I took one of Peter Winkler’s puzzle books on holiday recently.  After dinner each night I intended to impress my friend with an amazing math puzzle.  I had done this before.

The book dissapeared on the flight out.  After dinner each night my friend impressed me with an amazing math puzzle.  I haven’t seen the book since.

Serves me right!

This is one of those puzzles, you will understand why I have to do it from memory.

I really like Jeff’s post  A Fun Trick – Guess the Polynomial.  You might want to look at it first.

If you relax the conditions a bit you have a similar sounding puzzle with a very different solution.

So my puzzle is this:

I am thinking of a polynomial.  All of the co-efficients are fractions.   You may use any number as your test number.  When you give me a test number I will tell you the result.

How many test numbers do you need to identify the polynomial?

Here’s an old puzzle, I think from Martin Gardner though I can’t immediately find a reference, adapted as a fun trick for spring break — if you have just the right sort of friends.

Invite a friend to invent a polynomial of any order, but require that it have only positive integer coefficients. Next, you explain that you have to get something of a feel for the polynomial so you provide an integer and ask them to apply the polynomial to it, telling you the value. You may have to do this more than once. You then close your eyes and dramatically tell them what their polynomial is.

What strategy would you use and how many values would you have to provide?

There has been a theme in some of the recent posts and problems. It’s a little buried but almost enough to say its another of those Mathfactor agendas when we try to sneak some knowledge to you buried in the fun.  Never one to miss such an opportunity I will jump in with a post, and a problem.  This is a slight change to a classic problem that comes out of the work of one of my mathematical heroes:  Leonardo of Pisa, also known as Fibonacci.  He is responsible for changing how we count! Not many people can claim that. He introduced the system base value, also known as Arabic numerals that we still use today into Europe.  He is more famous however for talking about rabbits:

Imagine that you have immortal rabbits, Bugs Bunny’s version of Olympus perhaps.  Even if they are immortal however rabbits are famous for one thing.  They breed like, well rabbits.  Some of the rabbits are children and some adults and are divided into pairs.  Each month any child pairs become adults and any adult pairs breed to produce a new child pair.  They are immortal so no pair ever dies.  These rabbits are also a little odd.  They live on a line (don’t complain, this is no more ludicrous than that they are immortal!), but can shuffle along.  Also if you are worried about inbreeding, the male rabbit leave the family hutch and shuffle along the line past others until they find a suitably unrelated mate.  Why we would be worried about inbreeding in immortal rabbits living on a line escapes me!

Anyway we start with one pair of children.  Lets put a c.  After a month they become adults, a.  Another month passes and they now have a pair of children, but are still there themselves.  We therfore have the original pair and a pair of children: ac.  Next month the adults have another pair of children and the children become adults: aca.  Can you see how this will work?  Each month the children become adults so we replace every c with and a, each pair of adults has a new pair of children but stays as adults, so we replace every a with ac.  We can continue to get longer and longer sequences of rabbits on this line:

aca  to acaac to acaacaca to acaacacaacaac….

Now some puzzles.  Given a line with 21 adult pairs and 13 child pairs, how many pairs of adults and children would there be after one month?

Given p adults and q children how many adults and children will there be after one month?

Finally a more difficult one.  How will the ratio of adults to children behave month on month?  Will it

a) Get closer and closer to a particular number?

b) Keep on changing without pattern?

In either case can you say more?

You have a sack of coins of three types: brass, silver and gold. You know that the majority of the coins are gold, though they’ve been painted and partially hollowed so that you can’t actually determine the type of a particular coin. Fortunately, you have a machine into which you can insert two coins and the machine will tell you whether the two coins are of the same type or different types.

Your task is to locate a gold coin.

You will compare the coins in “passes” with each coin not being compared more than once in a pass. In a pass, every coin can be a member of a comparison or not, but any particular coin can’t be part of more than one comparison during the pass. Your goal is to minimize the number of passes required to be sure that you locate a single gold coin.

You should be able to describe how many passes (at most) your solution will require rather than the number of passes increasing arbitrarily with the number of coins that turn out to be in the bag.

Here’s a quick puzzle that I don’t think has appeared here previously.

A man leaves at exactly 6:00 AM to climb a mountain.  He may not climb at a constant rate.  In fact, he may stop to rest, or even backtrack a bit to see something interesting.  He arrives at the summit at exactly 6:00 PM and camps for the night.

The next day, he starts down at exactly 6:00 AM, again meandering unevenly, but following exactly the same path and reaching the bottom at exactly 6:00 PM.

The question is, was there some point on the path he was sure to have visited at exactly the same time on the two trips?

I have just published a (rather long) article on mathematical surfaces, their models and links to art over at Maxwell’s Demon.  Here is a sneak preview.

Minimal Möbius, Benjamin Storch

Edmund Harriss, sometime contributor to the Math Factor, makes his first appearance in this early segment, from February 29, 2004.

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Sometimes rules set up to achieve one result can have exactly the opposite result.

You are a financial trader. 

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Inside every hexagram there is a cube trying to get out but, just like the thin woman hiding inside Eddie, it isn’t that easy to spot. 

New Scientist magazine has a regular math puzzle called Enigma.  A few years ago they posed this intriguing problem.

‘Assign the numbers 1 to 12 to the twelve vertices of a hexagram so that the sum along each line is the same.’

At first sight this is the worst possible kind of puzzle.  You have to try lots of ways of assigning numbers until you stumble across the right answer.  Very boring!

The best kind of puzzle is when you spot a neat insight that makes everything easy.

Guess what, that is exactly what we are going to do here!

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For me the holidays end on Monday so I just have time to post this seasonal question.

Why is Christmas the same as Halloween?

Specifically why is Oct. 31 = Dec. 25.

Hint: you need to look at exactly how this is written.

A short problem with a long pedigree.

I found this in Martin Gardner’s book ‘The Colossal Book of Short Puzzles and Problems’, problem 3.9 in my copy.  He credits Solomon Wolf Golomb, the inventor of Polyominoes which inspired tetris.

Isaac Asimov based a whole story on this puzzle,  ‘A Curious Case of Income Tax Fraud’, part of his Black Widowers series.

I asked some work colleagues and had some amusing answers, none of which were maths related.  Maybe you have your own?

Enjoy!

Steve