The Cosmic Number

four is cosmic

4 is cosmic.

1 is 3, 3 is 5, 5 is 4, and 4 is cosmic.
8 is 5, 5 is 4, and 4 is cosmic.
2 is 3, 3 is 5, 5 is 4, and 4 is cosmic.
15 is 7, 7 is 5, 5 is 4, and 4 is cosmic
9 is 4 and 4 is cosmic.
18 is 8, 8 is 5, 5 is 4, and 4 is cosmic.
17 is 9, 9 is 4, and 4 is cosmic. 
100 is 10, 10 is 3, 3 is 5, 5 is 4, and 4 is cosmic.

If you think you recognize the pattern, submit an example.  If you need more examples, I can give you more.

Running Out of Riddles

Well I thought I could make it a whole year providing a new riddle every week, but it appears I am almost out.  I will continue posting puzzles as I find them, but I can't guarantee a weekly riddle any more at this point.  For now, an old one my dad once told me:

What did the left eye say to the right eye?

The Hateful Neighbors

The Boggis, Bunce, and Bean families were once united in their pursuit of a common enemy, but have since developed a bitter and irreconcilable hatred for one another.  Each family lives in its own house in its own part of town, and all is well and good, so long as members of different families don't cross paths - if they do, they will start brawling until the poor sheriff has to come out and break them up.  They are very civil indoors, however. 

Each family needs to be able to access the post office, the general store, and the sheriff's office without encountering members of other families.

So the sheriff has come up with a great plan; draw up plans for each family to have its own three paths, traveling from each home to a each of the three municipal buildings.  In this way, for example, the Boggis family has three paths, where each path leads from its front door to the front doors of the post office, general store, and sheriff's office.  Can he do this without letting the paths cross, and without digging any tunnels or building any bridges - in other words, working in a two-dimensional plane?

1000 Bottles of Wine

Borrowed from folj.com. 
You are the ruler of a medieval empire and you are about to have a celebration tomorrow. The celebration is the most important party you have ever hosted. You've got 1000 bottles of wine you were planning to open for the celebration, but you find out that one of them is poisoned.
The poison exhibits no symptoms until death. Death occurs within ten to twenty hours after consuming even the minutest amount of poison.

You have over a thousand paid caterers to help with the testing and just under 24 hours to determine which single bottle is poisoned.

You have a handful of prisoners about to be executed, and it would mar your celebration to have anyone else killed.

What is the smallest number of prisoners you must have to drink from the bottles to be absolutely sure to find the poisoned bottle within 24 hours?

SEND MORE MONEY

One time in college I emailed my dad asking for money and he said I could have some if I solved the following equation:

   SEND
+MORE
_______
MONEY

Each letter represents its own digit (0-9) and multiple occurrences of the same letter represent the same digit (eg if one of the E's represents a 3, they all do).

A Boat in a Tank

Imagine you are in a small rowboat floating in a swimming pool.  There's a big rock in the boat and you drop it overboard. Does the water level rise or fall?  Why?

The Magic Square

This is one I come back to when I'm bored and all I have is pen and paper.  I also read in a biography of Benjamin Franklin that he used to do this when he was stuck in boring meetings.  Construct a 3x3 grid of numbers, using numbers 1 through 9, and arrange the numbers in the square such that every row, column, and diagonal (diagonals through the center) adds up to 15. 


Too easy?  Now construct a 4x4 grid made out of numbers 1 through 16, such that every row, column, and diagonal adds up to 34.  This one is killing me because I figured it out once, but can't seem to rediscover the solution.  There are ways to do this for grids 5x5, 6x6, and up, though they no doubt get very difficult.

No Riddle This Week

I can't think of any and I'm too busy with work, sorry everybody. 

4=5

The other day, someone in our creative services department tried convincing me that 4=5.  He offered a convincing proof, which is displayed below.  But what is wrong with it? 

Pirates, Part 2

Again, taken from folj.com. 

The five pirates mentioned previously are joined by a sixth, then plunder a ship with only one gold coin.


After venting some of their frustration by killing all on board the ship, they now need to divvy up the one coin. They are so angry, they now value in priority order:

1. Their lives
2. Getting money
3. Seeing other pirates die.


So if given the choice between two outcomes, in which they get the same amount of money, they'd choose the outcome where they get to see more of the other pirates die.


How can the captain save his skin?

The Blind Date Bachelor

(Modified from a puzzle about a sultan and his harem I heard from Jon Huang.) The Blind Date Bachelor is the newest dating show, in which where there is 1 bachelor and 4 contestants trying to win his heart. He will meet each contestant for the first time on a blind date. At the end of the date, he must choose whether to marry her or never see her again. If he marries her, the game is over. If he rejects her, he is set up on a date with the next contestant and repeats the process. If he rejects the first 3, he marries the last one automatically. He is able to compare and rank contestants that he has already met, but will not know for sure who he likes best until he has met them all. It's very important that he marry the best one, or he will spend the rest of his life wondering what could have been. What strategy will maximize his chances of finding the best mate-for-life?

3 Bonus Questions: What is the probability of winning using the best strategy?  What if there are 5 contestants, not 4?  And finally, what if there are n contestants? 

Pirates

I've seen this one before and haven't solved it.  Go ahead and comment answers in the blog, but not on Buzz or else  people will see them.  I quote this one from folj.com, but I've seen it before in other places as well. 

Five pirates have obtained 100 gold coins and have to divide up the loot. The pirates are all extremely intelligent, treacherous and selfish (especially the captain).

The captain always proposes a distribution of the loot. All pirates vote on the proposal, and if half the crew or more go "Aye", the loot is divided as proposed, as no pirate would be willing to take on the captain without superior force on their side.
If the captain fails to obtain support of at least half his crew (which includes himself), he faces a mutiny, and all pirates will turn against him and make him walk the plank. The pirates start over again with the next senior pirate as captain.

What is the maximum number of coins the captain can keep without risking his life?

Dots and Lines

This is a very old one, and many of you may have seen it before.  I try to avoid posting puzzles with outside-the-box solutions, but I made an exception for this one.  Using 4 straight, connected lines, connect all 9 dots.

Heads and Tails

This is borrowed from folj.com and I don't know the solution.  Submit ideas in the comments section - maybe you'll get it before I do. 

There are twenty coins sitting on the table, ten are currently heads and ten are currently tails. You are sitting at the table with a blindfold and gloves on. You are able to feel where the coins are, but are unable to see or feel if they are heads or tails. You must create two sets of coins. Each set must have the same number of heads and tails as the other group. You can only move or flip the coins, you are unable to determine their current state. How do you create two even groups of coins with the same number of heads and tails in each group?  [Note: I assume you need a guaranteed method of getting even groups, rather than method that will likely work.]

The Lantern and the Bridge

A small family is being pursued by an unknown enemy in the middle of a dark, dark night.  It comes to a deep chasm spanned by a narrow bridge. 

The family is composed of a father, mother, grandfather, and child.  The father is athletic and can cross the bridge in 1 minute; the mother can cross in 2 minutes; the child can cross in 5 minutes; and the grandfather, the slowest, takes 10 minutes to cross.  They have a lantern with them. 

Since it's pitch dark, the bridge can't be crossed without the lantern.  The bridge is so narrow that only two can cross at a time, and each pair can only move as quickly as its slowest member. 

Their pursuer is likely not far behind.  What is the quickest way to get everyone across the bridge?

Dots and Rows

This is an old one my dad told me, though I've also heard it as a CarTalk puzzler before:

Above is a straight row of 4 dots: it is a 4-dot row.
Next, we have a set of 13 dots that creates 5 different 4-dot rows.

 

Now again create 5 different 4-dot rows, but using only 10 dots.  Rows must be straight.  Oh, and before you tricksters try giving me a single long string of dots, a 5-dot row does not count as 2 4-dot rows.  You can use a dot multiple times, but rows can't overlap.

Cubic Calendar

From http://www.folj.com/puzzles/

A corporate businessman has two cubes on his office desk. Every day he arranges both cubes so that the front faces show the current day of the month.

What numbers are on the faces of the cubes to allow this?

Note: You can't represent the day "7" with a single cube with a side that says 7 on it. You have to use both cubes all the time. So the 7th day would be "07".

A Lost and Hungry Vagabond

This is from the Car Talk Puzzler:
A lost and hungry vagabond happened upon a pair of travelers one of whom had three loaves of bread while the other had five. All of the loaves were the same size and weight.

The two travelers decided to share their bread with the vagabond, and that the eight loaves should be shared equally among the three of them. When they had finished, the vagabond reached into his pocket and pulled out eight coins. He handed three coins to the traveler who had had the three loaves and five to the other one and disappeared into the inky shadows.

The next morning, right after no breakfast, the one who had received the three coins said to the other one, 'I don't think he should have given three coins to me and five to you. It's not fair.' And he was right. How should the coins have been split up?

The Camels

Borrowed from http://www.folj.com/puzzles/

Four tasmanian camels traveling on a very narrow ledge encounter four tasmanian camels coming the other way.

Tasmanian camels never go backwards, especially when on a precarious ledge. The camels will climb over each other, but only if there is a camel sized space on the other side.

The camels didn't see each other until there was only exactly one camel's width between the two groups.

How can all camels pass, allowing both groups to go on their way, without any camel reversing?

Hint: to help visualize, use paper clips or coins.  

Apples and Oranges

This also came from Wu Riddles. 

There are three closed and opaque cardboard boxes. One is labeled "APPLES", another is labeled "ORANGES", and the last is labeled "APPLES AND ORANGES". You know that the labels are currently misarranged, such that no box is correctly labeled. You would like to correctly rearrange these labels. To accomplish this, you may see only one fruit from one of the boxes. Which box do you choose, and how do you then proceed to rearrange the labels?

The Yukiad Contraption

Thanks to Jon Campbell for telling me about this one, found at Steven Landsburg's blog, The Big Questions.  The original puzzle is from the novel The Yukiad, by David Snaith. 


Consider a  glass contraption—a perpetual motion machine, really—consisting of a clear glass hula hoop on the ground containing several colored beads, which travel through the hoop, some clockwise, some counterclockwise, all at the same speed, bouncing off each other in perfectly elastic collisions whenever they collide. Whenever two beads collide, they instantly bounce off each other and proceed in the opposite of their original directions, still at the same speed. Take a snapshot of this system at, say, 12PM. Must there be some time in the future when another snapshot of the system will look identical? In other words, does the history of the system repeat itself?

(Hint: for simplicity, start with a few beads, then generalize)

Cut the Cake

Happy Birthday, Kyle! 
With 3 straight cuts through a cylindrical cake, make 8 equal-sized slices. 

There are two solutions.

(Hint: you are allowed to move some or all of the cake)

The Prisoner, the Liar, and the Truth, Part II: Enter the Other Guy

Recall from a couple weeks ago, the door that leads to the electric chair in the Prison for Creative and Unusual Punishment.  It is located in a depressing, windowless basement hallway near the back, right next to an identical door that leads to an unguarded fire exit.

A prisoner with similar lamentable circumstances to the guy in part 1 was being brought in for execution, and the warden wanted to give him one last chance at freedom.  This time, however, would be more complicated.  Rather than two guards, there were three guards with them, named Al, Bob, and Carl.  The warden held a brief huddle with the three guards, out of earshot of the prisoner.  He then told the prisoner some of what went down in the huddle. 

"I've instructed one guard to tell nothing but lies when asked a yes-or-no question.  I've instructed another guard to tell only the truth when asked a yes-or-no question.  I then repeated one of those two instructions to the remaining guard.  Unfortunately, I'm not going to tell you whether there are two liars an a truth-teller, or two truth-tellers and a liar.  And I'm certainly not going to tell you which is which.  You have two yes-or-no questions to ask, choosing one guard at a time to respond (no asking a question of the whole crowd).  From the information you glean, you may choose a door.  Best of luck."

What two yes-or-no questions can the prisoner ask whose answers will lead him to freedom?

Please submit answers in the comments section of the blog, or to me directly (no spoilers in Google Buzz).

The Catenary Chain

This week's post on Steve Strogatz's New York Times column reminded me of this puzzle.  In it he re-explains math in a creative and intuitive way, from basic counting through imaginary numbers, functions, and more.  The post reminded me of this because in it he talks about how mathematical functions can explain every-day shapes, such as how water at a drinking fountain forms a parabola.  A hanging chain forms a catenary, but its shape can be approximated with a parabola.  It is a great new one I found at Wu Riddles, and apparently it originated at a Microsoft Interview.  BUT: don't be intimidated by all the math, I have faith that you can solve this one. 

You have a 6-foot long chain that is suspended at its ends, tacked to a wall. The tacks are parallel to the floor. Due to gravity, the middle part of the chain hangs down below the ends, forming a 'U'-type shape; the height of this 'U' is 3 feet from top to bottom. Find the distance in between the tacks.

The Prisoner, the Liar, and the Truth

The door that leads to the electric chair in the Prison for Creative and Unusual Punishment is located in a depressing, windowless basement hallway near the back.  It just so happens that it is right next to an identical door that leads to an unguarded fire exit. 

One day the warden was bringing a prisoner to be executed, but was feeling sorry for him.  The prisoner had been convicted on dubious charges and DNA evidence had been unearthed that might have exonerated him, but the court threw it out.  So the warden wanted to give the prisoner one more chance at freedom.  There were two guards with them and the warden whispered in their ears, out of earshot of the prisoner.  The warden instructed one to tell nothing but lies when asked a yes-or-no question and the other to tell only the truth when asked a yes-or-no question.  Otherwise they were to remain silent.  The prisoner couldn't tell which was the liar and which was the truth-teller. 

The warden then told the prisoner that he could ask one yes-or-no question to one of the guards.  If, from the answer, he could pick the correct door to freedom, the guards would look the other way. 

What question should the prisoner ask?

Brothers and Sisters, I Have None

This is one of the first riddles I was able to remember consistently and I got it from my Dad.  A man looks at a portrait and says, "Brothers and sisters, I have none.  But that man's father is my father's son."  What is the relationship of the man in the portrait to the speaker? 

Solution to the Scale, Part 3 is up

I remembered the solution to the scale riddle with 12 weights and only 3 weigh-ins.  It's located in the expanded "The Scale, Part 3" post. 

The Scale, Part 3

Told to me by Tommy Dickie:


This time you have 12 identical-looking gold coins.  11 are equal weight, 1 is counterfeit and you don't know whether it's heavier or lighter than the others.  Using an old-fashioned balance scale only three (3) times, find the counterfeit coin and determine whether it is lighter or heavier than the others.  

The Monks

There are 40 monks living on a deserted island, at the foot of a mountain with a monastery at the top. Each monk has either blue or green eyes--there is at least one monk of each eye color, but none of them knows which color he has. They can see each other's eyes, but cannot in any way communicate to each other anyone's eye color (ie, everyone knows everyone ELSE's eye color, but not their own). They gather every day for supper, but otherwise spend their time alone in their individual secluded huts. One day, God appears and tells them all that green-eyed monks are welcome to hike up the mountain to the monastery; but ONLY green-eyed monks are allowed. He says that one morning, every green-eyed monk will wake up knowing that he is a green-eyed monk, and when that day comes, they will each ascend the mountain during sunrise, leaving just the blue-eyed monks to gather for supper later that day. This does indeed happen. How do the green-eyed monks accomplish this?

Sorry for the Delay

I'm sorry I didn't post anything last week. To compensate, I offer two riddles: one easy, one hard. I guess the easy one is whichever you get first. Feel free to post ideas in the comments section.

The first I got from Wu Riddles and the second from Tommy Dickie. The second is similar to a riddle about cheating husbands.

The Penny, the Cork, and the Bottle

Take a penny, an empty wine bottle, and a cork.  Put the penny in the wine bottle and cork the bottle.  Now remove the penny without pulling out the cork and without breaking the cork or bottle.

The Glass

You have a perfectly cylindrical glass completely full of water (no rounded edges on the bottom, same diameter all the way down).  How do you empty out exactly half of the water?  There are no markings on the glass and you have nothing else to work with.

2 Ant Riddles

1. An ant is on the corner of the floor of a perfectly cubic empty room and wants to get to the farthest corner of the room, on the ceiling.  What is the shortest route?
2. Four ants are on each corner of a square, 2x2" Post-It note.  Each ant is facing the ant immediately to his left.  All four ants begin marching at once, each aiming for the ant immediately to his left.  Eventually the four ants spiral into the middle of the Post-It note, where they meet.  How far does each ant travel?

Lions and Lambs

There are 17 perfectly rational lions and 1 lamb on a magical island.  Lions only eat lambs; they will not eat each other.  The magical thing about this island is that if a lion eats the lamb, it becomes the lamb by magical transformation.  The lions are very intelligent and live by two rules:

1. Don't get eaten
2. Eat a lamb only if it doesn't result in a violation of rule number 1.  

 A biologist observes these 17 lions and 1 lamb for a little while and then leaves for several years.  When the biologist returns, how many lions and how many lambs will remain?

The Scale, Parts 1 and 2

Part 1: You have 8 identical-looking gold coins; 7 are equal weight, 1 is counterfeit and slightly heavier.  Using an old-fashioned balance scale only twice, find the counterfeit coin. 

Part 2: Consider 8 gold coins again.  However, this time it is unknown whether the counterfeit coin is heavier or lighter.  You can use the scale 3 times.  Find the counterfeit coin.

The Egg Drop

A Phoenix Egg is best served hot. 

From Neel Tiruviluamala, via Tommy Dickie:

You've been assigned to determine how much force it takes to break a Phoenix egg and you are permitted the use of a 100-story building.  Phoenix eggs are hard and may break if dropped from the first floor or may not even break if dropped from 100th floor.  You need to find the highest floor from which the eggs will not break.  Phoenix eggs are very rare - you only have 2 identical eggs to work with.  Of course you could simply start by dropping an egg from the first floor, then the second floor, and so on until it breaks, but who has time for that?  In the worst-case scenario, in which the egg does not break even from the 100th floor, this would require 100 drops.

What strategy could you employ that requires the fewest number of drops?  Obviously the number of drops required depends on where the egg will break, so you will be judged according to the worst-case scenario.  That is to say, you will be judged according to the maximum number of drops your strategy requires.  Feel free to submit solutions to charlesdguthrie@gmail.com. 

On Sunday, if I remember, I will provide a hint.

Jugs, Part 2 and a Bonus!

You're at the ocean again and you need exactly 4 gallons of water this time.  Unfortunately, all you have is an opaque 3-gallon jug and an opaque 5-gallon jug.  What will you do?

Belated Bonus:
A man walks out of his house and heads due south one mile.  He then turns left and heads due east one mile.  Finally, he turns left and heads due north one mile and arrives back home.  Who is this man?