Constructing magic knight tour hypercubes of dimension n and order 4 A magic knight tour hypercube is a magic hypercube in which the path from the cell (a) to the cell (a+1) of the hypercube, where a = 1,2,..., is always (2-dimensional) knight jump. This page deals with magic knight tour hypercubes of dimension higher than 2 and order 4. When n is odd and greater then 2, we can construct a magic knight tour hypercube of dimension n and order 4 by using a binary matrix as follows: n = 3 : A hypercube M to find is given as follows: M[2x0 + x1, 2x2 + x3, 2x4 + x5] = 1 + 20a0 + 21a1 + 22a2 + 23a3 + 24a4 + 25a5, a0 0 1 0 1 0 1 x0 a1 0 1 1 1 0 1 x1 + 1 a2 = 1 1 1 0 1 0 x2 (mod. 2). a3 1 1 1 0 1 1 x3 a4 1 0 0 1 0 1 x4 a5 1 1 1 1 1 1 x5 where each of ak, xk (k = 0, 1, ..., 5) is either 0 or 1. (This formula uses x1 + 1 insted of x1 in order to make the knight tour closed, that is, the path from the maximum integer to 1 is also knight jump.) M is a 1,3-agonal magic knight tour cube. Note: Although M is also pan-3-agonal (pantriagonal), that is due to the speciality of dimension 3. If n > 3, M is not pan-n-agonal. In general,  a pan-n-agonal magic knight tour hypercube of dimension n and order 4 exists only for n = 3. n = 5 : A hypercube M to find is given as follows: M[2x0 + x1, 2x2 + x3, 2x4 + x5, 2x6 + x7, 2x8 + x9] = 1 + 20a0 + 21a1 + 22a2 + 23a3 + 24a4 + 25a5 + 26a6 + 27a7 + 28a8 + 29a9, a0 0 1 0 1 0 1 0 1 0 1 x0 a1 0 1 1 1 0 1 0 1 0 1 x1 + 1 a2 0 1 1 1 1 1 0 1 0 1 x2 a3 0 1 1 1 1 1 1 1 0 1 x3 a4 = 1 1 1 0 1 0 1 0 1 0 x4 (mod. 2). a5 1 1 1 1 1 0 1 0 1 0 x5 a6 1 1 1 1 1 1 1 0 1 0 x6 a7 1 1 1 1 1 1 1 0 1 1 x7 a8 1 0 0 1 0 1 0 1 0 1 x8 a9 1 1 1 1 1 1 1 1 1 1 x9 where each of ak, xk (k = 0, 1, ..., 9) is either 0 or 1. M is a 1,3,5-agonal magic knight tour hypercube. n = 7 : A hypercube M to find is given as follows: M[2x0 + x1, 2x2 + x3, 2x4 + x5, 2x6 + x7, 2x8 + x9, 2x10 + x11, 2x12 + x13] = 1 + 20a0 + 21a1 + 22a2 + 23a3 + 24a4 + 25a5 + 26a6 + 27a7 + 28a8 + 29a9 + 210a10 + 211a11 + 212a12 + 213a13, a0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 x0 a1 0 1 1 1 0 1 0 1 0 1 0 1 0 1 x1 + 1 a2 0 1 1 1 1 1 0 1 0 1 0 1 0 1 x2 a3 0 1 1 1 1 1 1 1 0 1 0 1 0 1 x3 a4 0 1 1 1 1 1 1 1 1 1 0 1 0 1 x4 a5 0 1 1 1 1 1 1 1 1 1 1 1 0 1 x5 a6 = 1 1 1 0 1 0 1 0 1 0 1 0 1 0 x6 (mod. 2). a7 1 1 1 1 1 0 1 0 1 0 1 0 1 0 x7 a8 1 1 1 1 1 1 1 0 1 0 1 0 1 0 x8 a9 1 1 1 1 1 1 1 1 1 0 1 0 1 0 x9 a10 1 1 1 1 1 1 1 1 1 1 1 0 1 0 x10 a11 1 1 1 1 1 1 1 1 1 1 1 0 1 1 x11 a12 1 0 0 1 0 1 0 1 0 1 0 1 0 1 x12 a13 1 1 1 1 1 1 1 1 1 1 1 1 1 1 x13 where each of ak, xk (k = 0, 1, ..., 13) is either 0 or 1. M is a 1,3,5,7-agonal magic knight tour hypercube. In general, we can construst a 1,3,...,n-agonal magic knight tour hypercube of dimension n and order 4 for any odd n > 2 by a similar way. Note: If n is even greater than 3, the hypercube M constructed by a similar way is only semimagic because n-agonals of M is not magic. It is an open problem whether a magic knight tour hypercube of dimension n and order 4 for even n greater than 3. Remark: The inverse formulae of these formulae are as follows: n = 3 : x0 0 0 1 0 1 1 a0 x1 + 1 1 0 1 0 0 1 a1 x2 = 1 1 0 0 0 0 a2 (mod. 2). x3 0 0 0 1 0 1 a3 x4 0 1 1 0 1 0 a4 x5 0 0 1 1 0 0 a5 n = 5 : x0 0 0 0 0 1 0 0 0 1 1 a0 x1 + 1 1 0 0 0 1 0 0 0 0 1 a1 x2 1 1 0 0 0 0 0 0 0 0 a2 x3 0 0 0 0 1 1 0 0 0 0 a3 x4 = 0 1 1 0 0 0 0 0 0 0 a4 (mod. 2). x5 0 0 0 0 0 1 1 0 0 0 a5 x6 0 0 1 1 0 0 0 0 0 0 a6 x7 0 0 0 0 0 0 0 1 0 1 a7 x8 0 0 0 1 1 0 0 0 1 0 a8 x9 0 0 0 0 0 0 1 1 0 0 a9 n = 7 : x0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 a0 x1 + 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 a1 x2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 a2 x3 0 0 0 0 0 0 1 1 0 0 0 0 0 0 a3 x4 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a4 x5 0 0 0 0 0 0 0 1 1 0 0 0 0 0 a5 x6 = 0 0 1 1 0 0 0 0 0 0 0 0 0 0 a6 (mod. 2). x7 0 0 0 0 0 0 0 0 1 1 0 0 0 0 a7 x8 0 0 0 1 1 0 0 0 0 0 0 0 0 0 a8 x9 0 0 0 0 0 0 0 0 0 1 1 0 0 0 a9 x10 0 0 0 0 1 1 0 0 0 0 0 0 0 0 a10 x11 0 0 0 0 0 0 0 0 0 0 0 1 0 1 a11 x12 0 0 0 0 0 1 1 0 0 0 0 0 1 0 a12 x13 0 0 0 0 0 0 0 0 0 0 1 1 0 0 a13 By using these formulae, we can show that the hypercube M is normal and every path from (a) to (a+1), where a = 1,2,..., is knight jump. August 26, 2016 Mitsutoshi Nakamura Magic Cubes and Tesseracts http://magcube.la.coocan.jp/magcube/en/ Copyright © 2004-2016, Mitsutoshi Nakamura. All rights reserved.