### Knight Moves 5

##### November 2023 : Solution

This month’s puzzle was an open-ended challenge: to find a knight’s tour from a1 to h8 on a constantly shifting 3-dimensional board.

In the end we received a wide variety of tours. We’ll mention a few of the most interesting ones below. (Their “finishing states” are all shown above.)

The longest1 valid tour came from Brandon Cage, which takes 1547 minutes.

The second-longest2, at 1482 minutes, came from Calvin Pozderac. This tour also set the records for “lowest finishing altitude” (-32) and “longest individual sink” (364 minutes).

The most efficient tour3 came from Sondre Rogde, who managed to get from a1 to h8 in just 33 jumps.

The first tour4 (out of just two) we received that managed to visit every point at least once came from Anton 3 Terekhov.

Those were some of our favorites, but congrats to everyone who managed to find a valid tour!

Some open problems we were pondering:

• What’s the longest valid tour? (Can 2000 minutes be achieved?)
• What’s the highest possible finishing altitude? (The highest we saw was 19; could it be as high as 25?)
• Does there exist a valid tour that does not repeat a single point? (I.e. the unclaimed “U” designation from the tourism board.)
• Is it possible to finish a tour with every point at the same altitude? If not, what’s the largest group of same-altitude points that can be obtained?
• Does there exist a valid tour with “max sink” < 10? (The lowest “max sink” we saw across any valid tour was 27.)

If you’re able to solve any of these problems, feel free to let us know! (We might update the leaderboard for especially interesting submissions.)

1. (0, b1), (0, b3), (0, c3), (0, a3), (0, a4), (0, a5), (0, c4), (0, c6), (0, b6), (0, b7), (0, c7), (3, e8), (0, f8), (0, e6), (0, e5), (0, g5), (7, f5), (0, h5), (6, h6), (0, h7), (0, g7), (3, g6), (4, e6), (0, e7), (0, f7), (9, f5), (0, e5), (0, d5), (0, c5), (0, c4), (0, d4), (0, d2), (0, c2), (0, b2), (0, b3), (16, c3), (0, c2), (0, b2), (16, c2), (12, d2), (7, e4), (0, d4), (0, c4), (0, c5), (0, d5), (0, f4), (0, h5), (0, h6), (0, g8), (0, f8), (0, e8), (0, c7), (0, e7), (0, f7), (0, g7), (0, g6), (8, g7), (4, f7), (12, e7), (2, c8), (20, d6), (0, d8), (5, b7), (28, d6), (58, f6), (10, h7), (39, g5), (47, h7), (49, f6), (0, g4), (0, h2), (0, f3), (0, h2), (0, g4), (0, f6), (0, e8), (54, c7), (0, e6), (0, f4), (0, h5), (57, g3), (0, e2), (0, c1), (126, a2), (0, c3), (0, b1), (62, d2), (0, b3), (122, c1), (0, a2), (0, b4), (120, c6), (0, d4), (0, b5), (118, a7), (0, c6), (0, a5), (116, b7), (0, c5), (0, a6), (114, b4), (0, d5), (0, b6), (56, d7), (0, b8), (110, d7), (0, b6), (0, c8), (108, d6), (0, e4), (0, f2), (0, f1), (0, g1), (0, g3), (0, h3), (0, h4), (0, g4), (0, g5), (0, g6), (0, g8), (0, f8), (19, h8)

2. (0, b1), (0, c1), (30, a1), (0, b1), (35, c1), (6, c2), (3, a1), (0, c1), (18, d3), (0, d4), (0, c4), (0, a5), (0, a4), (0, a3), (0, c2), (0, d2), (0, d1), (0, e1), (4, f1), (5, h2), (0, h3), (7, g5), (0, f3), (14, f1), (60, h2), (0, h1), (0, g1), (0, e2), (0, d4), (0, c4), (0, a5), (0, a4), (0, a3), (0, c2), (0, d2), (0, d1), (2, e1), (0, f3), (0, g5), (0, e4), (0, c5), (0, c6), (0, d6), (5, d7), (6, e7), (0, c7), (16, c5), (0, c6), (1, d6), (5, f5), (0, f3), (2, h4), (3, h3), (0, g5), (0, g4), (0, f4), (0, h5), (0, h6), (0, h7), (0, f7), (24, d8), (0, d6), (0, d5), (0, f4), (0, h5), (0, h6), (0, h7), (0, g7), (1, g8), (0, f8), (14, e8), (12, f8), (12, h7), (45, f8), (19, g6), (364, f6), (20, f7), (112, f6), (20, f7), (112, f6), (20, g6), (112, f4), (0, h3), (0, h2), (5, f2), (27, e4), (26, g3), (25, h5), (0, g3), (0, e4), (0, c5), (0, d5), (0, e5), (24, d5), (1, b6), (9, b5), (116, b6), (12, b5), (116, a7), (0, b5), (0, b6), (12, b7), (0, d8), (0, c6), (0, e7), (0, g6), (0, h8)

3. (0, b1), (0, b3), (0, c3), (0, a2), (9, a3), (0, a4), (0, a5), (0, c4), (0, b4), (0, b6), (0, b7), (0, c7), (6, a8), (18, b6), (0, d5), (0, f4), (0, g2), (0, h4), (0, f3), (68, d3), (0, b3), (95, a3), (0, a4), (0, a5), (0, c4), (0, e4), (2, e5), (0, e6), (0, e7), (0, f7), (0, g7), (1, h7), (0, h8)

4. (0, b1), (0, b3), (8, c3), (0, e3), (0, d3), (0, d4), (0, c4), (0, e4), (0, f2), (0, f3), (0, g5), (0, h5), (6, h3), (0, g5), (0, e5), (0, c5), (0, c6), (0, a6), (0, a4), (3, a2), (0, c3), (0, e3), (0, d3), (0, d4), (0, c4), (0, b4), (40, a4), (0, a2), (0, c2), (0, b2), (0, d2), (0, d4), (0, c4), (0, c6), (0, b6), (0, b8), (0, c8), (0, d8), (0, f7), (0, g7), (0, g5), (0, g6), (10, g7), (2, g8), (0, h6), (0, g6), (6, h4), (10, g4), (0, h4), (32, f5), (54, f3), (7, h4), (0, g2), (0, f4), (70, f2), (4, h1), (0, g1), (0, e2), (0, e1), (42, g1), (0, g3), (0, e4), (0, d6), (0, b7), (0, c7), (0, c5), (1, b5), (0, a7), (0, c8), (0, e7), (0, d5), (120, e5), (0, e6), (0, f6), (0, f5), (0, g3), (0, h1), (0, h2), (0, f1), (0, d1), (0, d3), (0, c1), (0, c3), (0, a3), (0, b5), (0, b4), (0, b3), (0, a5), (0, b3), (0, a1), (0, c2), (0, e1), (0, e2), (0, e4), (2, e6), (0, f8), (0, h7), (0, f8), (0, d7), (0, b6), (0, a8), (0, b6), (0, c8), (0, d6), (0, e8), (0, d6), (0, f7), (0, h8)