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Infintiely many polygons coinciding with P_0

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November 21, 2010

combinatorics proposedcombinatorics

Problem Statement

Let

\mathcal{P}_0=A_0A_1\ldots A_{n-1}

be a convex polygon such that

A_iA_{i+1}=2^{[i/2]}

for

i=0, 1,\ldots ,n-1

(where

A_n=A_0

). Define the sequence of polygons

\mathcal{P}_k=A_0^kA_1^k\ldots A_{n-1}^k

as follows:

A_i^1

is symmetric to

A_i

with respect to

A_0

,

A_i^2

is symmetric to

A_i^1

with respect to

A_1^1

,

A_i^3

is symmetric to

A_i^2

with respect to

A_2^2

and so on. Find the values of

n

for which infinitely many polygons

\mathcal{P}_k

coincide with

\mathcal{P}_0

.

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