MathDB

Let

n

be a natural number. A sequence

x_1,x_2, \cdots ,x_{n^2}

n^2

numbers is called

n-<span class='latex-italic'>good</span>

if each

x_i

is an element of the set

\{1,2,\cdots ,n\}

and the ordered pairs

\left(x_i,x_{i+1}\right)

are all different for

i=1,2,3,\cdots ,n^2

(here we consider the subscripts modulo

n^2

). Two

n-

good sequences

x_1,x_2,\cdots ,x_{n^2}

and

y_1,y_2,\cdots ,y_{n^2}

are called

<span class='latex-italic'>similar</span>

if there exists an integer

k

such that

y_i=x_{i+k}

for all

i=1,2,\cdots,n^2

(again taking subscripts modulo

n^2

). Suppose that there exists a non-trivial permutation (i.e., a permutation which is different from the identity permutation)

\sigma

\{1,2,\cdots ,n\}

and an

n-

good sequence

x_1,x_2,\cdots,x_{n^2}

which is similar to

\sigma\left(x_1\right),\sigma\left(x_2\right),\cdots ,\sigma\left(x_{n^2}\right)

. Show that

n\equiv 2\pmod{4}

Problem Statement