= Solution
An <interval map> $F:I\to I$ has a <horseshoe for an interval map>[horseshoe] if there are two <closed real intervals>[closed subintervals] $J_0,J_1\subseteq I$ with disjoint interiors such that
$$
F(J_i)\supseteq J_0\cup J_1
\qquad(i=0,1).
$$
It is <Glendinning chaos>[chaotic in Glendinning's sense] if some positive <iteration of a map>[iterate] $F^n$ has a horseshoe.
Suppose $a<b<c$ are the points of a <cycle of an interval map>[3-cycle] and put $J_0=[a,b]$, $J_1=[b,c]$. There are two possible cyclic orders. If
$$
a\mapsto b\mapsto c\mapsto a,
$$
then the <intermediate value theorem> gives
$$
J_0\longrightarrow J_1,
\qquad
J_1\longrightarrow J_0\cup J_1.
$$
Consequently both $F^2(J_0)$ and $F^2(J_1)$ contain $J_0\cup J_1$. If instead
$$
a\mapsto c\mapsto b\mapsto a,
$$
then
$$
J_0\longrightarrow J_0\cup J_1,
\qquad
J_1\longrightarrow J_0,
$$
and again both second images contain $J_0\cup J_1$. Thus in either cyclic order $F^2$ has a horseshoe.
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