MathDB

Pairwise distinct points

P_1,P_2,\ldots, P_{16}

lie on the perimeter of a square with side length

4

centered at

O

such that

\lvert P_iP_{i+1} \rvert = 1

for

i=1,2,\ldots, 16

. (We take

P_{17}

to be the point

P_1

.) We construct points

Q_1,Q_2,\ldots,Q_{16}

as follows: for each

i

, a fair coin is flipped. If it lands heads, we define

Q_i

to be

P_i

; otherwise, we define

Q_i

to be the reflection of

P_i

over

O

. (So, it is possible for some of the

Q_i

to coincide.) Let

D

be the length of the vector

\overrightarrow{OQ_1} + \overrightarrow{OQ_2} + \cdots + \overrightarrow{OQ_{16}}

. Compute the expected value of

D^2

.
Ray Li

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