A tale of touching tubes

Mathematicians find way to put seven cylinders in contact without using their ends

NO IFS, ASHES OR BUTTS  Seven cylinders can each touch each other using only their sides, not their ends, mathematicians have discovered.

D. Mackenzie

ATLANTA — Over 50 years ago, the popular mathematics writer Martin Gardner and readers of Scientific American pondered a challenge: Can you place seven cigarettes so that each cigarette touches every one of the others?

Gardner had a solution, but it was unsatisfying because some of the cigarettes’ ends touched others’ sides. If that end of a cigarette were lit, it would no longer touch its neighbor. Mathematicians wondered whether an arrangement could be found with only side-to-side contacts using infinitely long cylinders.

On March 20 at Gathering 4 Gardner, a conference to celebrate Gardner, mathematician Sándor Bozóki of the Hungarian Academy of Sciences in Budapest presented a solution of this more challenging problem, which was also posted online last summer at arXiv.org. Bozóki and colleagues used three months of computer time to solve 20 equations with 20 variables. Then they used a “certification” method developed by Isaac Newton, and updated by computer scientists in 2011, to prove that their solution was not an artifact of computer rounding errors. Finally, they built a wooden model to demonstrate their answer — although Bozóki notes that the model doesn’t verify the result because manufacturing errors are much greater than any errors the computer could have made.

Editor’s Note: This story was updated April 14, 2014, to correct the year in which the certification method was updated.

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