# Virtual Cubes

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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.

#### 2 H's3 Chessboards

Algorithm Walter Randelshofer 2015
U · F B' · R · F B · U' R' D' L U' · L R · D' B R' · F2 B · R (19 ltm, 19 ftm, 20* qtm)

Algorithm Walter Randelshofer 2015
U · L R' · B2 U2 B D · L2 R2 · U D' · L D F2 D' L' U' (17 ltm, 17* ftm, 22 qtm)

Algorithm Walter Randelshofer 2015
CF CR · R MF L2 D2 L U MF2 MD L D F2 D' L' U' (14* ltm, 17* ftm, 22 qtm)

Note
Creates a solid face on the top, 2 H's on the front and on the back and 3 Chessboards on the remaining faces.

#### 1 E2 Fish1 K1 Chessboard

Algorithm Walter Randelshofer 2015
L F U R B' · R2 L · F' L' F L B · U' D · B' R' F2 R (18 ltm, 18* ftm,20* qtm)

Algorithm Walter Randelshofer 2015
CF' CR · MF R MF' U2 MR D F' B2 D MF' D' F' MR2 U R' (15* ltm, 20 ftm,24 qtm)

Note
Creates as much single colored faces as possible: While the face on top has a single color, the right face is solid except for 1 edge, the faces on the front and on the back are solid except for 2 edges, the left face is solid except for 3 faces and the face on the bottom is solid except for 4 edges.

Algorithm
MR MF' MR' MF (4* ltm, 8* ftm, 8* qtm)

Algorithm
MD' MR' MD MR (4* ltm, 8* ftm, 8* qtm)

Algorithm
(MR · CD')4 (4* ltm, 8* ftm, 8* qtm)

Note
On the Four Color Cube, the 6 Dots pattern creates a Gift-wrapped Cube.

#### 6 H's With Dots(4 Small Edge Triangles)

Algorithm Michael Reid
R L · F' D · F B' · U D' · F' D B U' · F B' · U D' · B U' · R L (20 ltm, 20 ftm, 20* qtm)

Algorithm Tomas G. Rokicki 2003
R' D' · R L' · U' · F B' · L' F' U' F' R' B' L' U' R' · U D' · R' U' (20 ltm, 20 ftm, 20* qtm)

Algorithm Tomas G. Rokicki 2003
U' · F' B' · U' · R L · U' · F2 B2 · R2 L2 · D' · F B · D' · R' L' · D' (18 ltm, 18* ftm, 22 qtm)

Algorithm Michael Reid
L' · U' D' · L' · F B · L' · U2 D2 · F2 B2 · R' · U D · R' · F' B' · R' (18 ltm, 18* ftm, 22 qtm)

Algorithm Herbert Kociemba 2010
CD2 MF2 · D2 MR2 U' B' MR MD' B R' MF' MD R U' (13* ltm, 19 ftm, 24 qtm)

Note
On the Four Color Cube, the 4 Small Edge Triangles pattern creates 6 H's With Dots.

#### 6 H's(4 Small Edge Triangles With 6 Dots)

Algorithm Walter Randelshofer 2015
U D · R' B · U' D · F' B · U' L D R' · F' B · R' L · B U' · R L (20 ltm, 20 ftm, 20* qtm)

Algorithm Walter Randelshofer 2015
U2 F U R B L' U2 · F B · L2 U2 · F' B' · D' F U L B (18 ltm, 18* ftm, 22 qtm)

Algorithm Walter Randelshofer 2015
CF' CR · MF2 D2 MR2 U' F' MR' MD' F R' MD MF' R U' (13* ltm, 19 ftm, 24 qtm)

Note
On the Four Color Cube, the 4 Small Edge Triangles With 6 Dots pattern creates 6 H's.

#### 6 H's(Irregular Pattern)

Algorithm Walter Randelshofer 2015
F' R' · U D' · L' R · U B' · L R' · B' F · R U' (14 ltm, 14* ftm, 14* qtm)

Algorithm Ralph Jones 2015
CR CD' · U' F' MR' MD' F R' MD MF' R U' (10* ltm, 14* ftm, 14* qtm)

Note
On the Four Color Cube, this irregular pattern creates 6 H's. Compared to the 6 H's (4 Small Edge Triangles With 6 Dots) it requires less moves in all metrics.

#### The Color-Labyrinth of Minos(2 Small Edge Triangles)

Algorithm Ronald Flettermann
D' R U2 B' U2 R' · U' D · B D2 R D2 B' U (14 ltm, 14* ftm, 18* qtm)

Algorithm Tomas G. Rokicki 2003
U L' D2 F D2 L · U' D · F' U2 L' U2 F D' (14 ltm, 14* ftm, 18* qtm)

Algorithm Mirek Goljan 1982
MR U2 R2 F' MD L' MD2 L MD F R2 U2 MR' (13 ltm, 18 ftm, 24 qtm)

Algorithm Herbert Kociemba 2010
MD R' U2 B L' MD2 L MD2 B' U2 R MD' (12* ltm, 16 ftm, 22 qtm)

Note
On the Four Color Cube, the 2 Small Edge Triangles pattern creates The Color-Labyrinth of Minos.