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Chessboard Puzzles Part 4 - Other Surfaces and Variations

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Chessboard Puzzles Part 4 - Other Surfaces and Variations

  1. 1. Chessboard Puzzles: Other Surfaces and Variations Part 4 of a 4-part Series of Papers on the Mathematics of the Chessboard by Dan Freeman May 19, 2014
  2. 2. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar Table of Contents Table of Figures .............................................................................................................................. 3 Introduction ..................................................................................................................................... 4 Knight’s Tour Revisited .................................................................................................................. 4 2 Knight’s Tour on a Torus ............................................................................................................ 5 Knight’s Tour on a C ylinder ....................................................................................................... 6 Knight’s Tour on a K lein Bottle .................................................................................................. 7 Knight’s Tour on a Möbius Strip ................................................................................................ 8 Rooks and Bishops Domination on a Torus ................................................................................... 9 Kings Domination and Independence on a Torus ......................................................................... 11 Knights Domination on a Torus.................................................................................................... 13 Queens Domination on a Torus .................................................................................................... 13 The 8-queens Problem on a Cylinder............................................................................................ 14 Domination and Independence on the Klein Bottle ...................................................................... 16 Independent Domination Number................................................................................................. 19 Upper Domination Number .......................................................................................................... 20 Irredundance Number ................................................................................................................... 22 Upper Irredundance Number ........................................................................................................ 23 Total Domination Number ............................................................................................................ 26 Conclusion .................................................................................................................................... 28 Sources Cited ................................................................................................................................ 32
  3. 3. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar Table of Figures Image 1: Knight Movement ............................................................................................................ 4 Image 2: Chessboard on a Torus..................................................................................................... 5 Image 3: C losed Knight’s Tour on 2x7 Torus ................................................................................ 6 Image 4: Closed Knight's Tour on 3x8 Torus ................................................................................. 6 Image 5: Closed Knight's Tour on 4x9 Torus ................................................................................. 6 Image 6: Klein Bottle ...................................................................................................................... 7 Image 7: Closed Knight's Tour on 6x2 Klein Bottle....................................................................... 8 Image 8: Closed Knight's Tour on 6x4 Klein Bottle....................................................................... 8 Image 9: Möbius Strip .................................................................................................................... 9 Image 10: Rook Movement .......................................................................................................... 10 Image 11: Bishop Movement ........................................................................................................ 10 Image 12: There Are n Distinct Diagonals (Shown in Red) on an nxn Torus .............................. 11 Image 13: King Movement ........................................................................................................... 11 Image 14: Nine Kings Dominating a Regular 7x7 Board ............................................................ 12 Image 15: Seven Kings Dominating a 7x7 Torus ......................................................................... 12 Image 16: Queen Movement......................................................................................................... 14 Image 17: Eight Queens Fail to Be Independent on 8x8 Cylinder ............................................... 15 Image 18: Row, Column and Negative Diagonal Labeling of 8x8 Chessboard .......................... 16 Image 19: Fifteen Kings Dominating a 7x14 Klein Bottle ........................................................... 18 Image 20: Thirteen Kings Dominating a 14x7 Klein Bottle ......................................................... 18 Image 21: Queens Domination Number, Independent Domination Number and Independence Number on 4x4 Chessboard ........................................................................... 20 Image 22: Queens Domination Number, Independent Domination Number, Independence Number and Upper Domination Number on 6x6 Chessboard.............................. 21 Image 23: Maximal Irredundant Sets of 8 Kings and 9 Kings on 7x7 Chessboard ...................... 23 Image 24: Maximum Irredundant Set of 16 Kings on 7x7 Chessboard ....................................... 24 Image 25: Maximum Irredundant Set of 12 Rooks on 8x8 Chessboard ....................................... 24 Image 26: Maximum Irredundant Set of 18 Bishops on 8x8 Chessboard .................................... 25 Image 27: Maximum Irredundant Set of 7 Bishops on 6x6 Chessboard ...................................... 25 Image 28: Totally Dominating Sets of 5 Queens and 8 Rooks on 8x8 Chessboard ..................... 26 Image 29: Totally Dominating Sets of 10 Bishops and 14 Knights on 8x8 Chessboard .............. 27 Table 1: Knights Domination N umbers on Regular C hessboard and Torus for 1 ≤ n ≤ 8 ........... 13 Table 2: Queens Domination N umbers on Regular Chessboard and Torus for 1 ≤ n ≤ 10 .......... 14 Table 3: K ings Total Domination N umbers for 1 ≤ n ≤ 12........................................................... 27 Table 4: Upper Bounds for K ings Total Domination N umbers for 13 ≤ n ≤ 25........................... 28 Table 5: Domination Number Formulas on the Torus by Piece ................................................... 29 Table 6: Chessboard Number Formulas by Piece ......................................................................... 31 3
  4. 4. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar Introduction 4 In this fourth and final paper in my series on mathematical chessboard puzzles, I will build on the three major topics in domination, independence and the knight’s tour that I examined in my first three papers by looking at them in the context of irregular surfaces such as the torus, cylinder, Klein bottle and Möbius strip. I will then explore some other concepts related to domination and independence such as the independent domination number, upper domination number, irredundance number, upper irredundance number and total domination number. Since bringing such irregular surfaces and other variations to the table results in a vast amount of combinations that one could potentially explore, this paper is by no means exhaustive. Rather, it should serve as an overview of the mathematical properties and formulas of these variations to the concepts of domination, independence and the knight’s tour and how these variations compare and contrast with the original concepts. My hope is that this paper will provide the reader with a sense of the vast diversity among mathematical chessboard puzzles. Knight’s Tour Revisited Recall that knights move two squares in one direction (either horizontally or vertically) and one square in the other direction, thus making the move resemble an L shape. Knights are the only pieces that are allowed to jump over other pieces. In Image 1, the white and black knights can move to squares with circles of the corresponding color [4]. Image 1: Knight Movement
  5. 5. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar 5 Also, recall from my preceding paper that a knight’s tour is a series of moves made by a knight that visits every square on an mxn1 chessboard once and only once. A knight’s tour may fall into one of two categories, closed or open, defined as follows:  A closed knight’s tour is one in which the knight’s last move in the tour places it a single move away from where it started.  An open knight’s tour is one in which the knight’s last move in the tour places it on a square that is not a single move away from where it started. Knight’s Tour on a Torus Like any of the surfaces we will be studying in this paper, a torus is a topological surface with a very specific definition. However, for our purposes, we can simply view a torus as a donut-shaped surface in which both the rows and columns wrap around on their edges. That is, when moving to the right beyond the right edge of the board, one returns to the left edge and when moving up beyond the top edge of the board, one returns to the bottom edge [1, p. 65]. A toroidal chessboard is illustrated in Image 2 [5]. Image 2: Chessboard on a Torus In 1997, John Watkins and his student, Becky Hoenigman, proved the remarkable result that every mxn rectangular chessboard has a closed knight’s tour on a torus! Images 2, 3 and 4 show examples of closed knight’s tours on a 2x7, 3x8 and 4x9 tori, respectively [1, p. 67]. The 1 Throughout this paper, m and n refer to arbitrary positive integers denoting the number of rows and columns of a chessboard, respectively.
  6. 6. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar up arrows on either side of the board and the right-pointing arrows above and below the board are commonly used notation to indicate that the board is on a torus. Image 3: Closed Knight’s Tour on 2x7 Torus Image 4: Closed Knight's Tour on 3x8 Torus Image 5: Closed Knight's Tour on 4x9 Torus 1 Knight’s Tour on a Cylinder 6 1 Unlike a torus, a cylinder only wraps in one dimension, not both. Since the knight’s tours on the 3x8 and 4x9 tori in Images 4 and 5 above only make use of horizontal wrapping, these tours would also work on a cylinder. In 2000, John Watkins proved that a knight’s tour exists on an mxn cylindrical chessboard unless one of the following two conditions holds:
  7. 7. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar 7 1) m = 1 and n > 1; or 2) m = 2 or 4 and n is even [1, p. 71]. It is easy to see why the above cases are excluded. If m = 1, a knight can’t move at all. If m = 2 and n is even, then each move would take the knight left or right by two columns and so then only at most half of the columns would be visited. Lastly, if m = 4 and n is even, then the coloring argument by Louis Pósa from my previous paper on knight’s tours holds [1, p. 71]. Knight’s Tour on a Klein Bottle The Klein bottle, due to German mathematician Felix Klein in 1882 [6], operates like a torus, except when wrapping horizontally, the rows reverse order due to the half-twist in the construction of the surface [1, p. 79]. A Klein bottle is represented in Image 6 [7]. Image 6: Klein Bottle John Watkins proved that, just like with a torus, every rectangular chessboard has a knight’s tour on a Klein bottle [1, p. 81]. Examples of closed knight’s tours on 6x2 and 6x4 Klein bottles are shown in Images 7 and 8. The up arrow on the left side of the board, the down arrow on the right side of the board and the right-pointing arrows above and below the board are commonly used notation to indicate that the board is on a Klein bottle.
  8. 8. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar Image 7: Closed Knight's Tour on 6x2 Klein Bottle Image 8: Closed Knight's Tour on 6x4 Klein Bottle Knight’s Tour on a Möbius Strip 8 A very famous one-sided surface known as the Möbius strip shares properties of both the cylinder and the Klein bottle. It is like the cylinder in that it only wraps in one dimension but is like the Klein bottle in that it makes a half-twist when wrapping, thereby reversing the order of the rows [1, p. 82]. A Möbius strip is shown in Image 9 [8].
  9. 9. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar 9 Image 9: Möbius Strip John Watkins proved that a closed knight’s tour exists on a Möbius strip unless one or more of the following three conditions hold: 1) m = 1 and n > 1; or n = 1 and m = 3, 4 or 5; 2) m = 2 or 4 and n is even; or 3) n = 4 and m = 3 [1, p. 82] Rooks and Bishops Domination on a Torus Now, let’s turn our attention to the concept of domination, which we explored in my first paper in this series, applied to a torus. First, let’s look at the uninteresting case of rooks domination on a torus. Recall that rooks are permitted to move any number of squares either horizontally or vertically, as long as they do not take the place of a friendly piece or pass through any piece (own or opponent’s) currently on the board. In Image 10, the white rook can move to any of the squares with a white circle and the black rook can move to any of the squares with a black circle [4].
  10. 10. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar 10 Image 10: Rook Movement Since it doesn’t make any difference whether a rook is on a regular board or on a torus, it follows that γtor(Rnxn) = γ(Rnxn) = n, where the subscript “tor” indicates domination on a torus [1, p. 144]. Almost as boring as the rook’s case is the bishop’s case. Recall that bishops move diagonally any number of squares as long as they do not take the place of a friendly piece or pass through any piece (own or opponent’s) currently on the board. In Image 11, the white bishop can move to any of the squares with a white circle and the black bishop can move to any of the squares with a black circle [4]. Image 11: Bishop Movement Since the number of distinct diagonals in either direction drops from 2n – 1 on a regular square chessboard to n on a torus, it is easy to see that γtor(Bnxn) = γ(Bnxn) = n [1, p. 144]. Image 12 illustrates this fact.
  11. 11. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar Image 12: There Are n Distinct Diagonals (Shown in Red) on an nxn Torus Kings Domination and Independence on a Torus 11 Recall that kings are allowed to move exactly one square in any direction as long as they do not take the place of a friendly piece. In Image 13, the king can move to any of the squares with a white circle [4]. Image 13: King Movement Kings domination on a torus is more interesting than that of rooks or bishops because the kings domination number on a torus is quite distinct from the formula on a regular chessboard2. 2 Throughout this paper, the symbols ‘└’ and ‘┘’ will be used to indicate the greatest integer or floor function and the symbols ‘┌’ and ‘┐’ will be used to indicate the least integer or ceiling function.
  12. 12. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar While γ(Knxn) = └(n + 2) / 3┘ 12 2, γtor(Knxn) = ┌(n / 3)*┌ n / 3 ┐┐ [1, p. 147]. The former uses the floor function while the latter uses the ceiling function. However, both share a factor of 1/3 because regardless of whether a king is on a torus or not, it can control squares on at most three rows and at most three columns (in fact, on a torus, a king always covers squares on exactly three rows and three columns, since a torus technically has no edges). The kings domination number on a torus can be generalized to a rectangular board using the following formula: γtor(Kmxn) = max{┌(m / 3)*┌ n / 3 ┐┐, ┌(n / 3)*┌m / 3 ┐┐} [1, p. 149]. It is worth pointing out that, in general, fewer kings are needed to cover a torus than a regular chessboard of the same size. For example, 9 kings are required to cover a 7x7 regular board while only 7 kings are needed to cover a 7x7 board on a torus. This fact is illustrated in Images 14 and 15. Image 15: Nine Kings Dominating a Regular 7x7 Board Image 14: Seven Kings Dominating a 7x7 Torus
  13. 13. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar 13 The formulas for the kings independence number on a torus take a similar form to those for the kings domination number except that the floor function is used instead of the ceiling function and a minimum value, not a maximum value, is sought in the rectangular formula. The formulas are as follows: βtor(Knxn) = └(½*n)*└½*n┘┘ and βtor(Kmxn) = min{└(½*m)*└ ½*n ┘┘, └(½*n)*└ ½*m ┘┘}. Note that the factor of ½ appears here in the toroidal formulas just as it does in the formula on the regular chessboard (recall that β(Knxn) = └½*(n + 1)┘ 2). This is due to the fact that any 2x2 block of squares can contain at most one independent king, regardless of surface [1, p. 194]. Knights Domination on a Torus The knights domination numbers on both a regular board and a torus for 1 ≤ n ≤ 8 appear in Table 1 [1, p. 140]. Note that, shockingly, the value for γtor is lower for n = 8 than it is for n = 7! Also, each value of γtor is unique up to n = 8. This may or may not be the case in general. Table 1: Knights Domination Numbers on Regular Chessboard and Torus for 1 ≤ n ≤ 8 n γ(Nnxn) γtor(Nnxn) 1 1 1 2 4 2 3 4 3 4 4 4 5 5 5 6 8 6 7 10 9 8 12 8 Queens Domination on a Torus Recall that queens move horizontally, vertically and diagonally any number of squares as long as they do not take the place of a friendly piece or pass through any piece (own or opponent’s) currently on the board. In Image 11, the queen can move to any of the squares with a black circle [4].
  14. 14. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar 14 Image 16: Queen Movement The queens domination numbers on both a regular board and a torus for 1 ≤ n ≤ 10 appear Table 2 [1, p. 140]. Note that the only case where the two numbers differ is n = 8. In this case, one fewer queen is needed to dominate the board on a torus than is required to cover the regular board. Table 2: Queens Domination Numbers on Regular Chessboard and Torus for 1 ≤ n ≤ 10 n γ(Qnxn) γtor(Qnxn) 1 1 1 2 1 1 3 1 1 4 2 2 5 3 3 6 3 3 7 4 4 8 5 4 9 5 5 10 5 5 The 8-queens Problem on a Cylinder Unlike the queens independence number on a regular chessboard (recall that this is just n on an nxn square board for all n other than 2 or 3), a formula for this number on the cylinder, denote it βcyl(Qnxn), has not yet been found. While βcyl(Q5x5) = β(Q5x5) = 5 and βcyl(Q7x7) = β(Q7x7) = 7, βcyl(Q8x8) = 6 ≠ β(Q8x8) = 8 [1, pp. 193-194]. Image 17 illustrates why one particular arrangement of eight queens that would be independent on a regular 8x8 board fails to be
  15. 15. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar independent on an 8x8 cylinder [1, p. 192]. The queen in the first column from the left resides in the same negative diagonal as the queen in the fourth column while the queen in the second column lies on the same positive diagonal as the queen in the seventh column. The right-pointing 15 arrows above and below the board are commonly used notation to indicate that the board is on a cylinder. Image 17: Eight Queens Fail to Be Independent on 8x8 Cylinder As it turns out, none of the twelve fundamental solutions to the 8-queens problem on a regular 8x8 chessboard are independent on an 8x8 cylinder. One can simply check this by brute force. E. Gik provided an alternative proof by labeling each square of an 8x8 board with an ordered triple (i, j, k), where i, j and k represent the following:  i – the row number starting with 1 from the bottom of the board,  j – the column number starting with 1 from the left side of the board and  k – the number of the negative diagonal in reverse order from 8 through 1starting on the diagonal right below the main diagonal [1, p. 192]. This (i, j, k) labeling on an 8x8 board is shown in Image 18. By design, the sum of each of the three coordinates i, j and k is divisible by 8 for each of the 64 squares on the board. Now suppose that we have 8 independent queens on the board. Then all three of the coordinates must be distinct for each square, otherwise at least two queens would share the same row, column or
  16. 16. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar negative diagonal, contradicting the fact that the set of queens is independent. Therefore, the i coordinates for these 8 queens sum to 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36, and the j and k coordinates sum to 36 as well. But then the sum of all three coordinates for the 8 queens would be 108, which is not divisible by 8, contradicting the fact that i + j + k for each square is divisible by 8. Therefore, our assumption that the 8 queens are independent is false [1, p. 192]. 16 Image 18: Row, Column and Negative Diagonal Labeling of 8x8 Chessboard One can then check by exhausting all possibilities that no arrangement of 7 independent queens exists on an 8x8 cylinder. In the 1 5 8 6 3 7 2 4 solution to the 8-queens problem on the regular 8x8 chessboard shown in Image 17, one can simply remove one of the two queens lying on the same positive diagonal and one of the two queens lying on the same negative diagonal to arrive at an independent set of 6 queens on an 8x8 cylinder. Therefore, βcyl(Q8x8) = 6 [1, p. 193]. Domination and Independence on the Klein Bottle Kings domination on a Klein bottle is somewhat convoluted so I will merely state the formula rather than prove it. The formula on a square nxn chessboard is as follows [1, p. 151]:
  17. 17. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar γKlein(K γKlein(K = 17 (1/9)*n2 if n = 3k (1/3(n + 1))2 if n = 3k + 2 (1/3(n + 2))2 – (1/6)*(n – 1) if n = 6k + 1 (1/3(n + 2))2 – (1/6)*(n + 2) if n = 6k + 4 nxn ) = The preceding formula can be generalized to a rectangular mxn chessboard as below. While it looks nothing like the formula for the square board, this does in fact reduce to the square formula when m = n [1, p. 152]. ┌m/6┐*┌2n/3┐ – ┌(n – 1)/3┐ if m ≡ 1, 2 or 3 mod 6 ┌m/6┐*┌2n/3┐ if m ≡ 4, 5 or 6 mod 6 m xn ) I find it fascinating that, unlike with a regular chessboard and even a torus, it does actually matter whether one chooses to orientate the board horizontally or vertically as far as the domination number is concerned on an mxn rectangular Klein bottle. That is, γKlein(P m xn ) need not be the same as γKlein(P nxm ) for some chess piece P. This is certainly the case with kings, as γKlein(K 7x14 ) = 15 while γKlein(K 14x7 ) = 13. This phenomenon, exhibited in Images 19 and 20, is due to the half-twist that the Klein bottle makes as it wraps horizontally. The 7x14 board has one 3x28 band (the 3x14 band in dark orange at the top of the board merged with the 3x14 band at the bottom in Image 19) that requires 10 kings to dominate. An additional 5 kings are needed to cover the single row in the middle of the board (highlighted in gray in Image 19) for a total of 15 kings. However, on the 14x7 board, five kings are needed to cover the two 3x14 bands (the 3x7 band in dark orange at the top of the board merged with the 3x7 band at the bottom and the two 3x7 bands in light orange just below and above the dark orange bands in Image 20) and only three kings are needed to cover the two rows in the middle of the board (highlighted in gray in Image 20) for a total of 13 kings [1, pp. 160-161].
  18. 18. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar 18 Image 19: Fifteen Kings Dominating a 7x14 Klein Bottle Image 20: Thirteen Kings Dominating a 14x7 Klein Bottle
  19. 19. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar βKlein(K βKlein(K 19 A formula is known for the bishops domination number on an nxn Klein bottle. Since a bishop is able to cover twice as much ground on a Klein bottle as it can on a regular chessboard (due to wrapping), it should come as no surprise then that γKlein(Knxn) = ┌½*n┐. More specifically, if n is even, then γKlein(Knxn) = ½*n and if n is odd, then γKlein(Knxn) = ½*(n + 1) [1, pp. 153-155]. In 2002, John Watkins and B. McVeigh provided a formula for the kings independence number on an mxn rectangular Klein bottle, as follows [1, p. 196]: └½*m┘*└½*n┘ if n even n*└(1/4)*m┘ – 1 if n odd, m ≡ 0 mod 4 n*└(1/4)*m┘ if n odd, m ≡ 1 mod 4 n*└(1/4)*m┘ + └½*n┘ if n odd, m ≡ 2 or 3 mod 4 ) = m xn As you can see, the above formula considers of four different cases for m and n. The formula simplifies a little bit (but maybe not as much as we would like) when considering only square nxn chessboards. Since it is impossible to have n odd and n ≡ 0 mod 4, the second case in the above formula disappears, reducing the formula for the kings independence number on an nxn square Klein bottle to the following: └½*n┘ 2 if n even n*└(1/4)*n┘ if n ≡ 1 mod 4 n*└(1/4)*n┘ + └½*n┘ if n ≡ 3 mod 4 nxn ) = Independent Domination Number The independent domination number for a given piece P and a given mxn chessboard is the minimum size of an independent dominating set, denoted i(Pmxn). This quantity need not equal γ(Pmxn) or β(Pmxn), as shown in the arrangements of queens placed on 4x4 chessboards in Image 21 [1, p. 198].
  20. 20. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar 20 Image 21: Queens Domination Number, Independent Domination Number and Independence Number on 4x4 Chessboard From Image 21, we see that γ(Q4x4) = 2, i(Q4x4) = 3 and β(Q4x4) = 4, and so in this case, the domination number, independent domination number and independence number are all distinct from one another. This strict ordering is somewhat rare. In fact, it has been conjectured by G. H. Fricke and others that i(Qnxn) = γ(Qnxn) for sufficiently large n. However, the following inequality does hold for all P, m and n: γ(Pmxn) ≤ i(Pmxn) ≤ β(Pmxn). The first comparison in the inequality is clear because the minimum independent dominating set is always at least as large as the minimum dominating set overall. Also, since we have already seen that a maximum independent set is also a dominating set, it then immediately follows that the minimum size of an independent dominating set is no larger than a maximum independent set, that is, i(Pmxn) ≤ β(Pmxn) [1, p. 198]. For rooks, bishops and kings, the independent domination number is always the same as the independence number. The reasons for these equalities are straightforward. Rooks that are placed along the main diagonal both are independent and dominate the board. Bishops that are placed down a central column also are independent while covering the board. Lastly, in our construction of minimum dominating sets of kings from my first paper in this series on mathematical chessboard puzzles, no two kings were adjacent to each other, hence the kings were independent [1, p. 199]. Upper Domination Number A dominating set, call it D, of chess pieces of type P is said to be a minimal dominating set if the removal of any of the pieces from the board makes it so that the remaining pieces fail to
  21. 21. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar dominate the board. This implies that no proper subset of D is a dominating set. The upper domination number is defined to be the maximum size of a minimal dominating set of pieces P, denoted as Γ(Pmxn). Since a maximum independent set is also a minimal dominating set, we can expand our chain of inequalities to the following: γ(Pmxn) ≤ i(Pmxn) ≤ β(Pmxn) ≤ Γ(Pmxn) [1, pp. 201-202]. 21 From Image 22, we see that γ(Q6x6) = 3, i(Q6x6) = 4, β(Q6x6) = 6 and Γ(Q6x6) = 7 [1, p. 201]. The minimal dominating set of 7 queens in Image 22 was found by Weakley, who showed that for n ≥ 5, it follows that Γ(Pnxn) ≥ 2n – 5. The largest value of n for which Γ(Qnxn) is known is 7 and Γ(Q7x7) = 9 [1, p. 202]. Image 22: Queens Domination Number, Independent Domination Number, Independence Number and Upper Domination Number on 6x6 Chessboard McRae produced a minimal dominating set of 37 kings on a 12x12 board; whether this is maximum or not has yet to be verified. However, this shows that β(K12x12) = 36 ≤ Γ(K12x12) [1, p. 202].
  22. 22. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar 22 For rooks, bishops and knights, it turns out that Γ = β. A minimal dominating set of rooks will have exactly one rook in each row or exactly one rook in each column; hence, Γ(Rnxn) = n [1, p. 202]. Thus, overall, we have that γ(Rnxn) = i(Rnxn) = β(Rnxn) = Γ(Rnxn) = n. A minimal dominating set of bishops can have no more than one bishop on each of the 2n – 1 positive diagonals and cannot have bishops occupying both the upper left-hand corner and the lower right-hand corner. Therefore, Γ(Bnxn) ≤ 2n – 2. Since Γ(Bnxn) ≥ β(Bnxn) = 2n – 2, it follows that Γ(Bnxn) = β(Bnxn) = 2n – 2. As for knights, the fact that Γ(Nnxn) = β(Nnxn) stems from a graph theory argument using bipartite graphs due to Cockayne, Favaron Payan and Thomason [1, p. 203]. Irredundance Number An irredundant set of chess pieces is one in which each piece in the set either occupies a square that is not covered by another piece or else it covers a square that no other piece covers. A maximal irredundant set is one that is not a proper subset of any irredundant set. The irredundance number for a given piece P and a given mxn chessboard is the minimum size of a maximal irredundant set, denoted by ir(Pmxn). Since a minimum dominating set doesn’t have any pieces that are redundant (otherwise, it wouldn’t be minimum) and must be maximal (otherwise, it wouldn’t be dominating), it follows that a minimum dominating set is a maximal irredundant set. Therefore, we can expand our chain of inequalities to the following: ir(Pmxn) ≤ γ(Pmxn) ≤ i(Pmxn) ≤ β(Pmxn) ≤ Γ(Pmxn) [1, pp. 204-205]. In order to illustrate the concept of irredundance, Image 23 contains two maximal irredundant sets of kings on 7x7 chessboards. Neither set is the proper subset of any irredundant set. However, the set of eight kings on the left fails to cover the entire board while the set of nine kings on the right is a minimum dominating set. Thus, ir(K7x7) ≤ 8 < γ(K7x7) = 9. This example shows that ir(Pmxn) ≠ γ(Pmxn) in general [1, p. 204].
  23. 23. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar 23 Image 23: Maximal Irredundant Sets of 8 Kings and 9 Kings on 7x7 Chessboard As one might expect, the irredundance number for both rooks and bishops on a square nxn chessboard is n. However, no formulas are available for kings, queens or knights [1, p. 205]. Upper Irredundance Number The upper irredundance number of a chess piece P on an mxn chessboard is the maximum size of an irredundant set of such pieces, denoted by IR(Pmxn). For example, the irredundant set of 16 kings in Image 24 is a maximum irredundant set because there exists no irredundant set of kings on a 7x7 board that is larger. Thus, IR(K7x7) = 16. Since a minimal dominating set is irredundant by definition, it then follows that such a set will be no larger than a maximum irredundant set. This gives us the inequality at the far right of our chain of inequalities: ir(Pmxn) ≤ γ(Pmxn) ≤ i(Pmxn) ≤ β(Pmxn) ≤ Γ(Pmxn) ≤ IR(Pmxn) [1, p. 205].
  24. 24. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar 24 Image 24: Maximum Irredundant Set of 16 Kings on 7x7 Chessboard Upper irredundance formulas are known for rooks, bishops and knights. For rooks, unlike the other five chessboard numbers we’ve studied thus far, the upper irredundance number is not simply n. Hedetniemi, Jacobson and Wallis have proved that for n ≥ 4, IR(Rnxn) = 2n – 4. A maximum irredundant set of 12 rooks is shown in Image 25. For bishops, Fricke has shown that for n ≥ 6, IR(Bnxn) = 4n – 14. A maximum irredundant set of 18 bishops is shown in Image 26. Lastly, the same theorem due to Cockayne, Favaron Payan and Thomason that states that Γ(Nnxn) = β(Nnxn) also applies to the upper irredundance number for knights. Thus, IR(Nnxn) = β(Nnxn) as well [1, p. 206]. Image 25: Maximum Irredundant Set of 12 Rooks on 8x8 Chessboard
  25. 25. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar 25 Image 26: Maximum Irredundant Set of 18 Bishops on 8x8 Chessboard Much less is known about the kings and queens upper irredundance numbers. For kings, Fricke has supplied a pair of upper and lower bounds: for n ≥ 1, (1/3)*(n – 1)2 ≤ IR(Knxn) ≤ (1/3)* n2. The irredundant set of 16 kings displayed in Image 24 along with the upper bound (1/3)*72 = 49/3 proves that IR(K7x7) = 16, as stated above. Even less is known about the upper irredundance numbers for queens. Only a handful of values are known, including IR(Q5x5) = 5, IR(Q6x6) = 7, IR(Q7x7) = 9 and IR(Q8x8) = 11. It is worth noting that in the latter three cases, the upper irredundance number is larger than the corresponding independence number (recall that β(Qnxn) = n for all n other than 2 or 3). A maximum irredundant set of 7 queens on a 6x6 chessboard is shown in Image 27 [1, pp. 206-207]. Image 27: Maximum Irredundant Set of 7 Bishops on 6x6 Chessboard
  26. 26. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar Total Domination Number 26 W.W. Rouse Ball introduced the concept of total domination in 1987. The total domination number for a given chess piece P on a given mxn chessboard, denoted γt(Pmxn), is the minimum number of such pieces that are required to attack every square on the board, including occupied ones [1, p. 207]. Since occupied squares must also be under attack, total domination is a more restrictive notion than domination. By virtue of this, we have that γ(Pmxn) ≤ γt(Pmxn). Also, γt(Pmxn) ≤ 2γ(Pmxn) since a minimum dominating set of pieces of size k will require at most an additional k pieces to cover the k squares that are occupied by the dominating set [3, p. 2]. Ball showed the total domination number on an 8x8 chessboard to be 5 for queens, 8 for rooks, 10 for bishops and 14 for knights [1, p. 207]. Totally dominating arrangements of these pieces on an 8x8 board are shown in Images 28 and 29 [1, p. 212]. Image 28: Totally Dominating Sets of 5 Queens and 8 Rooks on 8x8 Chessboard
  27. 27. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar 27 Image 29: Totally Dominating Sets of 10 Bishops and 14 Knights on 8x8 Chessboard In 1995, Garnick and Nieuwejaar gave the kings total domination number for the first 12 values of n, shown in Table 3 [3, p. 2]. It is worth noting that the kings total domination number matches the kings domination number for n = 1, 4 and 7, while the upper bound of 2γ(Knxn) is attained for n = 6. Table 3: Kings Total Domination Numbers for 1 ≤ n ≤ 12 n γt(Knxn) 1 1 2 2 3 2 4 4 5 5 6 8 7 9 8 12 9 15 10 18 11 21 12 24 While the kings total domination number is unknown beyond n = 12, Garnick and Nieuwejaar have provided upper bounds for 13 ≤ n ≤ 25, as shown in Table 4 [3, p. 2].
  28. 28. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar 28 Table 4: Upper Bounds for Kings Total Domination Numbers for 13 ≤ n ≤ 25 n Upper Bound for γt(Knxn) 13 29 14 33 15 38 16 43 17 48 18 54 19 60 20 68 21 72 22 80 23 87 24 95 25 102 Conclusion One can take away from this paper that the knight’s tour is a better understood concept on irregular surfaces than domination and independence. After all, thanks primarily to John Watkins, the knight’s tour existence problem has been solved on all the major surfaces such as the torus, cylinder, Klein bottle and Möbius strip. Conversely, so much is still unknown regarding domination and independence on these surfaces. However, as I pointed out in my previous paper on the knight’s tour, much remains to be desired regarding the counting of the number of different knight’s tours on different size chessboards. It would be interesting to see how many more permutations emerge, say on the torus, as compared to the regular chessboard. It would also be good to find out the number of different knight’s tours on irregular surfaces for chessboards of sizes that don’t work on the regular chessboard. For example, how many closed knight’s tours are there on a 5x5 torus (recall that closed tour knight’s tours do not exist when both dimensions are odd)? Since so little is known regarding knight’s tour combinatorics on the general chessboard (recall that the number of permutations of closed and open tours is known only for chessboards up to size 8x8), it didn’t
  29. 29. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar surprise me that I was unable to find any material in the literature regarding knight’s tour combinatorics on other surfaces. Furthermore, since so many questions remain regarding magic and semi-magic knight’s tours on the regular chessboard, it naturally follows that these same questions hold on the torus, cylinder, Klein bottle and Möbius strip. 29 Moreover, domination and independence on irregular surfaces remain an active area of research. Table 4 provides a snapshot of what is known and unknown regarding the domination number formulas for the various chess pieces on the torus. While formulas have been supplied for the rook, bishop and king on both square and rectangular chessboards, much informat ion is lacking regarding knights and queens domination on the torus. Likewise, formulas are known for the toroidal rooks, bishops and kings independence number but not entirely for the knights independence number and not at all for the queens independence number. I failed to mention this earlier in this paper, but the formula for the knights independence number on the torus is the same as that on the regular chessboard for even n ≥ 4 (that is, βtor(Nnxn) = ½*n2). However, due to the wrapping nature of the torus, the alternating black and white color scheme fails when n is odd, hence the odd formula on the regular chessboard does not work on the torus. In addition, as we have observed that very little is known about the n-queens problem on the cylinder, it goes to show that even less is known about the n-queens problem on the torus (due to the even greater freedom of a queen’s ability to move on the torus, thereby making independence more difficult). Table 5: Domination Number Formulas on the Torus by Piece Piece (P) γtor(Pnxn) (Square) γtor(Pmxn) (Rectangular) Rook n min(m, n) [2, p. 13] Bishop n gcd(m, n) [2, p. 13] max{┌(m / 3)*┌ n / 3 ┐┐, King ┌(n / 3)*┌ n / 3 ┐┐ ┌(n / 3)*┌m / 3 ┐┐} Knight Unknown, though values up to n = 8 are known Unknown Queen Unknown, though values up to n = 10 are known Unknown Beyond the torus, an active research task remains to find a formula for the queens independence number on the cylinder. Further insight into the n-queens problem on the cylinder might aid with uncovering new information about the n-queens problem on the torus, Klein
  30. 30. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar bottle and Möbius strip. While ultimately arriving at a formula for the queens independence number on any of these surfaces would appear to be a tall task, once a formula for one of the surfaces is discovered, finding formulas for the other surfaces should be markedly easier. Also, I have found that perhaps not enough attention has been paid to domination and independence in general on the Möbius strip. This fascinating surface should definitely be a major focus of future research in mathematical chessboard problems. 30 Other areas that would benefit from further research are the variants to domination and independence that we studied in the latter part of this paper. Table 5 provides a summary of the known and unknown formulas for the irredundance number, domination number, independent domination number, independence number, upper domination number and upper irredundance number for each of the five chess pieces that we’ve been studying. This table not only illustrates that all six formulas are known for the rook and bishop, but also the relatively simplistic nature of these two pieces, as many of the formulas are simply n (five for the rook and three for the bishop). Things start to get a little murky when one considers the king, for which only three of the six formulas are known. The same holds true for the knight, but one glaring hole here that isn’t present with the king is that arguably the most fundamental of the formulas, the domination number, remains unknown. Without question – and this comes as no surprise – the queen remains the biggest mystery, as only the independence number has a known formula. Greater computing power and further analysis of more general mxn rectangular boards will yield new insights into the mathematics associated with the king, knight and queen.
  31. 31. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar 31 Table 6: Chessboard Number Formulas by Piece Piece (P) ir(Pnxn) γ(Pnxn) i(Pnxn) β(Pnxn) Γ(Pnxn) IR(Pnxn) Rook n n n n n 2n – 4 for n ≥ 4 Bishop n n n 2n – 2 2n – 2 4n – 14 for n ≥ 6 King Unknown └(n + 2) / 3┘ 2 └(n + 2) / 3┘ 2 └½*(n + 1)┘ 2 Unknown Unknown Knight Unknown Unknown Unknown 4 if n = 2; ½*n2 if n ≥ 4, n even; ½*(n2 + 1) if n odd 4 if n = 2; ½*n2 if n ≥ 4, n even; ½*(n2 + 1) if n odd 4 if n = 2; ½*n2 if n ≥ 4, n even; ½*(n2 + 1) if n odd Queen Unknown Unknown Unknown 1 if n = 2; 2 if n = 3; n for all other n Unknown Unknown All in all, I hope that this series of four papers on the mathematics of chessboard problems has enabled the reader to appreciate just how fun and fascinating this topic really is. These papers have covered a lot of ground in examining seven different chessboard numbers (the six in Table 5 above plus the total domination number), four different surfaces (the torus, cylinder, Klein bottle and Möbius strip) in addition to the regular chessboard, and the knight’s tour problem and its relation with other mathematical concepts such as magic squares and Latin squares. As these are only a small fraction of all the problems that have been studied and that one could potentially study on the chessboard, needless to say, the possibilities for further research and analysis in this area are practically endless. Fortunately, recreational mathematicians won’t have to worry about running out of work to do for quite some time.
  32. 32. Dan Freeman Chessboard Puzzles: Other Surfacs and Variations MAT 9000 Graduate Math Seminar Sources Cited [1] J.J. Watkins. Across the Board: The Mathematics of Chessboard Problems. Princeton, New Jersey: Princeton University Press, 2004. [2] J. DeMaio, W.P. Faust. Domination on the mxn Toroidal Chessboard by Rooks and Bishops. Department of Mathematics and Statistics, Kennesaw State University. [3] J. DeMaio, A. Lightcap. King's Total Domination Number on the Square of Side n. Department of Mathematics and Statistics, Kennesaw State University. [4] “Chess.” Wikipedia, Wikimedia Foundation. http://en.wikipedia.org/wiki/Chess [5] “Joint International Meeting UMI – DMV, Perugia, 18-22 June 2007.” Dipartimento di Matematica e Informatica. http://www.dmi.unipg.it/JointMeetingUMI-DMV/events-ughi.htm [6] “Klein bottle.” Wikipedia, Wikimedia Foundation. http://en.wikipedia.org/wiki/Klein_bottle [7] "Imaging Maths - Inside the Klein Bottle." Plus Magazine. http://plus.maths.org/content/os/issue26/features/mathart/WhiteBlue [8] “Möbius strip.” Wikipedia, Wikimedia Foundation. http://en.wikipedia.org/wiki/M%C3%B6bius_strip 32

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