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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $\\Delta$ be an $n$-simplex in $\\R^n$ with vertices $v_0, \\ldots, v_n$.\nLet $T$ be a triangulation of $\\Delta$ by a finite [[simplicial complex|V8GZ0W1ZZ5]].\nLet $\\lambda$ be a colouring of the vertices of $T$ by $\\set{0, 1, \\ldots, n}$.\nAssume the following for every set $S \\subseteq \\set{0, \\ldots, n}$.\nEach vertex of $T$ in the face $\\operatorname{conv}\\set{v_i : i \\in S}$ of $\\Delta$ has a colour in $S$, where $\\operatorname{conv}$ is the [[convex hull|0FQSDC2EQ2]].\nThen the number of $n$-simplices of $T$ whose vertices carry all $n + 1$ colours is odd; in particular there is at least one.\n\n## Remarks\n\nThe proof is by induction on $n$ with a parity count.\nCall an $(n-1)$-face of $T$ a door if its vertices carry exactly the colours $0, \\ldots, n-1$.\nCount pairs (an $n$-simplex of $T$, a door in it).\nA fully coloured simplex has one door, and any other simplex has zero or two.\nAn interior door lies in two simplices, while a boundary door lies in one.\nAll boundary doors lie in the face $\\operatorname{conv}\\set{v_0, \\ldots, v_{n-1}}$, where their number is odd by induction; so the number of fully coloured simplices is odd.\n\nExample: for $n = 1$, subdivide a segment from $v_0$ to $v_1$ and colour $v_0$ with $0$, $v_1$ with $1$, and the inner points freely.\nThe colour changes an odd number of times along the segment, so an odd number of small segments have both colours.\n\nThe boundary condition is needed: if every vertex of $T$ gets the colour $0$, there is no fully coloured simplex.\n\n## Explanations\n\n### L1 human\n\nCut a triangle into small triangles and colour every corner red, green or blue.\nColour the three big corners with the three colours, and colour each point on a big side with one of the two colours at the ends of that side.\nThen some small triangle has all three colours, and in fact the number of such triangles is odd.\n\n### L2 human\n\nSperner's lemma is a discrete form of the Brouwer fixed-point theorem: applying it to finer and finer triangulations and passing to a limit gives a fixed point.\nIts constructive proof by following doors gives algorithms for approximate fixed points and for economic equilibria.\nIt is also used for fair division problems, such as envy-free cake cutting.\n",
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