When two men get up ridiculously early to fire pistols at each other we call it a duel. Personally I prefer to lie in.

But what is the right term when three men skip breakfast to fire pistols at each other?

In a cold, misty field near the outskirts of Paris the sun peers over the horizon to see three men face each other with pistols.  Xavier is an expert shot, he never misses. Jean-Christophe is a very good shot, he will get you four times out of five. Francois only has a fifty/fifity chance of hitting his target.

They each take turns to fire their pistol.  

What is the best strategy for each of them and what odds would you give for the last man standing?

After the bodies had been cleared away I started to wander home only to hear the referee say ‘Maintenant, M. Galois et ami’

{ Hi, Steve here. Jeff asked me to post a solution and I’m more than happy to oblige. It’s a fun puzzle with some nice maths to explore. I learnt a lot about graph theory and a new theorem (new to me), Turan’s theorem. More on that later. }

In Yoak: Batteries, and the Problem-of-the Week Jeff posed a great problem from Stan Wagon’s Problem of the Week.

You have eight batteries, four good and four dead. You need two good batteries to work the device; if either battery is dead then the device shows no sign of life. How many tests using two batteries do you need to make the device work?

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In Living With Crazy Buttocks  I posed a problem where 20 party guests were each given an unusual book.  These books were placed in identical boxes.  The guests enter the room with the boxes one at a time and are allowed to open half of the boxes.  They leave by a different door and cannot communicate with the other guests.  The room is put back identically before the next guest enters.

If every guest finds their book then the whole group win a trip to Paris.

What is their best strategy?

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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.  

Hmm. Somehow Stephen Morris pulls off that rarest of Math Factor tricks– leaving Kyle and Chaim at a loss for words, with his sneaky clock puzzle.

Standard Podcast [ 6:38 ] Play Now | Play in Popup | Download

Kyle and Chaim get into trouble with their wives and Mathfactor correspondent, Stephen Morris, discusses the Kate Bush Conjecture and And The Clocks Struck Thirteen  

Standard Podcast [ 10:33 ] Play Now | Play in Popup | Download

Oh by the way, would you like a cool Math Factor Poster? Click on this to download:

How does an amatuer mathematician collaborate with a professional?  Through the internet of course!

We do it all the time on Math Factor.

Chaim pointed me at the Macalester Problem of the Week.  This led to my making a minor contribution to a published paper.  I can’t claim it’s a world changing paper, or that my contribution amounted to much, but I did get my name in print!  You can read an extract here.  {Just above is a review of a book on symmetry, I’m not sure that is real, one of the authors is called Chaim Goodman-Strauss, clearly a made up name.}

It certainly is a fun paper.  Stan Wagon is a bit of a legend, as you’ll see from the picture.   I’m campaigning for all cycle paths to be built for square wheeled bicycles!

Can you solve some of these problems?

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In working out the proof for World of Britain I came across a paradox.  Maybe smarter Math Factorites can help me out?  My sanity could depend on it.

In the puzzle you have five different tasks.  On each day one of these tasks is given at random.  How long do you expect it to take to get all five tasks?

First consider a simple case.  Suppose some event has a probability, p, of happening on any one day.  Let’s say that E(p) is the expected number of days we have to wait for the event to happen.  For example if p=1 then the event is guaranteed to happen every day and so E(p)=1.

How can we calculate E(p)? 

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“It was a bright, cold day in April, and the clocks were striking thirteen.”

opens George Orwell’s novel ‘Nineteen Eighty-Four’.

1.  By an amazing coincidence thirteen squared is 169 which is the number of times my clock read the right time recently in a single calander day.  Normally it only reads correctly 164 times in a calander day.  This is even more surprising as my clock has been stopped for several years.  How can this be?

My solution combines a number of different techniques.  If you can think of any way a stopped clock can read correctly more than twice a day please post in the comments.  If you can think of something I’ve missed then we may be able to get a bigger answer!

2. I have a second clock which runs slightly fast and I have no way of adjusting it.  How can I make my clock read the right time?

3.  I noticed recently that my third clock was two minutes fast.  It runs at one minute per minute.  It tells me the right time four times a day.  Why?

For inspiration you may want to listen to Peter Sellers (Bluebottle) and Spike Milligan (Eccles) discussing the stopped-clock problem way back in 1957. 

Thanks to New Scientist’s Feedback Column and it’s readers for some of the idea’s here.

This made me smile.  I hope it makes you smile too.

What is

If you think they are the same then why?  If you think they are different then why?