Knight's Tour Notes by George Jelliss. Note 5a.
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The 5×5 board of 25 cells is the smallest knight-tourable square board. There are 112 geometrically distinct tours, all open, of which 8 are symmetric. The symmetric solutions were all found by Euler (1759).
If you prefer larger numbers and wish to count the different diagrams possible, then each symmetric tour can be viewed in 4 distinguishable orientations, and each asymmetric tour in 8, thus the total becomes: 8×4 + (112 – 8)×8 = 864.
If further you wish to present the tours in numerical form this total has to be doubled since each tour can be numbered from either end, giving the total 1728. This was the total given by C.Planck in his "Chessboard Puzzles" series in Chess Amateur, puzzles 25 and 26, December 1908 p.83 and February 1909 p.147 (this is the earliest reference I have so far found to the complete enumeration of the 5×5 tours).
Analysis
Every 5×5 tour must have one end on a corner cell, since if the knight passes through all four corners it makes a short circuit of 8 moves. The other end can be any other cell of the same colour. Thus, classified by separation of end-points there are 8 types of 5×5 tour; the numbers of each type are: {0,2} 14, {0,4} 30, {1,1} 8, {1,3} 14, {2,2} 8, {2,4} 14, {3,3} 6, {4,4} 18.
The moves through the corners form three different patterns: (A) two pieces of 4 cells, (B) pieces of 6 and 2 cells, (C) one path of 8 cells. The moves through the other cells, excluding the centre, form a short circuit of 16 moves, and this is either toured in two pieces of (1) 8+8 cells, (2) 6+10 cells, (3) 4+12 cells (4) 2+14 cells or (5) one piece of 16 cells. Combining these classifications we get 13 classes: A1, A2, A3, A4; B1, B2, B3, B4; C1, C2, C3, C4, C5. The classes (1) to (5) are also characterised by the pattern at the centre cell: in (1) the moves go straight through, in (2) they form an acute angle, in (3) a right angle, in (4) an obtuse angle and in (5) the tour terminates at the centre.
Classes A1–A4 comprise the 18 tours from corner to opposite corner. A1 being the eight symmetric tours, with straight move through centre (given by Euler 1759). Murray (1942) noted that the maximum number of two-unit lines in a 5×5 tour is three, and there are two such tours, both symmetric. A2 four with acute centre; A3 two with right-angled centre; A4 four with obtuse centre.
Classes B1–B4 comprise 30 tours from corner to adjacent corner, with B1 straight centre 2 tours, B2 acute centre 8 tours, B3 right-angled centre 12 tours, and B4 obtuse centre 8 tours.
Class C1 has 8 tours with straight move through centre.
Class C2 has 16 tours with acute centre.
Class C3 has 16 tours with right-angled centre.
Class C4 has 16 tours with obtuse angle at centre.
Class C5 has 8 tours terminating at the centre.
Puzzles: Complete the following fully determined 5×5 tours (note an orthogonal/diagonal angle is one whose bisector is at 90°/45° to a board edge): (1) a1 ... a4-c3. (2) a1-c2 ... b2 with central moves forming (a) diagonal-acute angle (b) orthogonal-acute angle (c) right angle containing b2 (d) right angle excluding b2 (e) orthogonal-obtuse angle (f) diagonal obtuse angle (g) straight line. (3) a1-c2 ... d4 (a)–(f) as in (2). (4) a1 ... e3 with centre angle (a) orthogonal-obtuse (b) diagonal-obtuse. (5) a1 ... c1 with centre angle (a) diagonal-acute (b) orthogonal-acute. (6) Identify which are 'tours of inspection', i.e. they 'pass through' every unit square — not only the squares of the board but also the squares whose corners are the centres of the cells.
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