400px, Isometric projection and net of naive (1) and optimal (2) solutions of the spider and the fly problem
The spider and the fly problem is a
recreational
Recreation is an activity of leisure, leisure being discretionary time. The "need to do something for recreation" is an essential element of human biology and psychology. Recreational activities are often done for enjoyment, amusement, or pleasur ...
geodesics
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
problem with an unintuitive solution.
Problem
In the typical version of the puzzle, an otherwise empty
cuboid room 30 feet long, 12 feet wide and 12 feet high contains a spider and a fly. The spider is 1 foot below the ceiling and horizontally centred on one 12′×12′ wall. The fly is 1 foot above the floor and horizontally centred on the opposite wall. The problem is to find the minimum distance the spider must crawl along the walls, ceiling and/or floor to reach the fly, which remains stationary.
Solutions
A naive solution is for the spider to remain horizontally centred, and crawl up to the ceiling, across it and down to the fly, giving a distance of 42 feet. The shortest distance strictly abiding by the rules, 40 feet, is obtained by constructing an appropriate
net
Net or net may refer to:
Mathematics and physics
* Net (mathematics), a filter-like topological generalization of a sequence
* Net, a linear system of divisors of dimension 2
* Net (polyhedron), an arrangement of polygons that can be folded up ...
of the room and connecting the spider and fly with a straight line, but counter-intuitively this optimal path used five of the six faces of the cuboid and can easily be missed.
A
lateral thinking
Lateral thinking is a manner of solving problems using an indirect and creative approach via reasoning that is not immediately obvious. It involves ideas that may not be obtainable using only traditional step-by-step logic. The term was first u ...
solution involves the spider attaching
dragline silk to the wall to lower itself to the floor, and crawling 30 feet across it and 1 foot up the opposite wall, giving a crawl distance of 31 feet. Similarly, it can climb to the ceiling, cross it, then attach the silk to lower itself 11 feet, also a 31-foot crawl.
For a room of length ''l'', width ''w'' and height ''h'', the spider a distance ''b'' below the ceiling, and the fly a distance ''a'' above the floor, the optimal distance ''o'' is
while the naive distance ''n'' is
.
This table gives integer solutions for and sorted by ascending ''o'' then with the original values in bold.
History
The problem was originally posed by
Henry Dudeney
Henry Ernest Dudeney (10 April 1857 – 23 April 1930) was an English author and mathematician who specialised in logic puzzles and mathematical games. He is known as one of the country's foremost creators of mathematical puzzles.
Early life
...
in the English newspaper ''Weekly Dispatch'' on 14 June 1903, presented in ''The Canterbury Puzzles'' (1907) and described by
Martin Gardner
Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lew ...
.
References
{{DEFAULTSORT:spider and the fly problem
Recreational mathematics
