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A '''slice''' of the cube <math>[3]^n</math> is a set of the form <math>\Gamma_{a,b,c}</math> for some (a,b,c) in the triangular grid

:<math>\Delta_n := \{ (a,b,c) \in {\Bbb Z}_+^3: a+b+c = n \},</math>

where <math>\Gamma_{a,b,c}</math> is the set of all strings with a 1s, b 2s, and c 3s.  Thus for instance

:<math>\Gamma_{1,2,0} = \{ 011, 101, 101 \}.</math>

The cube <math>[3]^n</math> has <math>\frac{(n+1)(n+2)}{2}</math> slices, and each slice <math>\Gamma_{a,b,c}</math> has cardinality <math>\frac{n!}{a!b!c!}</math>.

Each slice <math>\Gamma_{a,b,c}</math> is incident to <math>2^a+2^b+2^c-3</math> combinatorial lines.

The [[equal-slices measure]] gives each slice a total measure of 1.

The k=2 analogue of a slice is sometimes known as a ''layer'', and plays a role in the proof of [[Sperner's theorem]].