Virtual Cubes

Four Color Cube

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Four Color Cube

The Four Color Cube only features stickers in four different colors. In the initial state the colors are arranged in such a way that no two adjacent stickers share the same color. This sticker arrangement follows the 'four color theorem', also known as 'four color map theorem'. In mathematics it states that, no more than four colors are required to color the regions of a map, so that no two adjacent regions have the same color.

The layout of the Four Color Cube was created in 2015 by Evgeniy Grigoriev. The color scheme was created in 2015 by Walter Randelshofer.

The four color theorem

The intuitive statement of the 'four color theorem', i.e. «that given any separation of a plane into contiguous regions, called a map, the regions can be colored using at most four colors so that no two adjacent regions have the same color», needs to be interpreted appropriately to be correct.

First, all corners, points that belong to (technically, are in the closure of) three or more countries, must be ignored. In addition, bizarre maps (using regions of finite area but infinite perimeter) can require more than four colors. Second, for the purpose of the theorem, every 'country' has to be a connected region, or contiguous. In the real world, this is not true (e.g. the Upper and Lower Peninsula of Michigan, Nakhchivan as part of Azerbaijan, and Kaliningrad as part of Russia are not contiguous). Because all the territory of a particular country must be the same color, four colors may not be sufficient.

This problem is sometimes also called 'Guthrie's problem' after F. Guthrie, who first conjectured the theorem in 1852. The conjecture was then communicated to de Morgan and thence into the general community. In 1878, Cayley wrote the first paper on the conjecture.

The four color theorem was proven in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer.

© Walter Randelshofer. All rights reserved.