The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.



(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A244951 Minimum number of colors needed to color the faces of the Platonic solids such that no two faces meeting at a common edge share the same color (in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron). 2
4, 3, 2, 4, 3 (list; graph; refs; listen; history; text; internal format)



Table of n, a(n) for n=1..5.

Felix Fröhlich, Illustration of colorings via Schlegel diagrams

Martin Gardner, The Five Platonic Solids, in Origami, Eleusis, and the Soma Cube: Martin Gardner’s Mathematical Diversions, Cambridge University Press, (see page 6).


a(1) = 4, since in the tetrahedron any face shares a common edge with any other face, so each face needs a distinct color.

a(2) = 3, since the cube has three sets of opposite faces. Any two faces that are not opposite share a common edge, so only opposite faces can have the same color.

a(3) = 2, since cutting the octahedron along its "equator" results in two square pyramids. The triangular faces of a single pyramid can be colored using two colors in an alternating fashion. Then the two pyramids are reassembled such that at the "equator" differently colored faces meet.

a(4) and a(5) are shown in illustration in the links.


with(GraphTheory): with(SpecialGraphs):

map(ChromaticNumber @ PlaneDual, [TetrahedronGraph(), HypercubeGraph(3), OctahedronGraph(), DodecahedronGraph(), IcosahedronGraph()]); # Robert Israel, Aug 24 2014


Cf. A098112, A198861, A158478 (analog for sides of polygons).

Sequence in context: A184338 A184412 A304240 * A110631 A333669 A159846

Adjacent sequences:  A244948 A244949 A244950 * A244952 A244953 A244954




Felix Fröhlich, Jul 08 2014


Corrected value of a(4) due to discovery of a new coloring for the dodecahedron.

Corrected value of a(5) due to discovery of a new coloring for the icosahedron.



Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 11 03:12 EDT 2021. Contains 342886 sequences. (Running on oeis4.)