he continued. "You must select a box by opening it, draw one bead from it and then present the bead to the court. Thus will your fate be decided".

Upon saying this a cruel smile appears on the King's face and you suddenly break into a cold sweat as you remember that the King is both very wicked and devilishly cunning.

"There are 100 beads, 50 black and 50 white. You will be allowed to draw one bead, whilst blindfolded of course. If it is black you will be condemned to death, if it is white you will be set free".

So far so good you think to yourself, at least I have a 50/50 chance.

"The beads will be distributed amongst four boxes by me," he continued. "You must select a box by opening it, draw one bead from it and then present the bead to the court. Thus will your fate be decided".

Upon saying this a cruel smile appears on the King's face and you suddenly break into a cold sweat as you remember that the King is both very wicked and devilishly cunning.

*Assuming that the King is incredibly smart, evil, thinks that you are a stupid, uneducated peasant and wants to minimise your chance of freedom, what strategy should you employ, and what is the probability of surviving?***Notes:**

- The King whilst evil won't cheat.
- The trick is to work out how he plans on distributing the beads to minimise your chance of success.
- As soon as you stick your hand in one of the boxes you must draw a bead from it.
- The boxes and beads light and portable, however you are not allowed to remove them from the area.
- The King thinks you are stupid.

### Boxes, Beads, and a Blindfold Puzzle Solution

The king puts 1 black bead in 3 of the 4 boxes and all the other beads (both black and white) in the fourth box.In the kings' view, you will just randomly pick a box because you are so stupid. This gives you barely 1 chance out of 8 to pick a white bead (1/4 to pick that one box containing white beads multiplied with almost 1/2 to pick a white bead out that box).

Assuming each of the four boxes are identical, by picking up each box in turn, you will be able to tell by weight or the rattling noises which one of the boxes contains the mixed beads. Picking the box with the mixed beads will mean that you have a slightly better than 50% chance of living.

Thanks to "Ben Leil" and "Kobold" for posting solutions in the forum

## Tuesday, August 11, 2015

## Which Chest Is Which

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One day Arthur came to Merlin and asked him, "Show me how to be a wise and good king." Merlin replied, "If you can pass a series of mental tests, I will teach you".

Merlin then showed Arthur three chests, one was labelled

**GOLD COINS**, the second was labelled

**SILVER COINS**, and the last,

**GOLD OR SILVER COINS**. He stated that all the three labels were all on the wrong chests. Given that one chest contained gold, one silver, and one bronze.

*How many chests must Arthur open to deduce which label goes on which chest?*

### Which Chest Is Which Puzzle Solution

Arthur does not need to open any chests.Since all labels are on the wrong chests, the chest labelled

**GOLD OR SILVER COINS**cannot contain either gold nor silver, so must contain bronze. Thus the chest labelled

**GOLD COINS**must contain silver coins, and

**SILVER COINS**must contain gold.

## Thursday, August 6, 2015

## U2 Gig

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Unfortunately they only have one torch between them which must be used to cross the bridge safely and may not be thrown, only carried across the bridge.

The bridge will hold up to two band members at any time.

Each member crosses at their own pace and two members must go at the slower members pace.

Bono can cross in one minute, The Edge in two, Adam in five, and Larry in ten.

*How do they make it in time?*

### U2 Gig Puzzle Solution

The trick is to get the two slowest people to cross at the same time. One solution is...- Bono and Edge cross the bridge for which they take 2 mins (Total time = 2)
- Then Bono comes back with the torch (Total Time = 2 + 1 = 3)
- Then Adam and Larry cross the bridge (Total time = 3 + 10 = 13)
- Then Edge comes back (Total = 13+2 = 15)
- Then both Bono and Edge cross the bridge (Time = 15+2=17)

## Saturday, August 1, 2015

## Duck Hunt

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Tim and Tom were playing their Dad's old favourite game machine - a classic
Nintendo Entertainment System (NES). The game was, surprisingly,

Tim pokes Tom, and whines, "let me play." Tom looks back, and Mum is looking at them, so Tom begins to feel generous.

"Tell ya what, Tim. If you can answer a riddle about ducks, you can play. Otherwise, I get a half an hour more."

Tim, who occasionally can be a dunce, happily says "OK!".

Tom asks his question, "How many ducks do you have if you have two ducks in front of two ducks, two ducks behind two ducks, and two ducks between two ducks?"

Tim is stumped. "15?"

"Nope," says Tom.

"12?"

"Nope."

Exasperated, Tim says, "Ok, what is the answer?"

When Tom tells him, Mom goes off smiling, Tim stomps off, and Tom gets the high score on the game.

*Duck Hunt*. Tom was at the controls, shooting innocent ducks with the lightgun.Tim pokes Tom, and whines, "let me play." Tom looks back, and Mum is looking at them, so Tom begins to feel generous.

"Tell ya what, Tim. If you can answer a riddle about ducks, you can play. Otherwise, I get a half an hour more."

Tim, who occasionally can be a dunce, happily says "OK!".

Tom asks his question, "How many ducks do you have if you have two ducks in front of two ducks, two ducks behind two ducks, and two ducks between two ducks?"

Tim is stumped. "15?"

"Nope," says Tom.

"12?"

"Nope."

Exasperated, Tim says, "Ok, what is the answer?"

When Tom tells him, Mom goes off smiling, Tim stomps off, and Tom gets the high score on the game.

*What it the least number of ducks to meet the conditions?*### Duck Hunt Puzzle Solution

Four ducks in a single row would do it.## Monday, July 27, 2015

## The Devil's Muse

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It's everybody's fate, but unfortunately for you today it's yours. You die
and wander through a long tunnel towards a light. But, wait a second, that's
not white light? It's crimson red!

Indeed, going through the gate at the end you face an enormous deity, cloaked in flames, holding a gigantic trident in one hand. On this trident you can make out two bodies speared to it... beads of sweat start to form on your forehead, and they're not from the heat that's omnipresent in this room.

"Hahaha", the Devil laughs, "don't be afraid. You won't necessarily end up like them... they're just the souls of Mickey and Tung. They sold them to me in exchange for the ability to create awesome websites, or something like that, never kept contact with them though... Now, the fact you're here is that this millennium it's my turn to decide who goes to Heaven and who goes to Hell. St. Peter really needed that 1000 year vacation, so I'm on duty now. Since I'm the boss, I get to decide who goes where. And, to be completely honest, this 'Good/Bad'behaviour thingy... it's a bit outdated, isn't it ?

Instead, I would like to read some poetry to you."

And so the Devil starts to recite :

Evil am I.

Evil,

as so to die not sane.

Menace I lay.

A stab mocks.

I revolt.

No din is still.

I kidnap and I kill.

It's sin I don't love.

Risk combat.

Say a lie.

Cane men.

A stone I do toss!

Alive,

I'm alive.

"Now", the Devil says. "If you can point out the single most peculiar thing about this poem to me, I'll let you go and you can take that elevator over there.... what ? ... yeah yeah, the one with hostesses dressed like poultry at its doors... now start thinking, 'cause I ain't got all day!".

Right at that moment you notice that there's still one spike unoccupied on Satan's trident...

Indeed, going through the gate at the end you face an enormous deity, cloaked in flames, holding a gigantic trident in one hand. On this trident you can make out two bodies speared to it... beads of sweat start to form on your forehead, and they're not from the heat that's omnipresent in this room.

"Hahaha", the Devil laughs, "don't be afraid. You won't necessarily end up like them... they're just the souls of Mickey and Tung. They sold them to me in exchange for the ability to create awesome websites, or something like that, never kept contact with them though... Now, the fact you're here is that this millennium it's my turn to decide who goes to Heaven and who goes to Hell. St. Peter really needed that 1000 year vacation, so I'm on duty now. Since I'm the boss, I get to decide who goes where. And, to be completely honest, this 'Good/Bad'behaviour thingy... it's a bit outdated, isn't it ?

Instead, I would like to read some poetry to you."

And so the Devil starts to recite :

Evil am I.

Evil,

as so to die not sane.

Menace I lay.

A stab mocks.

I revolt.

No din is still.

I kidnap and I kill.

It's sin I don't love.

Risk combat.

Say a lie.

Cane men.

A stone I do toss!

Alive,

I'm alive.

"Now", the Devil says. "If you can point out the single most peculiar thing about this poem to me, I'll let you go and you can take that elevator over there.... what ? ... yeah yeah, the one with hostesses dressed like poultry at its doors... now start thinking, 'cause I ain't got all day!".

Right at that moment you notice that there's still one spike unoccupied on Satan's trident...

*What is so special about the Devil's poem?*### The Devil's Muse Puzzle Solution

It's a palindrome.## Wednesday, July 22, 2015

## The Diagram

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Bob looked up from his book and noticed that Joe was staring at the VCR clock,
holding a pencil and a pad of paper. He knew the clock was set correctly, having
set it himself. But he noticed that Joe would occasionally write something down.

Finally Bob's curiosity got the best of him: "What are you doing, Joe?"

"Just another minute" came the reply.

"How long are you going to stare at the clock like that?" As Bob finished he noticed the clock advance a minute.

At this, Joe started scribbling some more until he produced this diagram:

"Why, just ten minutes, of course", Joe beamed as he showed the diagram
to Bob.

Finally Bob's curiosity got the best of him: "What are you doing, Joe?"

"Just another minute" came the reply.

"How long are you going to stare at the clock like that?" As Bob finished he noticed the clock advance a minute.

At this, Joe started scribbling some more until he produced this diagram:

8 | ||

6 | 8 | |

7 | ||

4 | 9 | |

7 |

*What was Joe doing, and what does his diagram mean?*### The Diagram Puzzle Solution

Joe counted how many times each of the 7 LEDs lit up for all 10 digits..._ |_| |_|

## Friday, July 17, 2015

## How Many Monks?

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One day, the head monk of the area comes to visit. He's a bit different - he's allowed to talk and has a life outside the monastry. He tells them that there is currently a plague ravaging the land. People everywhere are dying. The disease manifests itself as red blotches on the forehead. The blotches are the only manifestation of the disease for three months, whereupon the next stage starts, a horribly painful death.

The head monk tells them that at least one of their midst will have the disease, probably more. Anybody with the disease should kill themselves, to save all of the pain and suffering. By killing themselves, they will restrict the movement of the disease, and will go to heaven. Anybody with the disease will show the first symptoms within a month. The chief monk then leaves. He returns two months later, to find that all of the infected monks have killed themselves, and they all did it on the same day.

Bear in mind the following:

- They have no mirrors or any other way of seeing themselves
- The blotches appear only on the forehead and cannot be seen by the monk
- Infected monks feel no different - the only manifestation of the first part of the disease are the blotches.
- The monks cannot talk to each other or in any other way communicate.
- Any monks with the disease will display the blotches within a month of the head monk leaving.
- At least one monk definitely has the disease.
- The monks only see each other once per day, at lunch, when they are all sat round the round table.
- These monks are brighter than the average monk....

*How did they know whether or not they were infected and why did they all kill themselves on the same day?*

### How Many Monks? Puzzle Solution

The way they work this out and the reason it happens on the same day is as follows. It is important to note that not only are the monks intelligent but they all know all their fellow monks are as well.Start with one monk:

He knows he is the only one who can have the disease so he kills himself on

day 1.

Then two monks:

They know that either one of them has the disease or both do. If monk1

doesn't see the mark of the disease on monk2 then he will realise that he

must have it so will kill himself. If he sees the mark then he knows that

monk2 will following the same reasoning and kill himself. If the next day

the other monk is still alive then he realises that the other monk must have

seen the mark on him and so they both have the disease and he must kill

himself. Both follow the same reasoning and kill themselves on day 2.

Three monks:

If only one of the monks has the disease he will see no mark on the other

two and so diagnose himself. He will kill himself on day one. If two monks

have the disease then they will each see one monk with the mark and one

without. When they see each other again the next day they will deduce that

the monk they see with the mark would only not have killed himself if he

could see someone else with the mark. They know it is not the third monk so

it must be them. Both diseased monks follow the same reasoning and kill

themselves on day two. If all three monks have the disease then they are all

in the same position as the healthy monk in looking on in the previous

example. Each of them can see two diseased men. When they haven't both

killed themselves on day two there can be only one reason - the viewing monk

must have the mark as well. All taking the same reasoning they all kill

themselves on day three.

And so on...

*N*monks all with the disease will all kill themselves on day

*N*.

## Sunday, July 12, 2015

## The Geneaology

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Tom and Tim, time tested twice troublesome twins, entered the kitchen at ten
o'clock on a Tuesday evening. "Mom, mom, look what I found" said Tom,
waving a sheet of faded paper.

"No, I found it." said Tim.

"What is it, Tom?" Mom asks.

"I don't know, mom, but it talks about Genies."

"really?" she replied as she took the paper from him. It was a copy of the family geneaology she had been looking for so she could do some research. "where was it?"

"It was in that Bible on the mantle." said Tim, "Between page 588 and 589."

"No," said Tom, "it was between pages 1201 and 1202!"

Mom gave Tom a dirty look, and said to Tim, "thanks for finding this sweetie." She looks at Tom.

"Why did you lie to me, Tom?"

"No, I found it." said Tim.

"What is it, Tom?" Mom asks.

"I don't know, mom, but it talks about Genies."

"really?" she replied as she took the paper from him. It was a copy of the family geneaology she had been looking for so she could do some research. "where was it?"

"It was in that Bible on the mantle." said Tim, "Between page 588 and 589."

"No," said Tom, "it was between pages 1201 and 1202!"

Mom gave Tom a dirty look, and said to Tim, "thanks for finding this sweetie." She looks at Tom.

"Why did you lie to me, Tom?"

*How did Mom know that Tom was lying?*### The Geneaology Puzzle Solution

Page 1 in a book is the page just inside the front cover. Page 2 is the other side of the same sheet.Pages 1201 and 1202 are opposite sides of the same sheet of paper, so finding something between these two is highly improbable.

## Tuesday, July 7, 2015

## How Old is the Vicar?

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There once was a choirmaster.

One day three people came in and asked to join the choir.

The choirmaster, who believes that there should be age for his choir's members, asks their ages.

To that question, one of them replied: "We can't tell you our ages, but we can tell you the following: the product of our ages is 2450, and the sum of our ages is twice your age."

The choirmaster is puzzled: "That's not enough information!"

Just then, the vicar walked in and said: "But I'm older than all of them"

The choirmaster, who knew the vicar's age, then exlaimed: "Ah! Now I know."

One day three people came in and asked to join the choir.

The choirmaster, who believes that there should be age for his choir's members, asks their ages.

To that question, one of them replied: "We can't tell you our ages, but we can tell you the following: the product of our ages is 2450, and the sum of our ages is twice your age."

The choirmaster is puzzled: "That's not enough information!"

Just then, the vicar walked in and said: "But I'm older than all of them"

The choirmaster, who knew the vicar's age, then exlaimed: "Ah! Now I know."

*How old is the vicar?*### How Old is the Vicar? Puzzle Solution

The vicar is 50.The way to solve this puzzle, is to first of all write down all the possible permutations of three numbers whose product is 2450.

Starting Numbers | Product | Sum | Choirmaster |
---|---|---|---|

1, 1, 2450 | 2450 | 2452 | 1226 |

1, 2, 1225 | 2450 | 1228 | 614 |

1, 5, 490 | 2450 | 496 | 248 |

1, 7, 350 | 2450 | 358 | 179 |

1, 10, 245 | 2450 | 256 | 128 |

1, 14, 175 | 2450 | 190 | 95 |

1, 25, 98 | 2450 | 124 | 62 |

1, 35, 70 | 2450 | 106 | 53 |

1, 49, 50 | 2450 | 100 | 50 |

2, 5, 245 | 2450 | 252 | 126 |

2, 7, 175 | 2450 | 184 | 92 |

2, 25, 49 | 2450 | 76 | 38 |

2, 35, 35 | 2450 | 72 | 36 |

5, 5, 98 | 2450 | 108 | 54 |

5, 7, 70 | 2450 | 82 | 41 |

5, 10, 49 |
2450 |
64 |
32 |

5, 14, 35 | 2450 | 54 | 27 |

7, 7, 50 |
2450 |
64 |
32 |

7, 10, 35 | 2450 | 52 | 26 |

7, 14, 25 | 2450 | 46 | 23 |

Since the choirmaster, after being told that the product of the ages is 2450 and that the sum is twice his age, still can't work out the ages, we can deduce that there are two (or more) combinations with the same sum. Those combinations have been highlighted in the table above.

The vicar then claims to be older than all of them. The oldest of the three is 49 in the first remaining combination, and 50 in the other. The choirmaster knows the vicar's age, and after his claim, he deduces everyone's age. The only way he's able to do so is if the vicar is 50, leaving the combination 7, 7, 50 logically impossible (the vicar has to be older, that is at least 1 year older than the others), and therefore learning that the people's ages are 5, 10, and 49.

## Thursday, July 2, 2015

## The Maze

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After entering the maze the first junction you come to is a 'T' intersection where you may continue to the left or the right. You decide to turn right. A short while later you come to another seemingly identical intersection, this time you turn to the left.

Hours later, after arriving at several hundred identical intersections and more or less randomly choosing left or right you eventually decide to head back (it's being some time since your last meal).

Unfortunately you can't remember the way you came and you didn't bring anything useful with you such as chalk or string.

*What do you do?*

### The Maze Puzzle Solution

If there are only T junctions, then all you have to do to get back is take all of the side passages that you come across, as they are the T junctions that were taken from your original perspective.Effectively, every side passage you come across was a T junction that was taken earlier.

Of course, defining this the "ultimate" maze was a bit of an exaggeration on our part...

## Saturday, June 27, 2015

## Four Hats

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Four men have been buried all the way to the neck, only their heads
stick out. They cannot turn their heads, so they can see only in front
of them. A wall has been placed between

A hat has been placed on top of each man's head. All of them know that there are two black hats and two white hats, but no one is told the colour of the hat he's wearing.

They will all be saved if at least one of them can safely say what
colour is the hat he's wearing. Otherwise they'll all be decapitated.

**A**and**B**, so that**A**cannot see the other 3 (**B**,**C**,**D**), and viceversa. All of them know in which position the others have been buried. So, for example,**B**knows that**C**and**D**can see him, even though he can't see them.A hat has been placed on top of each man's head. All of them know that there are two black hats and two white hats, but no one is told the colour of the hat he's wearing.

*Which one of them saved the day? And, most importantly, how?*### Four Hats Puzzle Solution

**C**saves the day.

**D**clearly has the most information at his hands, but seeing one white and one black hat doesn't give him any certainty about his own hat's colour. Would

**B**and

**C**both have been wearing the same colour,

**D**would have been able to provide the answer.

But

**C**is one clever guy and he knows that if

**D**doesn't answer, it means that

**B**is wearing a different colour than him. Because

**B**is wearing white,

**C**knows he's wearing black.

Note that

**A**is redundant: the puzzle could have included only

**B**,

**C**,

**D**. That way, the hats would have been three, with two hats of the same unspecified colour, and one other hat of the opposite colour.

## Monday, June 22, 2015

## Alan and Bert

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I told Alan and Bert that I had two different whole numbers in mind, each bigger
than 1, but less than 15. I told Alan the product of the two numbers and I told
Bert the sum of the two numbers. I explained to both of them what I had done.

Now both these friends are very clever. In fact Bert, who is a bit of a know it all, announced that it was impossible for either of them to work out the two numbers. On hearing that, Alan then worked what the two numbers were!

Now both these friends are very clever. In fact Bert, who is a bit of a know it all, announced that it was impossible for either of them to work out the two numbers. On hearing that, Alan then worked what the two numbers were!

*What was the sum of the two numbers?*### Alan and Bert Puzzle Solution

Starting Numbers | Sum | Product | Product can also be made using |
---|---|---|---|

6, 5 | 11 | 30 | 10, 3 |

7, 4 | 11 | 28 | 14, 2 |

8, 3 | 11 | 24 | 6, 4 and 12, 2 |

9, 2 | 11 | 18 | 6, 3 and 9, 2 |

As mentioned in the question, he's quite clever - so he looks at the product that would appear if he used each of the four possible combinations. As shown in the table above, the product that appears can also be made from different numbers.

So, he announces that it is impossible for him or Alan to work out what the original numbers were. He seems to be on fairly safe ground.

Alan is a little bit more devious as well as being clever. Armed with this snippet of information he needs to look for a pair of numbers that give a non-unique sum and also give a non-unique product.

11 is the only non-unique sum which always gives a non-unique product. Alan is very clever and very smug.

## Wednesday, June 17, 2015

## Weighing an Elephant

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The visitor brought a present with him. It was an elephant.

The visitor gave it and said, "Can you measure the elephant's rough weight in a day?"

The king of the port city consulted with the retainers. "We just have beam scales weighing bags. Do you have any ideas?"

One vassal said, "I can make the relevant measuring equipment assembling large levers and pulleys, your majesty."

"Can you make it in a day?"

"......I can't."

Another vassal said, "How about weighing in pieces after killing the elephant?"

"I won't."

At last they found the method and measured the elephant's approximate weight without sophisticated devices.

*What was the method?*

### Weighing an Elephant Puzzle Solution

Load the elephant onto a boat large enough to carry it. The boat will sink slightly, and you mark the level of the water on the side of the boat. Then you offload the elephant and fill the boat with bags until the boat sinks to the level marked. The bags can be individually weighed using beam scales and the weight of the elephant is the sum of the weight of the bags.This puzzle is slightly cunning in that it the geographic location of the city is a small clue.

## Friday, June 12, 2015

## Carpet Layer

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Walter Wall is a carpet layer. He and his two apprentices are asked by a nightclub owner to give a quote on laying carpet.

The owner indicates an oblong dance floor (figure on the right) and tells them that he wants a square of carpet adjacent each of the sides and running its entire length, making four squares in all (figure below left).

Walter asks both apprentices how many measurements must be made to calculate the total area of carpet needed in order to give a quote.

Sam, the slower of the two, replies that eight measurements are needed: two sides of each square.

Walter reprimands him, reminding him that these are squares and therefore have all sides the same length, and that they are in identical pairs, "So we only need to take two measurements - one side of one of the large squares and one side of one of the smaller squares".

Brian, the bright apprentice, points out that they can give the quote after taking only one measurement.

The owner indicates an oblong dance floor (figure on the right) and tells them that he wants a square of carpet adjacent each of the sides and running its entire length, making four squares in all (figure below left).

Walter asks both apprentices how many measurements must be made to calculate the total area of carpet needed in order to give a quote.

Sam, the slower of the two, replies that eight measurements are needed: two sides of each square.

Walter reprimands him, reminding him that these are squares and therefore have all sides the same length, and that they are in identical pairs, "So we only need to take two measurements - one side of one of the large squares and one side of one of the smaller squares".

Brian, the bright apprentice, points out that they can give the quote after taking only one measurement.

*How can the total area (that is, the sum of the areas of the four red squares) be calculated by taking just one measurement?*### Carpet Layer Puzzle Solution

The only measurement that needs to be taken is the distance between opposite angles of the rectangular dance floor (figure on the left). That distance then will be squared (figure on the right) and doubled to get the sum of the areas of the 4 squares.## Sunday, June 7, 2015

## The Farmer's Problem

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Farmer John had a problem. There were a group of brigands that had
taken all he had... except for three things: his prized wolf, his goat,
and a box of cabbages. They were coming after him, to get the rest.
These brigands did not like water, so John went to the Blue River, a
deep, fast river that no one could swim, and it had no bridges. He
always kept a boat there, because he liked to fish, but it was small. So
small, in fact, that he and only one of his precious things could be in
the boat at the same time.

It sounds simple, right? Ferry one item across at a time, and come back for the others? Well, if John leaves the goat with the cabbages alone on one side of the river the goat will eat the cabbages. If he leaves the wolf and the goat on one side the wolf will eat the goat. If john is there, only he can seperate the wolf from the goat and the goat from the cabbage.

It sounds simple, right? Ferry one item across at a time, and come back for the others? Well, if John leaves the goat with the cabbages alone on one side of the river the goat will eat the cabbages. If he leaves the wolf and the goat on one side the wolf will eat the goat. If john is there, only he can seperate the wolf from the goat and the goat from the cabbage.

*How can farmer John keep his possessions safe from the brigands, without losing a single one?*### The Farmer's Problem Puzzle Solution

There are two solutions:Solution A:

1) John takes the goat to the other side, and leaves it there.

2) He then takes the wolf to the other side.

3) He brings the goat back.

4) He takes the cabbages across, leaving them with the wolf.

5) John Comes back for the goat.

Solution B:

1) John takes the goat to the other side, and leaves it there.

2) He then takes the cabbages to the other side.

3) He brings the goat back.

4) He takes the wolf across, leaving it with the cabbages.

5) John Comes back for the goat.

## Tuesday, June 2, 2015

## Loadsa Coins!

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Greed sidles up to you and whispers, "Actually, there are an infinite number. And it can all be yours!"

Upon seeing your raised eyebrow, he continues. "The rules are simple. 20 of the coins are heads, the rest are tails. All you have to do is split

*all*the coins into just

*two groups*so that the number of heads is the same in both groups."

Seeing the obvious difficulties, you begin to protest. "But..."

"You can turn over as many coins as you like," he interrupts. "And the groups do not have to have the same number of coins in them."

"Oh ok. Can I start now?"

"Sure, you have one day to complete this task. You can start as soon as I blind you for one day."

"How can I get through it all in a day...! What do you mean blind me...!?"

Although you cannot see him, you are quite certain that he is grinning.

*You cannot see. You cannot feel the difference between head nor tail. How can you accomplish this task with certainty in a single day?*

### Loadsa Coins! Puzzle Solution

You reach out and pick any 20 coins, and turn each of them over. The 20 coins is one group, and the remaining coins the other."Bugger," says Greed. "Why didn't I think of that?"

"It's so simple, yet so difficult to think of," you say. "Now can I have my eyesight back? Hello? Hello......!?"

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