© by George Jelliss, begun January 2001, revised November 2012 and August 2022.
Sections on this page: — Rook Tours — Magic Rook Tours — Non-Intersecting Rook Tours — Progressive Rook Paths — Integral Rook Paths
The chess-piece called a rook makes a single move that is like a series of wazir steps in a straight line. Some problems presented as rook tours are really wazir tours, that is single-step rook tours, this is the case if a cell is regarded as 'visited' by the rook even though it does not stop there but only rides through. Any non-intersecting (and non-overlapping) rook path is also a wazir path, except that the moves in the path are counted in terms of the straight sections of path rather than in unit lengths. For rook tours of this special type see the Wazir tours section.
By a rook tour we mean a tour in which each move from one cell to the next visited in the tour is a rook move of any length; the cells passed through by the rook en-route from one stopping place to another are not considered to be 'visited' during that move; they will have already been visited or will be visited later in the tour. Rook moves longer than a single step may overlap with others along the same line, so to distinguish them it is necessary to draw the longer moves as curved lines, otherwise they all merge into one straight line. These curves recognise the difference between the passage of the rider through intermediate cells at speed and its stationarity at the end points of the move.
The only tour by a wazir on a 1×n board is an open tour consisting of a straight walk from one end to the other, and a closed tour is possible only on the 1×2 board. A rook, however, can start on any cell and can enter the cells in any order. There are n! ways of numbering the cells of a 1×n board in sequence 1 to n, since there are n choices for the first cell, n−1 for the second, n−2 for the third and so on. Thus the number of open rook tour diagrams, each of which can be numbered from either end, is T(n) = n!/2. All of these open tours are reentrant, since the two end cells are always a rook move apart. The diagrams show the generic open tours of sizes 2 to 4.
Every one-rank tour has the rank, trivially, as an axis of symmetry, so strictly speaking all the tours are symmetric. However, we count as symmetric only those tours that have a vertical axis of symmetry when we draw the moves as curves above the horizontal line. When n is even the number of symmetric open tours is S(n) = [n.(n−2).(n−4)...6.4.2]/2 since there are n choices for first cell, but this also fixes the last cell, then there are n−2 choices for second cell, but this also fixes the next-to-last cell, and so on, until we reach 2 choices for the cell numbered n/2. When n is odd the centre cell of the board must be the centre cell of the symmetric open tour; taking this out we get the same formula for S(n) but with n−1 in place of n. If h = integral part of (n/2) we have S(n) = 2^(h−1)×h! in each case. The number of generic open tours is G(n) = [T(n) + S(n)]/2.
From one closed tour we can derive n open tours by deleting one move. So the closed tours total T(n) = (n−1)!/2 for n > 2. The diagrams show the generic closed tours of sizes 2 to 5.
The above diagrams show the symmetric tours of size 6.
The number of symmetric closed tours on an odd-length board is the same as the number of symmetric open tours on the same board, since they are formed by joining the ends of the open tours; this move being the only symmetric move in the closed tour: S(2h+1) = 2^(h−1)×h!. The number on an even-length board is n/2 + 1 times the total for the previous odd board: S(2h) = 2^(h−1)×(h+1)!. The added '1' in the factor counts the tours with no symmetric move, while the 'n/2' part counts those with two symmetric.
The generic tours are calculated as usual: G(n) = [T(n) + S(n)]/2.
These have not been studied to any extent. Obviously, in a closed tour the number of 'vertical' links must be even.
On the 3×3 board the rook can make 7 geometrically distinct closed tours. The first two illustrated below differ only in the stop at the centre being made on the vertical move or the horizontal move through the centre.
The first magic rook tour I know of is this diagonal magic one by Joachim Brügge, published in the German chess problem magazine Die Schwalbe August 1985. It uses 6 lengths of rook move (1, 2, 3, 4, 5, 6).
This inspired me to further research and I have a note that I found the tour on the right on 20 May 1986, and followed this with the other examples shown below on 21 May 1986, but then seem to have forgotten about this work, probably due to big changes in my life that took place later that year. My tours use rook moves of only 4 lengths (1, 2, 3, 4), and each quarter forms a bisatin of one circuit, using two cells in each rank and file. The circles mark the ends of the quarters where the numbers 1, 16, 17, 32, 33, 48, 49, 64 occur. These tours are all diagonally magic.
These biaxial tours remain magic in ranks and files when the origin of numbering is shifted to the next quarter, i.e. 17 becomes 1, 18 becomes 2 and so on. The three tours shown below also remain diagonally magic when renumbered in this way, but the tour above loses the diagonal property; this is because it has more than two numbers £ 16 in a diagonal.
An enumeration of the possible bisatins that could be used in this type of tour would be the next step, if anyone with computing ability would like to take it on.
The chess-piece called a rook makes a single move that is like a series of wazir steps in a straight line. Some problems usually presented as rook tours can also be regaded as wazir tours, this is the case if a cell is regarded as 'visited' by the rook even though it does not stop there but only rides through. Any non-intersecting (and non-overlapping) rook path is also a wazir path, except that the moves in the path are counted in terms of the straight sections of path rather than in unit lengths.
First we consider wazir tours with the minimum number of right-angled turns, which are equivalent to non-intersecting rook paths with the minimum number of rook moves. On a board n×n, if there are moves in every rank and file then we have at least 2n moves. On the other hand, if there is a rank or file without a move along it, then it must be crossed or met by n mutually parallel moves, and these will require a further n–1 rook moves to join them into a single path, or n to make a closed circuit. The minimum is therefore 2n rook moves for a closed tour and 2n–1 for an open tour. In an open tour the first and last moves must be parallel, since they are among the n parallel lines. We take these horizontal. There are always at least two full-rank moves, top and bottom. To solve the problem on a rectangle n×m (m > n) we just stretch the n parallel moves along the ranks, so we can confine our attention here to square boards.
The numbers of open tours: 2×2: G = S = 1, T = 4; 3×3: G = 2, S = 1, T = 12; 4×4: G = 5, S = 2, T = 32; 5×5: G = 13, S = 3, T = 92. On the 6×6 the total of open tours is not known, but there are 21 with end at corner (2 corner to corner, 1 symmetric).
The numbers of closed tours: 2×2: G = S = T = 1; 4×4: (the C-shaped tour) G = S = 1, T = 4; 6×6: (illustrated) G = 3, S = 2, T = 16; 8×8: (illustrated) G = 12, S = 3, T = 84; 10×10 G = 58?.
The problem of a 16-move non-intersecting rook closed tour of the standard chessboard was proposed and solved with one example (the 'four-pronged' pattern) by Sam Loyd [Chess Strategy 1881]. H. E. Dudeney gave a problem solved by a 15-move open tour c2-b2 [The Canterbury Puzzles 1904], ensuring a unique solution by a barrier between d1 and e1. T. R. Dawson gave complete solutions for two 8×8 cases in 15 moves, giving 7 tours a1 to a6 [Chess Amateur, i 1909, Puzzle 28] and 6 tours a1 to a4 [British Chess Magazine, vi 1943].
The problem of finding a closed tour with just two turning points in each rank and file is impossible; there must be more than two turns in at least one of the edges. There is however a pseudotour with this property: the set of n/2 concentric squares.
T. R. Dawson and Friends considered some paths of rooks in which the lengths of the moves increase in arithmetical progression. In the solutions L R U D stand for Left, Right, Up, Down.
T. R. Dawson, Fairy Chess Review, October 1933, problem 985: A rook moves 1, 2, 3, ... squares, never passing over the same square twice. What are the shortest closed tours. (a) On unbounded board. (b) Starting at a1 corner of unbunded quarter-plane. (c) Of rectangular shape. (d) Of rectangular shape, starting at a1. Solutions: (a)7 moves RULULDR. (b) 12 moves RUUURDRRULLD or URUURDRURDLL. (c) 8 moves LLUURRDL. (d) 20 moves 1-5R,6-14U,15L,16-20D. The following next cases of (d) are also given: 1-18,19-84,85-86,87-119; 1-29,30-84,85-89,90-119; 1-44,45-84,85-95,96-119; 1-174,175-491,492-522,523-696; 1-169,170-2870,2871-2875,2876-4059.
T. R.Dawson, Fairy Chess Review, October 1933, problem 986: Shortest circuit by rook in arithmetical progression 1, 3, 5, 7, ... Solution: 8 steps DRULULDR.
T. R. Dawson, Fairy Chess Review, 1934, problems 1200-1205: As for the first problem but with progression 2, 3, 4, ... Solutions: (a) 8 movesDLURURDL. (b) 9 moves RURRRULLD or RUUURULDD. (c) 12 moves RRRUULLLLDRR. (d) 2-9,10-43,44,45-61 with longer solutions beginning 2-16; 2-21; 2-34. Also given are solutions for the same problem starting at different numbers: 3-9,10-12,13-15,16-17; 4-5,6-8,9,10-11; 4-38,39-101,102-108,109-143; 5-28,29-130,131-133,134-184; 5-49; 5-64; 5-86; 5-103.
W. Jacobs, Fairy Chess Review, December 1943, problem 5799, p.69. Discover a rectangle which a rook can circumscribe exactly in two different arithmetical progressions, starting at a corner in each case. Solution (February 1944 p.78): 1-11R,12-14U,15-18L,19-20D and 12-14R,15-18U,19-20L,21-23D. Board 40x67.
H. Perkins, Fairy Chess Review, August 1944, problem 6039, p.98. Shortest possible arithmetical progression rook tour in rectangular form, starting at an angle. Solution (October 1944, p.110): 4,5D,6,7,8R,9U,10,11L.
Recent work (1990s) of a similar kind is on Golygons. I first came across them here: Golygons and Golyhedra. These are progressive rook circuits subject to the extra condition that every successive pair of moves must be at right angles (including where the last meets the first).
Another series of rook-path problems by T. R. Dawson concerned rooks playing only to points an integral distance from the origin (0,0) but not on the board edge.
T. R. Dawson Fairy Chess Review October 1934.
(a) Shortest closed rook path starting at (9,12).
Solutions: 10 moves (196 units) (9,12)-(16,12)-(16,30)-(40,30)-(40,9)-(12,9)-(12,16)-(30,16)-(30,40)-(9,40)-
or 8 moves (300 units) (9,12)-(35,12)-(35,84)-(63,84)-(63,16)-(30,16)-(30,40)-(9,40)-
(b) Shortest journey from (6,8) to (8,6).
Solution: 16 moves (338 units) (6,8)-(15,8)-(15,20)-(21,20)-(21,72)-(30,72)-(30,16)-(12,16)-
(12,9)-(40,9)-(40,30)-(72,30)-(72,21)-(20,21)-(20,15)-(8,15)-(8,6).
or 12 moves (364 units) (6,8)-(15,8)-(15,112)-(84,112)-(84,63)-(60,63)-(60,45)-(28,45)-(28,21)-(20,21)-(20,15)-(8,15)-(8,6).
also 14 moves (372 units) (6,8)-(15,8)-(15,20)-(21,20)-(21,28)-(96,28)-(96,40)-
(42,40)-(42,56)-(90,56)-(90,48)-(20,48)-(20,15)-(8,15)-(8,6).