MathDB

Let

n

be an integer greater than or equal to

2

. There is a regular hexagon

ABCDEF

with side length

n

, which is divided into equilateral triangles with side length

1

as shown in the left figure. We simply call the vertices of an equilateral triangle as vertex. A piece is placed in the center of a regular hexagon

ABCDEF

. For each of the vertices

P

inside

ABCDEF

(not including the perimeter), an arrow is drawn toward

4

vertices four of the six vertices connected to

P

by an edge of length

1

. When a piece is placed on the vertex

P

, this piece can be moved to any of the four vertices which the arrow is drawn. Regarding side

PQ

, even if a piece can be moved from vertex

P

to vertex

Q

, it is not necessarily possible to move a piece from vertex

Q

to vertex

P

. Then, show that there exists an integer

k

such that it can be moved the piece to vertices on the perimeter of the regular hexagon

ABCDEF

k

moves regardless of how the arrows are drawn and find the smallest possible value of

k

.
[asy] unitsize(0.3 cm);
pair[] A, B, C, D, E, F; int i; pair trans = (18,-4);
A[0] = 8*dir(120); B[0] = 8*dir(180); C[0] = 8*dir(240); D[0] = 8*dir(300); E[0] = 8*dir(0); F[0] = 8*dir(60);
for (i = 1; i <= 7; ++i) { A = interp(A[0],B[0],i/8); B = interp(B[0],C[0],i/8); C = interp(C[0],D[0],i/8); D = interp(D[0],E[0],i/8); E = interp(E[0],F[0],i/8); F = interp(F[0],A[0],i/8); }
draw(A[0]--B[0]--C[0]--D[0]--E[0]--F[0]--cycle);
for (i = 0; i <= 5; ++i) { draw(rotate(60*i)*(A[0]--(A[0] + 3*dir(-60)))); draw(rotate(60*i)*(A[1]--(A[1] + 3*(1,0)))); draw(rotate(60*i)*(A[1]--(A[1] + 3*dir(300)))); draw(rotate(60*i)*(A[2]--(A[2] + 4*(1,0)))); draw(rotate(60*i)*(F[6]--(F[6] + 4*dir(240)))); draw(rotate(60*i)*(F[7]--(F[7] + 3*dir(240)))); draw(rotate(60*i)*(F[7]--(F[7] + 3*dir(300)))); draw(rotate(60*i)*((A[1] + 3*(1,0))--(A[1] + 6*(1,0))), dotted); draw(rotate(60*i)*((A[2] + 4*(1,0))--(A[2] + 6*(1,0))), dotted); }
for (i = 0; i <= 2; ++i) { draw(rotate(60*i)*(3*dir(60)--3*dir(240))); }
for (i = 0; i <= 5; ++i) { draw(rotate(60*i)*(((2,0) + dir(60))--((-2,0) + dir(120)))); draw(rotate(60*i)*((3,0)--(5,0)), dotted); draw(rotate(60*i)*(((2,0) + dir(60))--((4,0) + dir(60))), dotted); draw(rotate(60*i)*(((2,0) + dir(-60))--((4,0) + dir(-60))), dotted); }
label("

A

", A[0], NW); label("

B

", B[0], W); label("

C

", C[0], SW); label("

D

", D[0], SE); label("

E

", E[0], dir(0)); label("

F

", F[0], NE);
draw(shift(trans)*scale(1.3)*((-3,0)--(9,0))); draw(shift(trans)*scale(1.3)*((-3*dir(60))--(9*dir(60)))); draw(shift(trans)*scale(1.3)*(((6,0) + 3*dir(-60))--((6,0) + 9*dir(120)))); draw(shift(trans)*scale(1.3)*(3*dir(120)--3*dir(300))); draw(shift(trans)*scale(1.3)*((6*dir(60) + (-3,0))--(6*dir(60) + (3,0)))); draw(shift(trans)*scale(1.3)*(((6,0) + 3*dir(60))--((6,0) - 3*dir(60)))); draw(shift(trans)*scale(1.3)*((-2,0)--(2,0)), linewidth(1.5*bp), Arrows(4)); draw(shift(trans)*scale(1.3)*(2*dir(60)--2*dir(240)), linewidth(1.5*bp), Arrows(4)); draw(shift(trans)*scale(1.3)*((6,0)--(8,0)), linewidth(1.5*bp), Arrow(4)); draw(shift(trans)*scale(1.3)*((6,0)--((6,0) + 2*dir(60))), linewidth(1.5*bp), Arrow(4)); draw(shift(trans)*scale(1.3)*((6,0)--((6,0) + 2*dir(240))), linewidth(1.5*bp), Arrow(4)); draw(shift(trans)*scale(1.3)*((6,0)--((6,0) + 2*dir(300))), linewidth(1.5*bp), Arrow(4)); draw(shift(trans)*scale(1.3)*(6*dir(60)--(6*dir(60) + (2,0))), linewidth(1.5*bp), Arrow(4)); draw(shift(trans)*scale(1.3)*(6*dir(60)--(6*dir(60) + 2*dir(120))), linewidth(1.5*bp), Arrow(4)); draw(shift(trans)*scale(1.3)*(6*dir(60)--(6*dir(60) + (-2,0))), linewidth(1.5*bp), Arrow(4)); draw(shift(trans)*scale(1.3)*(6*dir(60)--(6*dir(60) + 2*dir(240))), linewidth(1.5*bp), Arrow(4)); [/asy]

Problem Statement