MathDB

A rabbit is placed on a

2n\times 2n

chessboard. Every time the rabbit moves to one of the adjacent squares. (Adjacent means sharing an edge). It is known that the rabbit went through every square and came back to the place where the rabbit started, and the path of the rabbit form a polygon

\mathcal{P}

. Find the maximum possible number of the vertices of

\mathcal{P}

. For example the answer for the case

n=2

would be

12

. [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(2cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -11.3, xmax = 27.16, ymin = -11.99, ymax = 10.79; /* image dimensions */
/* draw figures */ draw((5.14,3.19)--(8.43,3.22), linewidth(1)); draw((8.43,3.22)--(11.72,3.25), linewidth(1)); draw((11.72,3.25)--(11.75,-0.04), linewidth(1)); draw((11.75,-0.04)--(11.78,-3.33), linewidth(1)); draw((11.78,-3.33)--(8.49,-3.36), linewidth(1)); draw((8.49,-3.36)--(5.2,-3.39), linewidth(1)); draw((5.2,-3.39)--(5.17,-0.1), linewidth(1)); draw((5.17,-0.1)--(5.14,3.19), linewidth(1)); draw((6.785,3.205)--(6.845,-3.375), linewidth(1)); draw((8.43,3.22)--(8.49,-3.36), linewidth(1)); draw((10.075,3.235)--(10.135,-3.345), linewidth(1)); draw((5.155,1.545)--(11.735,1.605), linewidth(1)); draw((5.17,-0.1)--(11.75,-0.04), linewidth(1)); draw((11.765,-1.685)--(5.185,-1.745), linewidth(1)); draw((5.97,2.375)--(10.905,2.42), linewidth(1)); draw((10.905,2.42)--(10.92,0.775), linewidth(1)); draw((10.92,0.775)--(9.275,0.76), linewidth(1)); draw((9.275,0.76)--(9.29,-0.885), linewidth(1)); draw((9.29,-0.885)--(10.935,-0.87), linewidth(1)); draw((10.935,-0.87)--(10.95,-2.515), linewidth(1)); draw((10.95,-2.515)--(6.015,-2.56), linewidth(1)); draw((6.015,-2.56)--(6,-0.915), linewidth(1)); draw((6,-0.915)--(7.645,-0.9), linewidth(1)); draw((7.645,-0.9)--(7.63,0.745), linewidth(1)); draw((7.63,0.745)--(5.985,0.73), linewidth(1)); draw((5.985,0.73)--(5.97,2.375), linewidth(1)); /* dots and labels */ dot((5.97,2.375),linewidth(4pt) + dotstyle); dot((5.985,0.73),linewidth(4pt) + dotstyle); dot((6,-0.915),linewidth(4pt) + dotstyle); dot((6.015,-2.56),linewidth(4pt) + dotstyle); dot((7.645,-0.9),linewidth(4pt) + dotstyle); dot((7.63,0.745),linewidth(4pt) + dotstyle); dot((9.275,0.76),linewidth(4pt) + dotstyle); dot((9.29,-0.885),linewidth(4pt) + dotstyle); dot((10.95,-2.515),linewidth(4pt) + dotstyle); dot((10.935,-0.87),linewidth(4pt) + dotstyle); dot((10.92,0.775),linewidth(4pt) + dotstyle); dot((10.905,2.42),linewidth(4pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]

Problem Statement