Puzzling

Inspired by Charlemagne (and who among us is not?), John Tierney poses some puzzlers and invites his readers to submit their own:

The emperor hired Alcuin, a renowned English scholar of the eighth century, to compile a collection of puzzles. Titled “Problems to Sharpen the Young,” the book was intended to get the youth of the day more interested in mathematics (a perennial challenge, obviously).

Here we go - the first is an old chestnut from the Alcuin collection:

A traveler comes to a riverbank with a wolf, a goat and a head of cabbage. To his chagrin, he notes that ere is only one boat for crossing over, which can carry no more than two passengers — the traveler and either one of the two animals or the cabbage. As the traveler knows, if left alone together, the goat will eat the cabbage and the wolf will eat the goat. The wolf does not eat cabbage. How does the traveler transport his animals and his cabbage to the other side intact in a minimum number of back-and-forth trips?

The second is "a modern classic devised by Boris Kordemsky":

A detachment of soldiers must cross a river. The bridge is broken, and the river is deep. The officer in charge spots two boys playing in a rowboat by the shore. The boat is so tiny, however, that it can only hold two boys or one soldier. All the soldiers succeed in crossing the river in the boat. How?

So far, Easy Cheesy.  Now let's tackle the readers suggestions found in the comments.  From Ryan L:

There are 100 people standing outside of a big room. Some of them have red hats on their head and some have green hats. Their task is to enter the room and situate themselves such that the red hats are all standing together and the green hats are all standing together. They may enter the room all together, one at a time, or anything in between.

Each of them can see everyone else’s hat, but cannot see the color of his own hat. Other than strategizing (before they have the hats on their heads) beforehand, they are not allowed to communicate with each other. No talking, no gestures, no pointing, nothing of the sort.

How do they do it?

This is from Alex:

The host of a party turned to a guest and said “I have three daughters.”

“Is that so?” the guest replied. “How old are they?”

“Well, the product of their ages is 72 and the sum of their ages is my house number.”

“Hmm,” said the guest, “excuse me a moment.” The guest rushed to the door, looked at the house number, and returned to the host. “I’m afraid I need a little more information.”

“Alright,” the host replied. “The oldest likes strawberry pudding.”

The guest smiled and announced the ages of the three girls.

And we will take two from Angus McCamant:

Two brothers own a herd of sheep. The sheep are sold, the price per head is the same as the number of sheep, that is if there were 4 sheep, the price was $4.00 per sheep. They receive a stack of ten dollar bills and some silver dollars. To divide the money the older brother takes the first ten, the younger the next, and so on until the older brother takes the last ten. The younger brother says, wait, you got one more ten. The older brother says “Take all the silver dollars”. Younger replies “There are less than ten silver dollars, you still owe me.” How much does the older brother owe to?

Second:

There are twelve balls, all appear identical but one weighs slightly more or less than the others. The difference is so small it can only be detected with a precision balance. You can place any combination of balls on the the balance and activate it three times. Describe how the balls can be found.

Hmmph - when I present the second puzzle the story is much more elaborate, with jewel thieves, powerful Sultans, and dancing girls.  Well, sometimes.  Anyway - it can be done, but good luck explaining it.  And to be clear, the weighing device is a balance, sort of like the Scales of Justice, with two pans; the heavier pan moves down and the lighter one moves up as objects are weighed.

[end of thread]

Comments

[Sue]

So far, Easy Cheesy.

Seriosly? mumble...grumble

[Sue]

I answered all of the riddles and typepad ate it. I swear. I really, really swear.

[daddy]

Here's a good one:

A President wishes to make America energy independent, so he prohibits all new domestic nuclear power options, bans coal plants and offshore drilling, and increases taxes so drastically on existing domestic oil producers that it's wiser for them economically to go drill overseas in garden-spots like Angola and Tajikistan rather than to struggle with onerous new Government Regulations and the Environmentalists.

Now that America is completely on its knees economically and energy-wise, and completely subject to the whims of OPEC, what does he do to make it all better?

A) Blame Rush Limbaugh

B) Blame George Bush

C) Treat Gordon Brown like s##t!

[Sue]

Oops...it regurgitated it and it appears I am caught in a blatant lie. ::grin::

[Jane]

Me too Sue.

[Sue]

I have a riddle for you...do my posts make any sense if you weren't here when it happened?

[ben]

The twelve balls on the first pass you weigh 6 and 6...whatever side is heavier you know there is a light ball among the other six. You then on the second pass weigh 3 and 3 and once again one side will be lighter. Now on the third pass and last pass you weigh any two of the remaining balls 1 and 1. If they are equal in weight the light ball is the remaining one. If one is lighter than the other you also know which one it is.

[sbw]

Three people are sitting in a circle. Each one holds a cane and each wears either a white or a black hat. They cannot see the color of their own hat, but can see the color of the hats worn by the other two.

The instructions are for each person to tap the cane should they see two white hats or a black and a white hat, but stop tapping should they see two black hats.

As soon as you know what color your hat is, explain how you got your answer.

[Rick Ballard]

Me too, Sue. I'm sure not going to risk retyping all of that.

[sbw]

Charlemagne tapped Alcuin to invent the Liberal Arts curriculum.

And Charlemagne attended class along with commoners.

Both pretty cool.

[sbw]

Eat the cabbage, take the wolf across the river. Poop. Go back and get the goat.

[mel]

Last scale- (And I am searching for the Light odd one, my choice, not yours, and this is a Persian puzzle, by the way)

Half on one side, half on the other. Throw the heavier set out. Do it again. Throw the heavy set out. You are left with three balls, pick two. Weigh them. If they are not balanced, the lighter one is your ball. If they are balanced, the one NOT weighed is the one you are looking for.

I'll do the others later. They are all either arabic, hindu, or persian puzzles and I can't remember them all right now. It's been almost thirty years since I taught that stuff.

(Are we in a new and improved format, while I was gone?)

[TM]

The twelve balls on the first pass you weigh 6 and 6...whatever side is heavier you know there is a light ball among the other six.

Ahh, but we don't know if the non-conforming ball is heavier or lighter, just that its weight is different.

[TM]

And let me add - the farmer struggling with the wolf, sheep and cabbage was easy for me because I try to fit three unruly kids into the car every day.

[Sue]

Well, since I'm an admitted liar already, I'll go ahead and admit I'm a cheat. I know the answer to the wolf, goat and cabbage puzzle, but only because google is my friend.

[MayBee]

Lick the salt, drink the tequila, then suck the lemon.

[bad]

OH yeah!!!

[Sue]

Lick the salt, drink the tequila, then suck the lemon.

What happens right before I start dancing on top of the tables. Finally, easy cheesy.

[kim]

42
==

[mel]

TM-

I defined my solution. You, however, chose the "prism puzzle", which has two solutions, like yours, and no definition.

Nice try. I used to play with proofs for a living, you just found some.

Thanks for letting us play though.

[ben]

"Ahh, but we don't know if the non-conforming ball is heavier or lighter, just that its weight is different"

Good point...weigh any three balls against any three balls. If they are the same the wayward ball is in the remaining 6. You can then weigh 3 of the balls you know are not odd against any three of the remaining 6. If they are equal you know the odd ball is in the last group of 3. If they are not equal then you know one of the groups of three you first weighed has the odd ball and the six remaining are equal. You then weigh any group of 3 you suspect of not being equal against the 3 you know are equal. In any event you will only be left with 3 suspect balls after 2 tries and you will know if they are heavier or lighter because you weighed them against equal balls. Take any 2 of the suspect balls and weigh them. If they match the one left over is the odd ball. If they don't it will either be the lighter one or heavier one.

[bad]

Are any of the balls bright blue?

[ben]

The heavier ones yes.

[Sue]

bad,

LOL. You are truly wicked. And I love it.

[RedHatRob]

put four balls on each side of the balance.

if they balance, the four remaining have the exception. weigh two. if they balance, it's one of the remaining two. if they don't, swap ONE of the balls out...

you can take it from there...

[ben]

"you can take it from there..."

The constraint is you can only do 3 weigh-ins.

[PaulL]

The traveler can manage it in 7 trips.

1) Traveler crosses river with goat
2) Traveler returns
3) Traveler crosses river with cabbage
4) Traveler returns WITH GOAT
5) Traveler crosses river with wolf
6) Traveler returns
7) Traveler crosses river with goat again (and boy is that goat getting fed up)

[bad]

I learned from you, Sue. LOL

Excellent, Ben!

[PaulL]

For the soldiers to cross the river:

The boys must split up, with one on either shore. A soldier can then cross the river in the boat by himself, and the boy there can take it back to the other side where the soldiers are waiting.

Then both boys have to cross again and drop off one boy so he can return the boat after the next soldier crosses over. Et cetera.

Sounds like a very time-consuming process.

[Sue]

Paul,

That is exactly what I had in my post that typepad ate. For real. Really.

[PaulL]

The ages of the daughters: 3, 3, and 8.

The guest has seen the house number (14) and so he knows the only possibilities are that the daughters are ages 2, 6, and 6 or else 3, 3, and 8. But he needs more info to know which is correct and the host obliges by mentioning his oldEST daughter. He wouldn't say oldest if he were talking about one of two 6 year olds.

If the guest had seen some other house number, he would have been able to figure out the ages without extra info, because there would be only one solution. For example, if the house number were 15, the ages would have to be 2, 4, and 9. If the house number were 18, the ages would have to be 1, 8, and 9.

[PaulL]

I believe you, Sue. Where is everybody?

[PaulL]

The sheep problem. In order for the older brother to get an extra ten, there have to be an odd number of tens to divvy up. That means that the tens place has to be an odd number.

For instance, 6 sheep at $6 = $36 with an odd number in the tens place. 16 sheep at $16 = $256 with an odd number in the tens place. 66 sheep at $66 = $4356 and since 5 is odd, there will be an extra ten to divvy.

The cool thing is that no matter how you slice it, the number in the ones place is 6. So the younger brother has 6 silver dollars.

You might think the older brother should give him 4, then, but were he to do that, the older would end up with only 6 himself. So the older brother should give his younger brother 2 dollars, so that they both are even with some amount plus $8.

[PaulL]

Maybe I'll dream about red hats and green hats. Good night.

[Ann]

Are any of the balls bright blue?

Sheesh, you crack me up. If laughing keeps you younger, than bad should be considered the fountain of youth.

[Ralph L]

The hatted people are told to walk into the room in pairs and push their partner into the correct corner.

I hope it was a power boat, or those boys were worn out.

[Ralph L]

Oh, I see pushing isn't allowed. Still paired up, each person must walk to the correct area for his partner, and then go to the area his partner first went to. Much more elegant.

[Carl Pham]

OK, here's a solution to the hat problem.

(1) The first two people (Al and Bert) walk in and stand together in one corner.

(2) The third guy (Chaz) walks in. If he sees that Al and Bert have the same color hat, he joins them. If Al and Bert have different hats, he goes to the opposite corner.

(3) Al and Bert now compare each others' hat with Chaz's hat. That is, Al compares Bert's hat with Chaz's. Bert compares Al's hat with Chaz's. The rule is that if the hats match, do nothing. If the hats don't match, go to the opposite corner.

For example, if Al sees that Bert's hat doesn't match Chaz's, he moves to the opposite corner. If Bert sees that Al's hat doesn't match Chaz's, he goes to the opposite corner.

After this shuffling, there are only two possibilities: either all three have the same hat and are standing in the same corner, or they are divided two and one, with the two that have the same hat in one corner and the odd man in the other. The important point is that in either case, the hats are now properly sorted.

(4 et sequens) Each new guy walks in and joins a group at random. Everyone compares the new guy's hat with the hats in the group he just joined. If it matches, no one does anything. If it doesn't match, everyone EXCEPT the guy that just walked in switches corners.

[bgates]

1) Barack Obama comes by, says "I do not believe in confiscating livestock," and takes the guys' goat. Then his science advisor reminds him the cabbage has carbon in it, so he makes the guy bury it to save the environment. As Obama's yacht pulls away, smashing the guy's boat, he suggests the guy extend an open hand to the wolf that is chewing his leg off.

2) The soldiers think for a minute about the boat and about the two kids. Then they realize that they are Thinking Of The Children, and that means they all have to desert. Of course, they murder the officer in charge first.

3) Trick question - eventually the green hats are revealed to be red hats.

4) The guest smiled and announced the ages of the three girls.
"I don't have any daughters," said the host.
"Sure you do. They're right here," said the guest.
"Begone, you loathsome, undereducated fool. Do you think a person of my breeding and class could fail to notice such elementary facts as the number and gender of my own children? You sicken me," said the host.
"OK, Mr Brooks," said the guest. ("Bye Sarah," said the hosts' 3 daughters.)

5) While they were haggling, fiat money began its collapse. As part of that process, the government seized all private gold and silver. The brothers were left with only worthless paper.

6) By my projections, every ball you put on the scale will be 3.2% higher by this time next year. They must all be lighter than the others.

[Cecil Turner]

I think I have the ball problem:

Weigh two sets of four.
• If they match, weigh two of the other group on one side, one on other with known ball. If they match it's the other ball, if not, remember which side went up, and weigh the two against each other. If they match it's the other ball, if not, it's the one that goes the same way as the previous weighing.
  
• If they don't match, remember which side went up, and weigh two on one side and one on the other with one ball from the other suspect group of four.
  • If they match, weigh two of the remaining suspect three against each other . . . if they match it's the odd one out, if not, it's the one that went the same direction as that group did in the initial weigh-in.
  • If they don't match, weigh the two from the same initial group against each other. Pick the one that goes the same way as the initial weigh-in, or, if they match, the ball from the other two that went the same way in both previous weigh-ins.

[Cecil Turner]

I don't like the hat problem (because I think it's an exercise in trying to find a sneaky way to communicate) but think this works:

Designate corners for each colored hat. Walk in two-by-two, and at the door, each guy starts walking to the correct corner for the other guy's hat. After five paces, if they're still together, they continue to the correct corner. If not, they each switch and walk to the opposite corners.

And I can't find any fault with PaulL's solutions to the math problems, so . . .

[hit and run]

Cecil:
I think I have the ball problem

Oh, gosh, sorry to hear that. I once got poison ivy on my ... wait ... what are you talking about? Oh, the puzzle.

[Jane]

believe you, Sue. Where is everybody?

Cowered by your brilliance.

[MayBee]

ha ha ha ha ha, everybody

What a great way to wake up this am

[Cecil Turner]

I once got poison ivy on my . . .

Heh. However, the real problem with that statement was use of the singular.

And bgates . . . superb.

[Sue]

Jane,

That looks great even though it is obvious to those who read the thread you are referring to Paul and not "Sue". Still, Sue and brilliant in the same sentence, works for me. ::grin::

Good morning brilliant people.

Sheesh...typepad has added another stumbling block in posting.

[hit and run]

Cecil:
However, the real problem with that statement was use of the singular.

Sure. My other line was...http://blogs.abcnews.com/politicalpunch/2008/05/carville-says-o.html>according to James Carville, if Hillary gave you one of hers, you'd both have two.

[bad]

Heh. However, the real problem with that statement was use of the singular.

Not to a one-hanger...

[MikeS]

This, my favorite puzzle, is not really a puzzle at all. It’s an example of the kind of deception the Clinton Administration used to employ.

Three salesmen often shared a room at a particular inn. One evening when they were checking in, one of the salesmen suggested to the innkeeper that he ought to give them a discount for being such good customers. The innkeeper politely declined and insisted upon the full $30 rate for the room, which the tree split equally.

Sometime later, after mulling the idea of a discount over, the innkeeper changed his mind. He gave the bellhop 5 $1 bills to deliver to the 3 salesmen as a discount.

On his way to deliver the money, the bellhop tried to figure out how to divide the $5 between the 3 salesmen. When he got to their room the bellhop gave each of the salesmen $1 and he kept $2 in his pocket for himself.

Earlier, each salesman had paid $10 for the room. Now that they had received a discount, each had paid $9 for his share of the room. 3 times $9 equals $27 dollars, plus we must remember that the bellhop has kept $2. So $27 + $2 = $29. Originally the room had cost $30. What happened to the other dollar?

[MikeS]

"tree split" ??
I uh uh. My keyboard was we we uh uh. You can't just uh ah ah say wha?

[Sue]

Mike,

I actually know the answer to this one. You aren't trying to reach $30, you are trying to reach $25, which is what the room now costs.$30 - 5 = $25. There is no extra dollar.

[bad]

They paid $30 and got a $5 dollar discount which brought the total of the room to $25.

Then they were each give $3 which brought the room cost to $28 dollars. 28/3 was the cost to each of the salesman, (they just don't know it was that cheap,) and the bell boy got a $2 tip.

The missing dollar is a fig newton of your imagination.

Or whatever one of the smart guys say....

[bad]

Ya see how it looked reasonable at first and seemed to make sense? Then ya thought about and went, "JUST ONE MINUTE"!!!!!

I give you governmental accounting.

[bad]

Okay, I posted a clever demonstration of government accounting using that setup and it's disappeared.

I hate when that happens....

[Porchlight]

What a fascinating little riddle. I totally couldn't figure it out until smarter JOM ladies helped me. :)

MikeS, you mentioned the Clintons using this kind of deception - can you give an example?

[bad]

Bold is another word showing up in O's speeches and the echoing commenters at Tapper.

[bgates]

If you liked Mike's, try Lou's. And here's a minute and a half of the Council of Economic Advisers, followed by a Wednesday Night Austerity Party that gets out of hand.

[DebinNC]

Joseph Goebbels used "bold" a lot too.

[bad]

Gosh bgates, I didn't realize the CEA was so intellectual.

[boris]

you are trying to reach $25

You can get to the $30 this way ...

Imagine the salesmen had exactly 30 one dollar bills (10 each). They give them all to the Innkeeper. After the refund the Innkeeper has 25 of the one dollar bills. The bellhop has 2 of the one dollar bills and the salesmen have 3 of the one dollar bills. 25 + 2 + 3 = 30

So the salesmen paid $27 (3x9) for the room and the Innkeeper got $25 and the bellhop got $2.

Clinton was always trying to call a reduction in any increase from projected spending (the mythical $30) a cut even though the spending was still growing.

[bad]

boris, the new administration is no different. They are claiming $1.6 trillion in savings because we won't be doing the surge again for each of the next 10 years.

[Carl Pham]

Very nice puzzle, Mike. The swindle happens in the accounting at the end. The correct statement is:

Earlier, each salesman had paid $10 for the room. Now that they have received a discount and paid a tax to the bellhop, each had paid $8.33 ($10 minus $5/3) for his share of the room and $0.67 ($2/3) in tax to the bellhop. Three times $8.33 equals $25, the new price of the room, what the innkeeper gets.

Three times $0.67 equals $2, the total tax paid to the bellhop. Added together $25 + $2 = $27, or $9 each, the total amount of money the men are out of pocket after the swindle. The difference between $27 and the original $30 they paid is $3, $1 each, and that of course is the $1 of his own money the bellhop refunded to each sucker.

[MikeS]

...you mentioned the Clintons using this kind of deception - can you give an example?

Yes.

[sbw]

Kim: 42

Great answer. But I forgot the question.

[bad]

lol MikeS

you funny

[MikeS]

Porchlight,
I do really and truly have some examples. They aren't readily at hand. I think I may have left them on an old laptop, but they were great examples.
You would have loved them!

[sbw]

42.

Great tribute to Douglas Adams.

[daddy]

Sort of related:

My favorite number is 12345679 (leave out the 8 in the sequence). Simply multiply that by any multiple of 9 up to 81 (ie) 9, 18, 27, 36, 45, etc, and see what it spits out on your calculator.

In many a Chinese market stall when they are pestering me to buy something, in order to combat their aggressive salesmanship routines, I will take their calculator, punch in that number, and multiply. Being sharp people, and fascinated with numbers, they will go "Ohhh" as if I am a magician, and call their co-workers over and start doing that trick on them, and by then everybody is friendly and not so aggressive.

[Porchlight]

MikeS, if you come across them, let me know. I am particularly fascinated by Clintonian deception. ;)

[Porchlight]

I just tried the calculator trick, daddy. That is really cool. I wish I could remember a joke my dad used to tell. You punched in numbers on a calculator to go along with it and the punchline is Shell Oil (77077345, upside down).

[Porchlight]

Oops, that's 71077345, sorry...

[daddy]

I vaguely recall that one Porchlight. There's a Bart Simpson one that's dirty.

Another good problem I bring up when I am chapperoning Fieldtrips and trying to entertain gradeschoolers, is the one where I think Hipparchus of ancient Greece was asked by one of the Ptolemy kings of Egypt to figure out how tall the Great Pyramid of Giza was. It was too high to be able to have any sort of horizontal stick poke out from the top far enough to where you could drop a plumbline down to the ground, so direct measuring wouldn't work. The ingenious answer he came up with was...(next post-down below).

[daddy]

He decided to use Sun shadows. He supposedly lay down in the sand, directly abeam the middle of the pyramid, and marked his exact height, from his foot to the top of his head. Then he stood up again, toed this foot line, and waited for the moment when the shadow of himself cast by the sun exactly matched that height he had previously marked in the sand. hH reasoned that the shadow of the tiptop of the pyramid, mnarked off from the exact middle of its base (the point he was standing abeam of) must also be the exact height of the Great Pyramid. Now all he had to do was measure it out at his leisure. Don't you wish we could put genius's like him in charge of figuring out this economy.

[daddy]

And since I'm bored...

You guys have probably all heard it before, but it has to do with the great Mathmatician Gauss. He was in some boring gradeschool where the teacher hated his students, and when the teacher got frustrated or upset, he would penalize the kids and have them do mindless math drills. The problem he gave one day was to add up every number from 1 to 100. In E.T. Bell's Men Of Mathmatic's, its written up very cool. While all the other kids start slowly tallying on their slates, young Gauss sits a minute lost in thought, then simply writes down the number _,___ ,and tosses the slate on the teachers desk, then he just sits there bored while all the other kids take hours to get the answer. Bell says the teacher thinks Gauss is a young, hotheaded snot, until he looks at the answer Gauss gave and see's it's correct. How did he do it?...(next post-down below)

[daddy]

E.T. Bell say's that what Gauss did was the following: Assume the problem was to add up all the numbers from 1 to 10. A simple way for brilliant brains to do that is as follows. In your mind pair the following:

1 + 10 =11
2 + 9 = 11
3 + 8 = 11
4 + 7 = 11
5 + 6 = 11

So now its simply a problem of multiplying 11 x half of the original amount of numbers to be added together. Half of those original 10 numbers is 5, so its simply 11 x 5, and the easy answer, = 55.

Gauss in his head simply saw that:

1 + 100 = 101...all the way down to
50 + 51 = 101.

So he simply took half of the original 100 numbers to be added together (50), multiplied that in his head by 101, and voila, 50 x 101 = 5050. That's what he wrote on his slate, thats what amazed his teacher, and according to E.T. Bell, that made his teacher instantly realize he had at least one brilliant student, and so he befriended him and mentored him and the rest is history. Apologies if you guys have all heard that before, but sometimes it works wonders with 6th graders on schoolbus Fieldtrips, so thats probably why I haven't forgotten it.

[bad]

Fascinating, Daddy.

Thank you so much for those puzzles.

Not that I'll remember anything but the calculator trick...

[MikeS]

I'm not sure that this is a puzzle. It came up during a nightmare I had recently. In my nightmare I'm having an argument with a close personal friend, when she says, "No you only call if it doesn't last 4 hours! Why the hell would anyone call the other way?"

[daddy]

Well its a puzzle to me MikeS but I'll be happy to pick you up some bootleg Viagra in Thailand if you think that would help.