Henry Ernest Dudeney/Modern Puzzles/220 - The Fly's Tour/Solution

Modern Puzzles by Henry Ernest Dudeney: $220$

The Fly's Tour

I had a ribbon of paper, divided into squares on each side.

I joined the ends together to make a ring, and tossed it down onto the table.

Then I watched a fly land on the ring and walk in a line over every one of those squares on both sides,

returning to the point where it started, without ever passing over the edge of the paper.

Its course passed through the centre of the squares all the time.

How was this possible?

Solution

Dudeney's answer is:

Before you join the ends give one end of the ribbon a half-turn, so that there is a twist in the ring.

Then the fly can walk over all the squares without going over the edge,

for we have the curious paradox of a piece of paper with only one side and one edge!

Dudeney has of course described a Möbius strip.

The author of this page is not convinced by his exposition, because at least part of the paper will end up flat on the table, and the fly may have trouble negotiating that bit.

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $220$. -- The Fly's Tour
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $517$. The Fly's Tour