Age is Just a Number

 

It has been said that mathematics and music are young people's games.  The best work seems to be done in the persons 20's or 30's.  Examples are Mozart in Music and Galois in Mathematics.  I never understood the analogy there though because Mozart died at 35 under mysterious circumstances and Galois at 21 in a silly dual after spending the prior night jotting down things he had in his mind, that had not been as yet published.

If they had lived long lives would they have done even more?  In any event older people are told they are not as productive as they were in their youth.  The ones that have discovered things may get a bit 'soft' and live on their laurels.  Sometimes there are wonderful exceptions

Recently, Avraham Trahtman, solved a famous and perplexing problem called the 'Road Colouring Problem'.  Joel Friedman, a math professor at the University of British Columbia, said "probably everyone in the field of symbolic dynamics had tried to solve the problem at some point, including himself. He said people in the related disciplines of graph theory, discrete math and theoretical computer science also tried."

It's not too important for our story that you understand what all that means.  But to help illustrate what it's all about, look at the graph below

You could think about it as the routes between the vertices which could be destinations on a road map with one way indicated by the arrows. Things  have been idealized and the distances removed.  Let's concentrate on the colours of the lines and the vertices.

Consider the vertex marked in yellow. No matter where in the graph you start, if you traverse the nine edges in the walk "blue-red-red-blue-red-red-blue-red-red", you will end up at the yellow vertex. Similarly, if you traverse all nine edges in the walk "blue-blue-red-blue-blue-red-blue-blue-red", you will always end up at the vertex marked in green, no matter where you started.

The road coloring theorem states that  for a certain category of directed graphs, it is always possible to create such a coloring.  This was a conjecture, but nobody really knew if it was true.

Trahtman, a mathematician who also toiled as a laborer after moving to Israel from Russia, succeeded where dozens failed, solving the elusive "Road Coloring Problem."  His is 63 years old.

One of our Saugeen Times readers pointed this out to us with the following story:

 

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Cliff Our Reader's Story

In California there are Peppermill restaurants, all of which are exactly alike on the inside, and very darn similar on the outside.  There was one in Cupertino, CA which I (Cliff) frequented regularly (the very low cut long clingy slinky dresses the very pretty cocktail waitresses wore might have had something to do with this).  

It was quite close to my house in Cupertino.  When I drank too much I would get extremely drowsy --- so I'd often ask the Cupertino bartender to phone for a cab, and to give the cab my address, which I would write down on a cocktail napkin.  I then proceeded to doze off, often drooling on the bar (I'm told).

On one occasion, I happened to be in an identical Peppermill in (I think) Reno, several hundred miles away from Cupertino.  I drank WAY too much and fell into old habits.

I woke up in the cab on a major Interstate, with a LOT of money on the meter.  Which of course doubled when I told him to take me back to my Reno hotel.

My Story

It was 1989 and I was flying into O'Hare Airport in Chicago.  The NBA finals were on with Portland vs. Detroit.  I usually took a taxi to Itasca which was about a 20  minute drive away.  This time a young man dressed in Chauffeur's attire, black cap and all approached me.  He said... "would you like to take my limo?"  I said no, too expensive.  He said "It's a slow night and I'll have to get back in line unless you jump in, now!  I'll charge you $25"  At that time I spotted a small TV in the Limo.... ah ha... the Pistons vs. Trailblazers.... I jumped inside.

As he exited O'Hare, I commented that I thought he was taking the wrong route.  He said, he had a shortcut.  I got engrossed in the game and was not paying attention.  At the end of the first quarter, I looked out and told the driver that I recognized nothing!  He did not comment.

I went back to the game, but when half-time came and we were still speeding along, I began to be alarmed.  Looking closer at the driver, I determined that he was just lost and even if he was some guy trying to abduct me, I could crack him on the head with no problem, so I went back to the game with the comment "Try stopping and asking directions!"  As an avid fan, I got lost in the game again and only looked up when a big sign approached on my side which read

 Welcome to Wisconsin

This time I insisted he pull over and let me get directions.  On second thought, I said  let's "both go in and get directions."  I feared he would take off without me.

Both stories illustrate that I should not have feared. There was a solution for getting me to Itasca and our reader Clff to his hotel.  We just did not realize it.  We were too young.