Seven Touching Cylinders

Theorem

It is possible to arrange $7$ identical cylinders so that each one touches each of the others.

The cylinders must be such that their heights must be at least $\dfrac {7 \sqrt 3} 2$ of the diameters of their bases.

Proof

It remains to be proved that the heights of the cylinders must be at least $\dfrac {7 \sqrt 3} 2$ of the diameters of their bases.

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Historical Note

Martin Gardner originally raised the similar Six Touching Cylinders, which was an old puzzle, in his Mathematical Games column in Scientific American.

It was originally written in the context of cigarettes.

On its publication, he received letters from about $15$ readers who had discovered this $7$-cylinder solution.

He credited George Rybicki and John Reynolds for the diagram.

Sources

• 1965: Martin Gardner: Mathematical Puzzles and Diversions: $12$: Nine More Problems: Answers: $1$.
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $312$. The Six Submarines