# A Beautiful Mind

An Israeli has solved an almost 4-decade-long mathematical mystery.

A mathematical mystery that has baffled the top minds in the esoteric field of symbolic dynamics for nearly four decades has recently been cracked — by a 63-year-old former security guard.

Avraham Trakhtman, a mathematician who worked as a laborer after immigrating to Israel from Russia, has succeeded where dozens have failed, solving the elusive “Road Coloring Problem.”

The conjecture essentially assumes that it is possible to create a “universal map” that would direct people to arrive at a certain destination, at the same time, regardless of their original location. Experts say this proposition, which seems to defy logic, could actually have real-life applications in the fields of mapping and computer science.

“In math circles, we talk about beautiful results — this is beautiful and it is unexpected. Even in layman’s terms it is completely counterintuitive, but somehow it works,” said Stuart Margolis, a colleague who recruited Trakhtman to Bar Ilan University near Tel Aviv.

He said the discovery was especially remarkable given Trakhtman’s age and background. “The first time I met him he was wearing a night watchman’s uniform,” he said.

The “Road Coloring Problem” was first posed in 1970 by Benjamin Weiss, an Israeli-American mathematician, and a colleague, Roy Adler, who worked at IBM at the time.

Weiss said he believed that given a finite number of roads, one should be able to draw up a map, coded in various colors, that would lead to a certain destination regardless of the point of origin.

For eight years, he tried to prove his theory. Over the next 30 years, some 100 other scientists attempted to as well. All failed, until Trakhtman came along and, in eight short pages, jotted the solution down in pencil last year.

Trakhtman said it took him a year to solve the problem. But that wasn’t nearly as impressive as the journey he took to get to his current lofty position.

Originally from Yekaterinburg, Russia, Trakhtman was already an accomplished mathematician before he came to Israel in 1992, at the age of 48. But like many immigrants in the wave that followed the breakup of the former Soviet Union, he too struggled to find work in the Jewish state and was forced into stints working maintenance and security before landing a teaching position at Bar Ilan in 1995.

The soft-spoken Trakhtman declined to discuss his arduous odyssey, saying those were the “old days.” He said he was “lucky” to be recognized, but played down his recent achievement as a “matter for mathematicians” and said it hasn’t changed him a bit.

“The solution is not that complicated. It’s hard, but it is not that complicated,” he said in heavily accented Hebrew. “Some people think they need to be complicated. I think they need to be nice and simple.”

Trakhtman’s solution is available for viewing on the Internet and will soon be published in the Israel Journal of Mathematics.

Weiss said it gave him great joy to see someone solve his problem, adding that Trakhtman’s solution “is something that is understandable.”

Joel Friedman, a math professor at the University of British Columbia, said probably everyone in the field of symbolic dynamics has tried to solve the Roadmap Coloring Problem at some point, including himself. He said people in the related disciplines of graph theory, discrete math and theoretical computer science have also tried.

“The solution to this problem has definitely generated excitement in the mathematical community,” he said in an e-mail message.

Trakhtman’s achievement is hardly the longest open problem to be solved recently. In 1994, British mathematician Andrew Wiles solved Fermat’s last theorem, which had been open for more than 300 years.

Margolis, Trakhtman’s colleague at Bar Ilan, said the solution could have many applications.

“Say you’ve lost an e-mail and you want to get it back — it would be guaranteed,” he said. “Let’s say you are lost in a town you have never been in before and you have to get to a friend’s house and there are no street signs — the directions will work no matter what.”

But even more exciting, he said, was Trakhtman’s personal history and advanced age, at least in the math world.

“The heartwarming part of it is here is a guy who had a good reputation for his work in the Soviet Union and couldn’t get work,” he said.

“Math is usually a younger person’s game, like music and the arts,” he said. “Usually you do your better work in your mid 20’s and early 30’s. He certainly came up with a good one at age 63,” he said.

Meanwhile, the only mathematical problem the palestinians have tried to solve is how to kill as many Jews as possible.

**Update**: You can view Avraham’s solution here. Here’s an excerpt:

The synchronizing word of deterministic automaton is a word in the alphabet of colors (considered as letters) of its edges that maps the automaton to a single state. A coloring of edges of a directed graph is synchronizing if the coloring turns the graph into deterministic finite automaton possessing a synchronizing word.

My head hurts.

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