The Knight's Tours Walkthrough

I highly dislike games with no walkthrough—especially since I invariably get stuck. Because I know others have similar problems, I'm providing a solution—not the solution (a 3×4 board alone has 16 ways to do it; 32 if you stand the board on end), but a solution— to almost every board in the dream. Also, I tried to follow two rules for these tours:

There is one exception to these rules: the 3×4 board. As it has an even number of squares, it doesn't have a middle. Neither does it have a closed tour, as is the case with all 4×n boards. As such, it is exempt from these self-imposed rules.

Only the 1×1 board has been left off this page; the solution is simply to pick up the Knight and drop it on the board. The board itself is only in the dream as a joke.

4×3 (12 squares)

Although a 4×4 board can't be solved, a 4×3 (or its other orientation, 3×4) board can. It's the smallest board that can be toured, and it's here and in the dream. I mentioned above that there are 32 solutions to this board. Here's the proof:

Since I can't follow my self-imposed rules posted above, I'll post the 16 tours 4×3 tours instead (the 3×4 tours have been omitted, for sake of space).

abcd
abcd
3107 2 5 3
21 4 9 122
18 116 3 1
abcd
abcd
33 6 118 3
2129 4 1 2
15 2 7 101
abcd
abcd
38 116 3 3
21 4 9 122
1107 2 5 1
abcd
abcd
35 2 7 103
2129 4 1 2
13 6 118 1
abcd
abcd
31 4 7 103
28 112 5 2
13 6 9 121
abcd
abcd
3129 6 3 3
25 2 118 2
1107 4 1 1
abcd
abcd
33 6 9 123
28 112 5 2
11 4 7 101
abcd
abcd
3107 4 1 3
25 2 118 2
1129 6 3 1
abcd
abcd
31 4 7 103
2129 2 5 2
13 6 118 1
abcd
abcd
3129 6 3 3
21 4 118 2
1107 2 5 1
abcd
abcd
3107 4 1 3
25 2 9 122
18 116 3 1
abcd
abcd
33 6 9 123
28 114 1 2
15 2 7 101
abcd
abcd
33 6 118 3
2129 2 5 2
11 4 7 101
abcd
abcd
3107 2 5 3
21 4 118 2
1129 6 3 1
abcd
abcd
38 116 3 3
25 2 9 122
1107 4 1 1
abcd
abcd
35 2 7 103
28 114 1 2
13 6 9 121

5×5 (25 squares)

The 1×1 board has a trivial solution, the Knight can't move at all on a 2×2 board, the Knight can either never reach or never move from the center square on a 3×3 board depending on whether or not it started there, and the 4×4 board has been proven impossible, so the 5×5 board is the smallest square board that can be toured. It's also the smallest in a series of boards whose size can be calculated in the equation (4n+1) × (4n+1), where n is an integer 1 or greater, and 4n+1 is the length of the side. There's three such boards in the dream and on this page:

n=1
5×5
n=2
9×9
n=3
13×13

These boards are special because they can be solved very simply by filling the outermost 2 rings of squares first, then moving inwards, resulting in the following sequence.

  1. Fill in the outermost 2 rings of a 13×13 board, and you're left with a 9×9 board.
  2. Fill in the outermost 2 rings of a 9×9 board, and you're left with a 5×5 board.
  3. Fill in the outermost 2 rings of a 5×5 board, and you're left with a 1×1 board.

If you set the current square to number 24 on the above board, you'll see what I mean.

abcde
abcde
57 1217225 5
418236 11164
3138 254 213
224192 15102
11 149 203 1

By the way, if n=0, that's the 1×1 board, so technically, you could say I have 4 boards of this sequence in the dream, but that's like a lapine breakup: splitting hares.

6×6 (36 squares)

We come to our first Knight's Circuit (also known as a Closed Tour, which means can legally move the knight from the last square of the tour to the beginning (if you have the JavaScript enabled, the and buttons allow you to do just that). In other words, unlike an open tour, which must be started at one end or the other, you can begin a closed tour whereëver you please. Closed tours are possible on boards unless:

abcdef
abcdef
61 12251435106
52615361124215
45 2 13229 344
316274 3120233
23 6 2918338 2
12817327 30191

7×7 (49 squares)

abcdefg
abcdefg
74726376 1728397
6365 4827387 186
5254613161140295
44 35104914198 4
3452415129 30413
2343 2243321 202
12344332 2142311

8×8 (64 squares)

This is the classic Knight's Tour, as it is performed on an 8×8 chessboard. This is also a classic solution, as performed by the famous chess-playing hoax The Turk, an automaton capable of playing—and winning—chess. It was actually a puppet, with a controller hidden inside.

abcdefgh
abcdefgh
86 1958374 1760478
757385 18594815627
6207 363 166146496
539562932352 63145
48 21341 302750454
355403128336413263
2229 4253241144512
141542310435225121

9 × 9 (81 squares)

The pattern mentioned above for the 5×5 board continues here—fill in the outside squares first, then solve a 5×5 board.

abcdefghi
abcdefghi
91326395211243750 99
84053122538511023368
727146368737761 8497
65441747862677235226
51528696481607648 75
44255797558716621344
329165770658059 6473
25643 2311845 433202
1 1301744 3321946 51

10 × 10 (100 squares)

One common technique for solving the larger boards is dividing the board into smaller ones. In this case, this is divided into quadrants, each quadrant being a 5×5 board. This is tricker when the board can't be evenly divided, as you'll see next.

abcdefghij
abcdefghij
10 32 49 36 41 30 51 64 59 70 5310
9 37 42 31 50 35 74 69 52 65 60 9
8 48 33 46 29 40 63 58 73 54 71 8
7 43 38 27 34 45 68 75 56 61 66 7
6 26 47 44 39 28 57 62 67 72 55 6
5 7 18 13 24 1 76 87 92 97 78 5
4 12 25 8 19 14 93100 77 86 91 4
3 17 6 23 2 9 88 83 98 79 96 3
2 22 11 4 15 20 9994 81 90 85 2
1 5 16 21 10 3 82 89 84 95 80 1

11×11 (121 squares)

This solution was a massive kludge, so far as I'm concerned. Unable to work a solution out or find one online, I finally divided the board into a 5×5 board, a 6×6 board, and 2 5×6 boards, looked up tours for them, and cobbled together a solution from there. Only later did I find an online solution, but it's not this one here.

abcdefghijk
abcdefghijk
11 57 44 37 52 59 42 63 80 75 70 6511
10 36 53 58 43 38 51 74 69 64 81 7610
9 45 56 47 32 41 60 79 62 83 66 71 9
8 48 35 54 39 50 33 84 73 68 77 82 8
7 55 46 49 34 31 40 61 78 85 72 67 7
6 12 21 30 3 10121110 89100 87112 6
5 29 8 11 20 1 96 99 86111 90101 5
4 22 13 2 9 4 109120 97 88113116 4
3 7 28 17 24 19 98 95106115102 91 3
2 14 23 26 5 16119108 93104117114 2
1 27 6 15 18 25 94105118107 92103 1

I don't know what it says about my mental state that I made sure that the solution finished in the middle.

12×12 (144 squares)

Quadrants again, this time solving four 6×6 boards.

abcdefghijkl
abcdefghijkl
12 19 8 23 34 21 10 39 64 43 54 41 6612
11 30 33 20 9 24 35 50 53 40 65 44 5511
10 7 18 31 22 11 14 63 38 51 42 67 7010
9 32 29 4 13 36 25 52 49 60 69 56 45 9
8 17 6 27 2 15 12 37 62 47 58 71 68 8
7 28 3 16 5 26 1 48 59 72 61 46 57 7
6129118133144131120 73 98 77 88 75100 6
5140143130119134109 84 87 74 99 78 89 5
4117128141132121124 97108 85 76101104 4
3142139114123110135 86 83 94103 90 79 3
2127116137112125122107 96 81 92105102 2
1138113126115136111 82 93106 95 80 91 1

13×13 (169 squares)

Currently, this is the last in the 4n+1 series in the dream. It's fairly easy—just time consuming. Like the 9×9 board, you just fill in the outer two rows first, which leaves you a 9×9 board—which is above. You can solve boards that are 17×17, 21×21, 25×25, and so on in this way.

abcdefghijklm
abcdefghijklm
13 19 40 61 82 17 38 59 80 15 36 57 78 1313
12 62 83 18 39 60 81 16 37 58 79 14 35 5612
11 41 20101114127140 99112125138 97 12 7711
10 84 63128141100113126139 98111124 55 3410
9 21 42115102151156161166149 96137 76 119
8 64 85142129162167150155160123110 33 548
7 43 22103116157152169148165136 95 10 757
6 86 65130143168163146159154109122 53 326
5 23 44117104145158153164147 94135 74 9 5
4 66 87144131 90119106133 92121108 31 524
3 45 24 89118105132 91120107134 93 8 733
2 88 67 2 47 26 69 4 49 28 71 6 51 302
1 1 46 25 68 3 48 27 70 5 50 29 72 7 1

14×14 (196 squares)

Aren't quadrants marvelous? In this case, four 7×7 boards.

abcdefghijklmn
abcdefghijklmn
14 68 79 58 91 70 81 60 9911212314210311412514
13 57 92 69 80 59 98 7112214310211312414110413
12 78 67 90 85 88 61 8211110013713413912611512
11 93 56 87 52 83 72 9714412113210113610514011
10 66 77 84 89 86 51 6213111013513813311612710
9 55 94 75 64 53 96 73120145108129118147106 9
9 76 65 54 95 74 63 50109130119146107128117 8
7 19 8 29 42 17 6 27158169180193160171148 7
6 30 43 18 7 28 49 16179194159170149192161 6
5 9 20 35 38 41 26 5 168157186181184151172 5
4 44 31 40 3 36 15 48195178183150187162191 4
3 21 10 37 34 39 4 25156167188185182173152 3
2 32 45 12 23 2 47 14177196165154175190163 2
1 11 22 33 46 13 24 1 166155176189164153174 1

15×15 (225 squares)

This one is the largest in the dream, and I really didn't see any reason to go 16×16 (for 256 tiles). This takes quadrants a step further: this board is divided into nine 5×5 boards, making it easy—if time-consuming—to solve.

abcdefghijklmno
abcdefghijklmno
1515516016517015313014914213713210512411711210715
1416617115415916414113613114814311611110612311814
1316115617315216915012914613313812510412110811313
1217216715816317413514012714414711011510211912212
1115716217516815112814513413912610312010911410111
10176187196193182207212217221205 80 91100 85 7810
9197192183188195218222206211216 99 86 79 90 959
8186177194181200213208225204220 92 81 96 77 848
7191198179184189223219202215210 87 98 83 94 897
6178185190199180201214209224203 82 93 88 97 766
5 7 12 17 24 5 32 37 42 49 30 51 68 75 62 575
4 18 23 6 11 16 43 48 31 36 41 74 63 58 67 724
3 13 8 21 4 25 38 33 46 29 50 69 52 73 56 613
2 22 19 2 15 10 47 44 27 40 35 64 59 54 71 662
1 1 14 9 20 3 26 39 34 45 28 53 70 65 60 551

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