A woodworker has a disc of wood, perfectly round, an inch thick and ten inches in diameter.  He wants to make it a wheel and so prepares to drill a one inch hole in the exact center.  Sadly, an ill-timed catastrophic sneeze causes him to drill the hole two inches off-center.  Undaunted, he pulls out his mathematically perfect laser saw (which can make perfect, zero-width cuts in wood) and his mathematically perfect glue (which can glue surfaces together with zero distance between them).  He cuts a piece of the wheel away, glues it back in a different position, and he has exactly the wheel he wanted to begin with.  How does he accomplish this?

Our band of intrepid pirates, having resolved previous squabbles over distributing booty amongst themselves and other issues have come across a treasure map fragment.  The picture has been destroyed, but the following text can be read:

Stand upon the gravesite and you’ll see two great palms towering above all others on the island.  Count paces to the tallest of them and turn 90 degrees clockwise and count the same number of paces and mark the spot with a flag.  Return to the gravesite and count paces to the second-tallest of the trees, turn 90 degrees counter-clockwise and count off that number of paces, marking the spot with a second flag.  You’ll find the treasure at the mid-point between the two flags.

Fortunately, our pirates knew which island the map referred to.  Sadly, upon arriving at the island, the pirates discovered that all evidence of a gravesite had faded.  The captain was preparing to order his men to dig up the entire island to find the fabled treasure when one of the more geometrically inclined pirates walked over to a particular spot and began to dig.  The treasure was quickly unearthed on that very spot.

How did the pirate know where to dig?

This is the solution to Morris: Trial/Trual/Whatever.  Please look there before reading the solution.

It turns out the right word is truel, first coined in 1954 by Martin Shubik.

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In Living With Crazy Buttocks, Stephen Morris told us of a rather interesting party. The story continues…

After winning their trip to Paris, the guests became elated and celebrated with the consumption of some adult beverages. Ever responsible, the host confiscated the keys to all cars to ensure that no one drove home drunk. Later on, when things started to calm down, party-goers started to request the return of their keys claiming to be sober enough for the drive home.

Having once been out-done by the guests, our host took another whack. He distributed all of the keys, but did so randomly. He then presented a challenge he felt sure they’d only be able to satisfy if they were indeed sober enough to drive. They were allowed to exchange keys, but only in rounds. During each round, each party-goer could either do nothing or pair up with another party-goer and exchange the sets of keys each was holding. (Each party-goer could be part of at most one pairing per round.) No one would be allowed to drive home unless everyone recovered their own keys.

The host wished to allow only a fixed number of rounds. To be fair, he wanted to be sure that it would indeed be possible to make the change. However, he also wanted to make it as difficult as possible for the party-goers. What is the minimum number of rounds must allow them to ensure that an exchange would be possible?

For clarity, all key recipients can discuss, share information such as who has the keys of whom, and agree upon a strategy. Also, careful readers will realize that there were 20 guests at the party originally. Sadly, it was a rather disorderly party and some guests did leave early, but many more appeared. Everyone present at the key ceremony had a key confiscated, and everyone with a key confiscated received a key for this challenge, but neither you nor the host knows just how many there are.

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So, I’m teaching a new course, Math 2033, Mathematical Thought, and it’s going great! I’d like to take a moment to write about it!

(This is one reason the MF has been kinda slow lately; another is that I’m chair) When it’s fully up and running, we’ll have about 150 students in one large section each semester (we’re starting with about 100). In a nutshell, it’s the Math Factor, as a course.

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Recently I discovered Stan Wagon’s Problem of the Week.  This is a delightful mailing list / site and some of the problems are in the vein of puzzles I post here.  Recent problem 1125 captured the attention of several Math Factor authors so I thought I’d post the puzzle here as an excuse to introduce you all to that list.

You have eight batteries and know that four are good and four are dead, but don’t know which are which.  Your only method of testing them is to insert two into a device that will work if you’ve put in two good batteries and not otherwise.  How many such “tests” are required in order to be sure that you’ve located two good batteries?

As of this posting, the answer to this question is not yet on the POTW website, but if you come to this later, the spoiler may be there, so be careful to avoid spoilers if you want to work this through.

Finding yourself chatting around the water cooler one afternoon, you and two co-workers agree that you would all like to know the average of your three salaries but none of you want your individual salary to be known to either of the other two.  Without need of involving any external person or machine as some sort of secret keeper, how can you achieve this end?

Janine is one of twenty guests at a Christmas party.  Each guest is given a book as a present.  Janines’s book is called ‘Living with Crazy Buttocks’.  She isn’t sure what to make of that.

The guests are invited to play a game.  Each book is put into an identical cardboard box.  The boxes can be opened and closed without leaving a mark.  The twenty boxes are piled up around the Christmas Tree.

The guests are told that they will each have the opportunity to open half of the boxes.  Their objective is to find their own book.  If they all succeed the group wins and they will win a trip to Paris.  If any one of them fails then the group fails but they will each get a Twinkie to keep for life.

The guests are taken to another room and then taken to the tree one at a time.  They cannot see what any other guest does at the tree.  They are not able to communicate once  the game starts.  The boxes are put back after each guest, as though they had never been there.

You would think that the chance of the group succeeding was 1/2^20 but they can do much better than that.

The group must come up with a strategy before the game starts.  What is the best strategy to get the group to Paris, and let Janine keep her ‘Crazy Buttocks’?

These books are all real.  They will be helpful to you if you have had any of the following thoughts:

We all know the Nazis killed millions of innocent people but what were they like on ecological issues?

I would like to speak Italian but can’t be bothered to learn any Italian words, can you help?

Aubergines are very flushed, just how angry are they?

I think I’m dead, how can I tell for certain?

I am rich but dead.  How should I pimp my coffin?

I am worried about running into large, slow moving objects; can you suggest any strategies to avoid this?

Just how boring was 1587?

I live thousands of miles from Versailles.  Will I get a good view?

I am English, am I human?

My buttocks are insane.  

There are five holes in a row in my yard.  A fox lives in them moving around as follows:  Each night, it abandons it current residence and moves to an immediately neighboring hole.  If I’m allowed to check one hole each morning, identify a sequence of holes that I can check in order to be sure to catch the fox.

I recently got back in touch with an old friend and puzzler and he reminded me of a puzzle that he once told me about that confounded me for weeks.  Faced with a restatement of it, again I couldn’t come up with an answer for the life of me.  The mechanism is painfully simple, but there is something about the particulars here that short my mind out.

Combine the four number 1,3,4,and 6 with operators of addition, subtraction, multiplication and division (and parenthesis to indicate order of operation) to yield an expression equal to 24.

I assure you that you can take this in the most straight-forward manner possible.  You aren’t mean to smoosh them together to get “13” out of 1 and 3.  You aren’t meant to use “1” as a problem number or something of that sort.  An answer will look something like this:

(4-1)*3/6

except that is equal to 1.5 .  Your expression must equal 24.

I’m interested to hear if this is as difficult for others as it was for me.