How to Cut a Cake
I picked up How to Cut a Cake, by Ian Stewart, at the library while looking for "emergency reading" for my car. If I'm on break or eating lunch, or have to wait for something, I like to have something to read, and sometimes whatever book I'm reading isn't available or is too involved. How to Cut a Cake was just right.
The book is adapted from a list of columns that the author wrote that detail mathematical puzzles and paradoxes. The first chapter describes the various methods that have been devised for cutting a cake for more than two people, so that each person feels like they have an equal share. Methods vary from having everybody cut a piece in sequence, to having a slowly moving knife move along the cake.
Many of the puzzles have real-world applications. Take the problem of map-coloring. A map of contiguous areas can be colored with only four colors and ensure that no two areas that are next to each other have the same color. This fact was not proven until 1976 with mathematics and computer models. A related problem is coloring a map of empires that have states on the Earth and the Moon. Each plane has a coloring number of four, but when you link the two planes, the number is somewhere between eight and twelve. The true answer is not known. The real-world application turns out to be testing circuit boards. The circuits on boards are connected in networks that are analogous to connected areas on a map. With some calculations, the number of tests for short circuits can be brought from thousands down to just a handful.
One surprise was re-discovering Pascal's triangle. It turns out it is related to Sierpinski's gasket, a triangle-shaped curve that touches itself at every point. The open nature of this curve is related to the fact that the numbers in Pascal's triangle (binomial coefficients), as they approach infinity in the limit, are nearly all even.
Other interesting items are the shapes of bubbles and minimum surface areas; the law of averages and what it really means; and fireflies that flash in sequence. Tangled phone cords are a study in topology and the similarities between a write and a twist. They also are related to DNA and how it twists into double helix's.
The book is an A-. It is entertaining and educational. I enjoyed reading about interesting math problems and their even more interesting solutions. Sometimes it got a little complicated, but it was always a good read.
The book is adapted from a list of columns that the author wrote that detail mathematical puzzles and paradoxes. The first chapter describes the various methods that have been devised for cutting a cake for more than two people, so that each person feels like they have an equal share. Methods vary from having everybody cut a piece in sequence, to having a slowly moving knife move along the cake.
Many of the puzzles have real-world applications. Take the problem of map-coloring. A map of contiguous areas can be colored with only four colors and ensure that no two areas that are next to each other have the same color. This fact was not proven until 1976 with mathematics and computer models. A related problem is coloring a map of empires that have states on the Earth and the Moon. Each plane has a coloring number of four, but when you link the two planes, the number is somewhere between eight and twelve. The true answer is not known. The real-world application turns out to be testing circuit boards. The circuits on boards are connected in networks that are analogous to connected areas on a map. With some calculations, the number of tests for short circuits can be brought from thousands down to just a handful.
One surprise was re-discovering Pascal's triangle. It turns out it is related to Sierpinski's gasket, a triangle-shaped curve that touches itself at every point. The open nature of this curve is related to the fact that the numbers in Pascal's triangle (binomial coefficients), as they approach infinity in the limit, are nearly all even.
Other interesting items are the shapes of bubbles and minimum surface areas; the law of averages and what it really means; and fireflies that flash in sequence. Tangled phone cords are a study in topology and the similarities between a write and a twist. They also are related to DNA and how it twists into double helix's.
The book is an A-. It is entertaining and educational. I enjoyed reading about interesting math problems and their even more interesting solutions. Sometimes it got a little complicated, but it was always a good read.
posted by Dylan Peters | 5/11/2007 08:44:00 PM
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