The worlds first riddle!

The following table lists the steps:

Starting Bank The Boat Finishing Bank
HHHEMM
HHHM EM
HHHM E M
HHH EM M
HHH E MM
HE HH MM
HE HM HM
HM HE HM
HM HM HE
MM HH HE
MM E HHH
M EM HHH
M E HHHM
EM HHHM
HHHEMM


Not lost again.

While crossing the desert, you become lost in a sandstorm. After wandering for days, your water supply nearly exhausted, you come upon an oasis. With plenty of fresh water and edible flora about, you replenish your supplies and contemplate your situation.

You know from experience that you can carry about three days worth of water and food. Your most efficient walking pace covers about twenty miles in a day. You have an accurate compass, so you can travel in a straight line, but you don't know where you are. Fortunately, you know that no part of the desert is more than a hundred miles in any direction from civilization.

Two questions:

a)How quickly can you guarantee escape from the desert

b)If the desert was more than a hundred miles in radius, what's the furthest you could travel given unlimited time

Assume that the oasis has unlimited supplies, you can always find any supplies you drop along the way, and that you can drop water and food in arbitrary amounts.

You can get all 100 feet of rope and you don't need the pocketknife.

The desert one sounds very similar to the rocket one somebody posted the other day. There the answer was 15 days but in this one you can place your supply bases anywhere you want. Hmmm will have to think.

The answer to part b) is infinite.

The first part is confusing.



So this is not radius 100 miles is it? It's diameter.
Very different answers depending which you mean.

OK, full solution for the rope one.

1. Climb up one of the ropes and untie the knot in the other one allowing it to fall to the floor.

2. Tie the ropes together giving you 100 feet hanging from one ring. NOTE: MAKE SURE that the knot is small enough to pass through the ring without fouling.

3. This is the hard part. Now you have to tie the loose end of the rope into a harness around your waist and climb up to the top again.

4. Hang from one ring while untieing the rope from the other. NOTE: DO NOT let the rope fall or you're buggered.

5. Thread the rope through the ring. This makes a basic pulley that you can use to lower yourself back to the ground.

6. You now have a 100 foot length of rope to tie around the bars in the window. Abscond at your leisure.

Answers.
Two questions:
How quickly can you guarantee escape from the desert?
If the desert was more than a hundred miles in radius, what's the furthest you could travel given unlimited time?
Assume that the oasis has unlimited supplies, you can always find any supplies you drop along the way, and that you can drop water and food in arbitrary amounts.

The strategy to solving this dilemma revolves around correctly dropping supplies on your exit path to reach safety, and then doubling back over your steps.

That said, the answer to the second question, "If the desert was more than a hundred miles in radius, what's the furthest you could travel given unlimited time?" is "An infinite distance". Go twenty miles, drop one days worth of food, and go back. By repeating this, you can effectively create a new oasis twenty miles away.

The first part of the question is trickier. In fact, so tricky the Minotaur missed it. You can escape from desert in less than twelve days. Here is how it might work...

On the last leg of your journey, you want to leave the desert with no food. I.e., you want to spend the last three days walking out without stopping or backtracking. To accomplish that, you'll need three days of food at a point 60 miles from the end. Let us say that's Camp Alpha.
How do you end up with three days of food at Alpha? Start one day away from Alpha (which is 80 miles from the edge of the desert) at a Camp Beta, walk to Alpha and drop off one day's worth, and return to Beta on the remaining day's supplies. On your next trip, you again arrive with two days supply. This, combined with the day's supply from the previous trip is enough to get you out of the desert. Note the trick is that you did not have to plan for a return trip to Beta.

Now you see that at Camp Beta you'll need six days of supplies. To get that, you will need a Camp Delta 12 miles (1/5 of a full load) from Beta. Walk from Delta to Beta and drop off 3/5 and return. Repeat and you now have 6/5 of a load. On your next trip, you once again plan to not return and arrive with 4/5 of a load making a total of 2 loads, or six days of supplies.

Therefore, Camp Delta is 92 miles and from freedom, and we are going to need nine days (three loads) worth of supplies there. This time we're only 8 miles from the oasis so creating another camp would be inefficient.

Getting from Camp Oasis to Delta and back requires 4/5 of a day of supplies. The remaining 2 1/5 days can be deposited. After two trips, we have 4 2/5 days at Delta. On the third trip we have 6 3/5 days. On the fourth trip we have more than 9 days worth. Over the first three round trips, you consumed 12/5 days of food. The last trip was a one-way, so that brings it up to 14/5 consumed. You now have the nine days of food at Delta required to leave the desert (you were trying to leave the desert, remember?).
Therefore, in your flight you consume 11 and 3/5 days of food so therefore that is how long it takes to escape.

All fine and good, until someone came up with a better way.

They cleverly observed that the problem states that no part of the desert is more than a hundred miles from civilization in any direction. Therefore, the desert does not have a hundred mile radius, it has a hundred mile diameter, because points just at the edge of the desert must be within 100 miles of civilization in all directions. Using this information, the following solution was devised:

Walk thirty miles in one direction. If you have not reached civilization, you now know that civilization is not more than a hundred miles in any direction from your new location.

Return to the oasis, re-supply, and walk ten miles in the opposite direction. Drop off supplies, and return. Then head out again along the same path using the food drop to extend your range to seventy miles. Seventy and thirty equals a hundred, so you must reach civilization within that range. Maximum time to escape the desert: 7.5 days.

Answer to the rope trick.

First, tie the two ropes together at the bottom. Next, climb one of the ropes. Once you are at the top, cut the other rope at the ring. You now have a hundred feet of rope tied together.

Feed the loose end of the rope through the ring, and pull through just enough slack that it touches the floor. Grab the ring that the rope feeds through, and cut the other end of the rope free. You may want to drop your knife at this point and grab both ends of the rope, or the momentum of the falling rope might be enough to drag the other end through.

You now have a hundred feet of rope looped through one ring. Holding onto both halves, slide down to safety and then pull the rope after you.


Over Columbus Day weekend yours truly received his ' Hang I' rating, the first rating for Hang Glider pilots. I did so flying a Falcon 195.
When the sun is directly overhead and the Falcon is on flat level ground, and with its wings level with the Earth, it casts a shadow measuring 195 square feet.

During one flight, when I was about 100 feet off the ground, I glanced down at my shadow and thought, "Gulp."

After landing, I wondered, what was the area of the Falcon's shadow when I was 100 feet off the ground?
Assume the sun was directly overhead, a level ground, and level plane of flight. What % larger or smaller



I spent ages on this. Only to be later told that it is an old ?'classic' some of you may enjoy the contest.

• You are a contestant on a TV show.
• You win a chance at the big car prize. The host shows you three curtains and tells you that there is a brand-new car behind one of the curtains.
• He asks you to choose a curtain.
• After much deliberation, you choose one of them.
• He then opens one of the two curtains you did not choose and shows you that it is empty. (He always opens an empty one because he knows where the car is.)
• He then turns to you and asks if you would like to switch from the curtain you chose to the other closed curtain.

a) What should your response be

b) What are the chances of the car being in the curtain you initially chose

c) What are the chances of the car being behind the curtain to which you are being invited to switch

Yes, I know. The questions are getting more difficult. However, all questions are easy if you know the answer. :wink:

Hand glide answer.

As it turns out, at any altitude the Falcon's shadow will be 195 square feet. Because of the extreme distance of the sun, the photons' paths are effectively parallel when they meet the glider.

The nit-pickers out there will recognize that this solution is theoretically not completely correct. Because the sun is actually a finite (though very large) distance away, the shadow should be slightly larger than the glider. But because the sun isn't a point, but a large sphere, light emits from many sources. And since these sources cover an area larger than the glider, the shadow should be smaller than the glider.

Which is correct? Both are: the glider's umbra, the part of the shadow where none of the sun's rays hit, is smaller. The glider's penumbra, the part of the shadow where some, but not all of the sun's rays are blocked by the glider, is larger. And both of these answers ignore the fact that photons don't necessarily travel in straight lines. But that's for another puzzle.

Old ?'classic'

It seems like the answer should be 1/2 but actually the chances of it being in the curtain that Monty offers you are 2/3 or 66.67%. Since there are 3 curtains and you pick one, you will be correct 1/3 of the time. The chances that it is in the two other curtains are 2/3. Because Monty always opens an empty curtain the chance of the final curtain that you can switch to does not change and is still 2/3. If he opened a random curtain then you would have the same odds if you switched or not.
If you always stick with your initial guess, when do you win? Only when you pick the car with your initial guess. This is 1/3 of the time.
If you always switch from your initial guess, when do you win? Only when you do not pick the car with your initial guess. This is 2/3 of the time.
If he was opening a random door then it would not make any difference, it would be 1/3 of the time either way because 1/3 of the time he would open the curtain with the car.

Here is another way of looking at it.
There are a million doors. You choose one of them. Monty opens 999,998 empty doors leaving the one you chose and another one. Do you really think you picked the car on your first try and should not switch.
From the sci.math.faq on the Monty Hall Problem, version 7.5, dated: 2-20-98.

There are several ways to convince yourself about why it pays to switch. Here is one. You select a door. At this time, assume the host asks you if you want to switch before he opens any doors. Even though the odds that the door you selected is empty are high (2/3), there is no advantage on switching as there are two doors, and you do not know which one to switch to. This means the 2/3 are evenly distributed, which as good as you are doing already. However, once Monty opens one of the two doors you selected, the chances that you selected the right door are still 1/3 and now you only have one door to choose from if you switch. Therefore, it pays to switch.

Believe me this is correct. Here are some other discussions about it.
http://www.io.com/~kmellis/monty.html
http://www.cut-the-knot.org/hall.shtml



Something for the weekend


a) A thief sneaks into a museum to steal an 18 ounce gold bar. The bar is on a pressure sensitive plate. Prior to the burglary, the thief stopped by a local science supply shop and purchased an 18 ounce lead weight. He swaps the two quickly, a la Indiana Jones, only to be surprised by the alarm.

Which weighs more 18 ounces of lead or 18 ounces of gold


b) When you flip on the light switch, how long does it take an electron to travel from the switch to the light bulb, traversing 10 feet of copper wire


c) Two red cards, and two black cards are shuffled, and placed face down. You select two- what is the probability that they are the same colour.


d) Fly due south from the capital of Florida until you are at the same latitude as the capital of Brazil.

What country or body of water is beneath you


e) What satellite has orbited the Earth for the greatest number of consecutive days


f) Can you think of an English word, which has all six vowels in alphabetical order


g) Name ten body parts that are spelled with three letters. No slang words.


h) What do the following have in common "Johnny B. Goode", "Rite of Spring", "Beethoven's 5th", "Dark Was the Night" and "Melancholy Blues" (Yes, I know they're all music, thank you.)


i) An assassin with a high-powered rifle shoots you in the foot from 50 feet away. The bullet travels at 1300 feet per second. You and the assassin are at sea level. What will be the first evidence to you of the attack


j) At the end of Star Trek II, Khan babbles some last words. Whom is he quoting


k) Name the seven wonders of the ancient world.

Have a good weekend

a) Your American measures always confuse me
b) Depends : DC - some time, depends - but always slower than speed of light. AC - never..
c) Deceiving. VERY. As in the cars problem.
d) Huh, I need a navigator over Pacific.
e) Moon
f) no.
g) Oko, uho, las, rit, kri, arm, leg, lip, toe, rib
h) Besides being listed in your sentence?
i) Light, sound, pain (depends on you attention & reflexe)
k) Easy for Civ players

Concernig the old 'probability' puzzle:
"If he was opening a random door then it would not make any difference, it would be 1/3 of the time either way because 1/3 of the time he would open the curtain with the car. "
If he was opening a random door, your probability would be 50-50 after seeing an empty space. It would be 0-0 after seeing a car.

It is interesting that if you don't actually know if the man uses random or not you can't decide correctly - to switch or not (it does not hurt if you switch - since it is 50-50 in random case. But imagine he is always choosing a 'full' curtain on purpose only showing blank when you already chose the car? In this case you must not switch!)

With the old problem, I think I am right in saying he always opened an empty curtain. You must be right with the odds. 1 in 3 to start. Then 1 in 2 with the one open curtain.

Anyway, this riddle contains no weights.

What do you know about, conditional probability'? Easy question difficult answer.

You are lost in the woods. You come to a T-junction and are wondering which way to go. You have a funny feeling that one way will get you out, the other will mean certain death. Weird, eh?

A pixie approaches you.

"A left turn will get you out of the woods," it advises.

A nearby sprite hears this and calls out, "Beware of him. He lies!"

"Exactly 4 times out of 10, I do!" the pixie retorts, and disappears.

You know from folklore that all sprites always tell the truth. You also know that throughout their life, an individual pixie tells the truth a certain number of sentences out of ten, determined by chance at birth. You also know this number is an integer.

Which way should you turn to maximise your chance of escape, and why



Oh, No. Not still lost?

Lost in the forest, you come across a clearing that diverges into 3 roads. There is a cabin here, and, not knowing which road to take, you knock on the door hoping to find the correct path out of the woods.

An old man answers and says that, while he no longer remembers the trails, maybe one of his children can help you. The old man says that out of his ten children, 5 of them always tell the truth and 5 of them always lie. However, he also adds that only 5 of them actually know the correct path.

Each child knows what type of child each of the others is, and what knowledge they possess. The ten children are lined up, and you ask each of them separately which road to take: the left, middle, or right. While each child whispers his answer to you, it is still loud enough that the next person in line can overhear what they said.

The ten replies were...

1) Take the left road
2) Take the right road
3) Don't take the middle road
4) Take the right road
5) Don't take the right road
6) Take the middle road
7) Don't take the left road
8) Take the left road
9) Take the middle road
10) Don't take the left road

Which road is the way out

No takers I see. Well, I do not blame you, they were not easy.
So, this riddle won't help. Only the best of the best would ever be able to get near the answer.

Great News! You've been nominated to be advisor to the
king. It's a great position meaning unbelievable
wealth and prestige. Not to mention being a Royal
Advisor is a great BABE MAGNET! Unfortunately, the
king is one of THOSE kings. You know the kind. They
select advisors based on the candidate's ability to
discern hat color. If you get this job one of your
first pieces of advice to the king will be to discard
this silly system.

The king's advisory board is a four member panel and
the king has decided that he wants a fifth advisor.
That's the position you hope to fill. You know nothing
about the rest of the advisors. They are cloaked in
mystery. When you meet with the king he explains the
rather complex rules to the 2004 Get A New Advisor
Test:

"It's really quite simple" he says "you will be
blindfolded and led into a room with the other four
sages. One who tells the truth at all times, without
fail. One who lies all the time without fail.
One who alternates between truth and lies and
One whose rules when he makes three statements are different from the
other three sages.
This final type of sage is also known as a mystic. "These sages are each wearing different coloured hats. The four colours are Red,
blue, Green and Black. These hats also signify the
power of each sage thusly" Here the king hands you a
little leaflet with "Chain of Power" written on it.

Chain of power
Red = Most powerful
Blue = Second most powerful
Green = Third most powerful
Black = Least powerful.

"You have to deduce which sage tells the truth/lies
etc., each sages position in the chain of power and
the colour of each of the sages hats" the king
continued "by listening to what each sage has to say.
You can tell the sages apart because they all have
very distinctive voices (convenient, eh?).

"What you have to do is deduce from the sages
statements the colours of each of their hats. If you
answer correctly then you will be my next high-paid
advisor. However, should you be wrong, you would be
tossed into the fiery furnace. If you cheat you will
be beaten, quartered, and then tossed into the
furnace--so don't do that." "That's a bit violent,"
you say. "I'm having second thoughts about this"

"Oh, by the way," The king says "If you back out now
you'll also get the furnace. I don't want any
quitters in MY kingdom"

You are about to tell the king to take his hats and
his test and stick them where the sun don't shine when
you notice a parade of lovely ladies-in-waiting stroll
slowly by. Hmm... OK...

"Bring it on!" you tell the king.



You are blindfolded and wheeled into the room where
the sages are. The sages are, according to the king,
labelled, A, B, C & D and you know that they will speak
in that order.

Sage A says

(1) B is a mystic
(2) D's second statement is false

Sage B says

(1) Only one of C's statements is true
(2) The lying sage is two places higher in the 'chain
of power' than the truth telling sage

Sage C says

(1) A does nothing but lie.
(2) D alternates consistently between truth and lies
and he is not a mystic.

Sage D says

(1) A always tells the truth
(2) C is higher in the 'chain of power' than A

When they finish you think. For a very..long..time.
You have to be right or else face certain death. It's
all down to you now, can you figure out which sage
tells the truth, which sage lies, which sage
alternates evenly and which sage is the mystic? Can
you tell their position in the chain of power and,
most importantly, can you deduce the colour of their
hats



Quickie.

There are four common English words that end in ?'dous'
Tremendous, stupendous, and horrendous, what is the fourth

Words ending in 'dous'

I query the fact that there are only four. 'Hazardous', and 'blizzardous', are perfectly good English words and I am sure there are others.

Three-letter body parts:
Arm, leg, toe, eye, ear, rib, gut, jaw, gum, hip, a$$, t!t, nut.

Three roads:
The right road

Hats of power:
C = red and alternates
A = blue and lies
D = green and is a mystic
B = black and tells the truth.

Waterlilies:
I wrote a while back that 'Monet clinched it for me'. That seems to have gone right past you, Try, so I guess I am being too smart for my own good again.

Water lilies:
I wrote a while back that 'Monet clinched it for me'. That seems to have gone right past you, Try, so I guess I am being too smart for my own good again.

Quote Lacomus,
Posted: Sat Feb 07, 2004 2:45 pm
"Also, and I make no other comment, Monet clinched it. "

I fully accept that you did indeed know the answer to the riddle. In fact, you were three days ahead of, Quote Adrian "Water Lilies. Good one Try. " (Feb 10th)

"Words ending in 'dous' I query the fact that there are only four. 'Hazardous', and 'blizzardous', are perfectly good English words and I am sure there are others."

You have a far better dictionary than I. Mine only lists ?'Hazardous'.

Body parts, last three are slang and not allowed.

Hats of power:

This problem was without a doubt the hardest I have, and you have answered it correctly. May I say very well done.

How about this:

During the Cold War, a submarine communications officer received the following message in Morse code which, when decoded, read:

TOP SECRET
Captain Mitchell
We have a spy in our midst and we believe he
is giving information to the Russians. We
have recently made a breakthrough and have
been able to narrow down our list of suspects
to one of the 252 members of your crew. We
have no exact information and no physical
description. We are convinced that you are
not the spy and as such we entrust you to
perform some covert investigation to aid us
in our attempt to catch the rogue operator.
We wish you good luck in this endeavour but
we must remind you to keep this secret. We
needn't remind you a refusal to follow these
instructions exactly to the letter will
result in an immediate court-martial.
Instructions for arrest:

1) If you fail to find the spy then, upon
docking you will ensure that none of your
crew can leave the ship.

2) If you do ascertain who the spy is then
you should arrest him, confine to the Brig
Immediately & place him under 24 hour guard.
From this point on you should cease all
Communications with us.

Naturally, the Captain was perturbed by this message. He was just about to destroy the message to ensure safety when he was presented with another message, which read.

Addendum to previous message:
D-C / T-H / A-A / O-H / C-A / X-K / W-A / O-F / X-D.

After the Captain read this message, he was a little less perturbed than he was before and soon decided on an appropriate course of action. What was it

Try

Either your 'hats of power' is not the hardest puzzle you have posted or I was fortunate to stumble on the right approach. It took me no great length of time and, for sure, that has not always been the case with all of your puzzles

I have sent a pm with my proposed solution to the 'spy' puzzle.

The 'a to k' list of questions:

a) Gold would be heavier as precious metals are weighed in Troy ounces, lead in ounces Avoirdupois. (And I agree fully with Relative about non-metrical units of measurement)

b) No precise figure but it would take surprisingly long. For example, the average snail would get there faster.

c) A tricky one. At first sight I was prepared to argue that the answer should be 'evens'. Only later did I realise/calculate that it would be one-in-three.

d) and e) Relative has already answered.

f) facetiously

g) I have answered this one elsewhere.

h) No idea, but I figure that variation in taste - i.e. a wide selection - has something to do with it.

i) This question is ambiguous with no clear answer. The light stimulus would reach the brain first, the sound second, the pain third. However, the first two could mean anything (and could well go unperceived). Only the pain would - possibly - say for sure that this is an attack. So only the pain would be 'evidence of an attack', the others would be, if evidence of anything, merely evidence of a flash of light and of a bang until connected with the injury.

j) What is a Star Trek?

k) Not without a lot of trawling through my brain, if then.
I would argue that this is an arbitrary list and it has changed more than once. Also, it is difficult to take a list seriously that does not include Asia or the Americas, where there are some astonishing ancient constructions. The Great Wall of China is not a wonder? Ptooie!

h) Music they sent up on Voyager.

i) You would hear it before you felt it.

k) Moby Dick.

Adrian

Sure you'd hear it before you felt it. And you'd see it before you heard it. But the question referred not to knowledge in general but 'evidence of an attack' and an attack, by definition, is not a flash of light or a loud bang.

Okay, here are the answers.

a) The 18 ounces of gold weigh substantially more than the 18 ounces of lead. Unlike the classic "pound of feathers vs. pound of bricks" trick, there is a difference- not in physics but units. Gold is commonly measured in troy ounces which is 31.103 grams metric. This is compared to a normal (avoirdupois) ounce which is 28.350 grams.


b) Several hours: While electrons travel at close to the speed of light, their path inside the wire is not direct. The electrons are constantly bouncing off each other. So while the electricity completes the circuit instantaneously, an individual electron takes a very long time. Related would be a photon generated near the centre of the sun- it might take close to a million years to reach the surface of the sun, and another 8 minutes to get to the Earth.


c) Assume the four cards were the four queens: there are two ways of getting the same suit: QS/QC and QH/QD, while six total possible combinations. So one in three.


d) The Pacific Ocean. Most of us accustomed to Mercator projection maps think of North America being more or less lined up directly above South America, but look on a globe and see that almost all of South America is east of North America.


e) The moon. (groan...)


f) There are two: facetiously and abstemiously.


g) Ear, eye, arm, toe, lip, leg, gum, hip, jaw, rib.


h) They were all on the first album to go "trans-solar", on Voyager.


i) If you happened to be looking straight at him, the flash from his rifle would reach you almost instantly. Assume for a moment that the he is a competent assassin who hid himself well (and for some reason was aiming for your foot). The speed of sound is approximately 1129 feet per second at sea level. The sound will reach your ears in .044 seconds. The bullet will hit your foot in .038 seconds. So much for dodging the bullet on this one. But I asked which would be the first evidence to you- and that would be the sound. Because once the bullet hits your foot, the pain impulse must travel your myelinated A-fibers to your brain at 330 feet per second. Assuming you're five feet tall, that's an extra .015 seconds onto the trip.


j) Captain Ahab, from Melville's "Moby Dick".

k) The Pyramid of Giza, The Gardens of Babylon, The Lighthouse of Alexandria, The Colossus at Rhodes, The Statue of Zeus, The Temple of Artemis, and the Mausoleum at Halicarnassus.


If you got eight or more of these questions right without using outside sources, you can consider yourself a master Riddler. If you got twelve, I don't believe you! :wink:

Congratulations to Adrian, Lacomus and Relative for ALL the correct answers.

Try this:

In the quiet village, almost every man is completely faithful to his wife. Why? Because, all the men are honourable gentlemen. Slightly germane as well is the long-standing tradition that if a woman ever learns her husband is unfaithful, she will kill him that night while he sleeps.

It also may be relevant that the women are notorious gossips. If a woman discovers a man to be untrue, either on her own or through heresy, this fact will quickly be known throughout the village within a matter of hours to all the married women except the wife of the unfaithful man.

One fateful day a fortune-teller came to town and revealed the terrible truth: some man or men had been unfaithful.

For a week, a great fear gripped the town as every woman wondered if her man was true, and as every philandering man wondered if his days were numbered.

Then, on the eighth morning after the fortune-teller came, the coffin maker received his orders. How many men were dead



China has been grappling with a population problem for some time.
For numerous social and cultural reasons, families strongly prefer male children to female.

Consider a hypothetical city somewhere in China where the practice has arisen that every family continues to procreate until a son is produced, at which point they stop having children.

Assuming that boys and girls are born with equal probability, what is the ratio of boys to girls after 100 generations


Simple riddle. (Part 1)

If you fear that I am near,
Here's a wise word in your ear,
You may look forth and aft for me,
But to do so simultaneously,
Ensures you'll just look straight through me,
Even if I'm right behind thee.

What am I

Finally I got some time for the ten brothers puzzle.. and solved it quicker than I thought.

You'd best take the MIDDLE road.

Relative

About the pixie : I'm no expert in conditional probability, but i'd say it is more likely that this pixie lies than not. I'd say this because he cannot tell 0 lies out of ten (the sprite told you so), which would be required to balance that he could be telling 10 out of 10 lies. Ergo, it is more likely that the right road will save you. Concerning his self-referential statement (that he lies 4 times out of 10) i believe it is not relevant at all. We can only say it is more likely to be a lie than not - but we cannot get any info out of it.

Does this make any sense ?

Relative, you 'win' again

Take the middle road.

If you break down the 10 answers, you have 2 people telling you to go right, left, and middle. You also have 2 people telling you NOT to take the left. Therefore, left cannot be the correct answer because 6 people would be lying.

To decide whether to go right or middle, you must determine how many children MUST have known the correct road ahead of time for each case. Remember, 5 children do not know the correct path, but can determine what to say or not to say by overhearing the child before them.

If the correct road is right, then children 1,5,6,8,9 are lying and children 1,2,4,7,9,10 (6 children) must have known the correct path without help from someone else.

The correct road is the middle, since only children 1,2,3,6,9 needed to know the road. The statements given by 4,5,7,8,10 can be deduced from the child's statement preceding them.


The solution to the Pixie problem is found through conditional probability.

To begin with, there are 11 possibilities for the number of times out of 10 the pixie lies. For each of these possibilities, find the probability that the pixie would be able to make the statement "I lie exactly 4 out of 10 times." For example, in the case that the pixie actually lies 7 of 10 times, there is a 7/10 probability that he is able to make the (false) statement "I lie exactly 4 out of 10 times."

The probability of the pixie being able to make the statement is the same as the probability of the pixie lying, except for the case that he is telling the truth (4/10); in that case, the probability of him being able to make that statement is 6/10 (which is equal to the probability of him being able to make a true statement).

Now, conditional probability says that the probability of the pixie lying X number of times out of 10 in his life is equal to the probability of him being able to make the statement (X/10, unless X is 4) divided by the sum of probabilities for all the cases. So, for our example, the probability that the pixie tells lies 7 of 10 times is:

(7/10) / (0/10 + 1/10 + 2/10 + 3/10 + 6/10 + 5/10 + 6/10 + 7/10 + 8/10 + 9/10 + 10/10) = 7/57

Similarly, the probability that the pixie lies X of 10 times is:

X/57, for X not equal to 4,
6/57, for X equal to 4.

Now we can find the probability of the pixie lying in his first statement. There is a X/57 chance (or 6/57 for the 4 of 10 case) that the pixie lies X of 10 times, and for each of these cases there is a X/10 chance that the pixie lied about the path. To find the odds that the pixie lied, simply add the products of probabilities for each case:

0/570 + 1/570 + 4/570 + 9/570 + 24/570 + 25/570 + 36/570 + 49/570 + 64/570 + 81/570 + 100/570 = 393/570 = 68.947%.

So, the probability the pixie lied is 68.947%, and you should go right.

Relative, your reply did answer the question.

The correct road was the middle road.

I said 'the right road'.

Sure enough, child 5 must have told the truth thus eliminating 'right'.

If only I could see where I went wrong as easily before I posted as I can after someone has told me the right answer