arXiv:2106.05353v1 [cs.GT] 9 Jun 2021 7--2143-/0$10 22 IEEE ©2020 978-1-7281-4533-4/20/$31.00 oFxGms[] o h ups fti ae,the paper, this of purpose the all simila For are [7]. eliminate games have Games to These Fox . to or the to corners, capture as com to to four encirclement encirclement trying are the the attacker’s attackers of the conversely the one while attackers, of reach defender out king the of the break goal The to variants. Tafl organizati is the the in attackers, differs by which surrounded of board the of middle vrl,teuprbudcmlxt o ovn h aeis game the solving g for the complexity for bound ones invalid around upper excluding account the and thoroughly states t on Overall, potential placed states Brandubh all been solving these for has strongly From effort of Great conditions. complexity derived. the state for final bound upper accepted genera a and positions, the set, piece of valid consider complexity will state-space g State-spaces overall to the used comple s are of for overall characteristics understanding tractable and its it make rules also should Brandubh’s which but solving. , players, of that two asym below the its well of for stu because goals chosen to varia metric was is Irish Brandubh associated paper an Brandubh. this of called games Tafl complexity of board state-space purpose combinatorial The strategy the Vikings. asymmetric the with of strongly family a Tafl, n (checkers). ae eems oua ni bu h 12 the about until popular most were games on,SaeSaeTransformation State-Space bound, hs stertoo icso ahsd.Teattackers The side. each on pieces of by defenders ratio the the outnumber is [4] players ’s for set rule histori balanced and [6]. their a places, [5], build both few to a for and in revision state recorded undergoing only continually were Th are [3]. games were 1700’s Tafl the they as to late though as rules [2], Lapland in chess played by being observed supplanted largely were they r eaieyukon hs ae r smerc though asymmetric, complexiti are state games These therefore unknown. relatively and be are perspective, will complexity Tafl or of [8] 1 Figure [5]. in considered shown variation Brandubh nUprBudo h tt-pc opeiyof Complexity State-Space the on Bound Upper An × a eest aiyo smercgmspae nan on played games asymmetric of family a to refers Tafl Terms Index Abstract htmksTflgmsaymti n ifrn from different and asymmetric games Tafl makes What h a aiyo ae sntwl tde rmatheoretic a from studied well not is games of family Tafl The n qaecekrdbad iia oces[] hs Tafl These [1]. chess to similar board, checkered square 10 14 Bfr hs aet otenErp hr was there Europe Northern to came chess —Before ira Compy Kiernan 1 tts ewe hto onc oraddraughts and four connect of that between states, colo lcrclEgneigadCmue cec,Was Science, Computer and Engineering Electrical of School Aymti,SaeSae togSle Upper Solve, Strong State-Space, —Asymmetric, .I I. 2 colo niern ple cec,GnaaUniversit Gonzaga Science, Applied & Engineering of School orsodn mi:hne.culuhwueu cranda [email protected], Email: Corresponding NTRODUCTION 1 1 : 2 ln Evey Alana , h eedr ei nthe in begin defenders The . 1 utrMcCullough Hunter , th etr when century Brandubh ie rule lized inof tion i an ain 7 game. trong plete ame. × xity ing cal on dy he es is 7 e s r - , oa opeiywl lob otlkl oalwfrfuture for Brandubh. allow solving to strongly likely Th or most weakly variations. be the to other also work as of will well understanding complexity as the total bound a expand for upper allow to the and will ability of set, simplicity calculation rule The accurate board. simplest more smallest the the has on it played as analysis state-space change. rules for as rule efficiently proposed and agents allowing quickly evaluating by trained balanced to be more moves to games the games make Tafl to changes on computer work in import especially time as becomes training This methods. reduce symmetri these should using The agents moves [11]. [10], the defender of perfor and nature to attacker able of be roles role-base should both rules, agents movement training Given piece and player. the symmetrization each of for nature symmetric play the game [9 advantageous decomposition determine symmetric using to analyzable be should they att center. are the Black in layout. King piece the initial with board defenders, Brandubh 1. Figure etA gnso ayn ifiute.W oet eventually to hope We difficulties. varying of agents AI test rnuhwscoe rmaogteohrTflvariants Tafl other the among from chosen was Brandubh u oiainfrsrnl ovn rnuhi obidand build to is Brandubh solving strongly for motivation Our 1 ida Allen Lindsay , igo tt nvriy ula,WA Pullman, University, State hington [email protected] I M II. 1 ,Soae WA Spokane, y, n ao .Crandall S. Aaron and , OTIVATION ces ht are white ackers, 2 ant m is d e c ] extend the agents to include other variations of Tafl board games as well. The importance of exploring asymmetrical analysis is ap- parent when considering its widespread applications. Not only is it relevant in developing game theory, but it is also proving useful for medical diagnosis based on neuro-imaging techniques such as MRI and CT scan results [12]. Figure 2. Game piece representations, in order from left to right: King, The work on Tafl games could be valuable when addressing defender, attacker. asymmetric game theory analysis. The most notable concern is in relation to cyber security. The U.S. Military has been creating AIs for weapon systems using asymmetry, which has further tempted the development of adversarial AIs related to it. These are intended to influence false predictions in their machine learning models that will result in unexpected, and possibly lethal behavior [13]. Exploring not only the total complexity of Tafl games, but continued mechanisms of how to build functional AI agents to compete in adversarial asymmetric situations can contribute to security applications.

III. RELATED WORKS This work focuses on estimating the total number of game states possible in a game of Brandubh. The most famous similar work was by Shannon [14] in estimating the number of games available in chess. In Shannon’s work, he looked at the number of possible moves, the length of average games, and any rules that would limit the state-space and Figure 3. Brandubh Initial Game Board Layout came to an initial number of 10120 games. That number has been reduced through eliminating invalid moves, position transposition, handling ties, and focusing on sensible moves. columns. It has a standardized setup that includes a king, four 40 This has brought the state-space to around 10 . Other games defenders, and eight attackers. The king always starts in the have similar analyses performed, such as Connect 4 with about very center of the board at position 4 × 4. This square of the 13 15 2 positions [15], domineering with 2 [16] positions, and game board will henceforth be referred to as the throne. The 20 draughts (checkers) at 10 positions [17]. This paper presents four defenders start the game adjacent to the king, and the a similar analysis for Brandubh, and by extension, the other 8 attackers fill the remaining spaces in row 4 and column Tafl games. 4. The king will be represented by a heart, the defenders There are few papers addressing the state-space analysis of by a diamond, and the attackers by a four point star. Each Tafl, or Brandubh specifically. One such analysis was done by representation is pictured in Figure 2. Slater, who roughly estimates Hnefatafl (played on an 11 × 11 board) at 1043 [18]. Slater’s work did not account for move B. Board Layout transpositions and corner states included in this Brandubh All the sub-cases for Brandubh can be described with the analysis. colored tiles used in Figure 4. Each colored tile represents A more complete complexity analysis of a similar game events in which the king is standing on that tile type. OT was done by Galassi, which addressed Tabult [19]. Tabult is (Red Tile) is abbreviated for On Throne; situations in which a Tafl game played on a board larger than Brandubh, with a the king is on the throne. ATT (Teal Tiles) is abbreviated 9×9 tile board. Galassi divided the move space into subspaces for Adjacent To Throne; situations in which the king is and handled more of the possible state space reductions than adjacent to the throne. OA is abbreviated for Open Area; Slater’s work. Galassi’s conclusion is that Tabult’s complexity situations in which the king is not adjacent to an edge, or is roughly the same as Draughts (checkers). The authors of horizontally/vertically adjacent to the center. This section is this work on Brandubh used some of Galassi’s approaches for divided up into 3 separate sub-cases: DOA (Purple Tiles) is handling state space reductions in Brandubh. abbreviated for Diagonal Open Area, COA (Orange Tiles) is

IV. BRANDUBH RULES USED A. Setup OT ATT DOA COA ADA OEC ENA ATC COR For all Tafl variations, the game board is a perfect square. The Brandubh game board, shown in Figure 3, can be thought of as a two dimensional array consisting of 7 rows and 7 Figure 4. Color Key for Game Board States abbreviated for Center Open Area, and ADA (Yellow Tiles) is abbreviated for Adjacent to Diagonal Open Area. OEC (Pink Tiles) is abbreviate for On Edge Center; situations in which the king is in the center of the edge of the board. ENA (Green Tiles) is abbreviated for Edge Non Adjacent; situations in which the king is on the edge of the board, not a center edge piece, and not adjacent to the corner tile. ATC Figure 6. Adjacent to Throne King Capture (Brown Tiles) is abbreviated for Adjacent To Corner; situation in which the king is adjacent to a corner tile. COR (Black Tiles) is abbreviated for Corner; situations in which the king is in the corner tile. Figure 7. Hostile Throne C. Movement

In Brandubh, movement is standardized. Every piece may 4) Off Throne, Off Edge: When any piece is not adjacent move any number of squares vertically or horizontally across to or on the throne and it is not on an edge or corner, it is in the board. Piece jumping is not allowed, and two pieces may an open area. In order for a capture to happen, the side trying not occupy the same space. The king may not return to the to capture must move into a position such that the piece being throne after leaving it, and both attackers and defenders are captured is between two capturing pieces occupying the same never allowed to rest on the throne. The rules of Brandubh row or the same column. state any piece may move through the throne at any time. However, to aid in the simplicity of calculations this was considered an illegal move.

D. Capture Figure 8. Open Area Capture A capture is defined as any move which results in the removal of a piece from the board. Across all Tafl games, 5) Hostile Corners: A corner piece may be considered captures primarily involve trapping a piece between two hos- hostile by both attackers and defenders. tiles, though there are game specific variations. Pieces may only be captured by hostiles which are adjacent to them, and diagonal captures are never allowed. In Brandubh, the edges of the board may not be considered hostile in captures. It should Figure 9. Hostile Corner Capture also be noted the side which is making the capture must be the side to initiate the move. The Brandubh variation allows for the king to participate in captures, slightly skewing the 2:1 E. Win Scenarios ratio. Brandubh also allows for the king to be captured like In Brandubh, and all Tafl games, there are three win any other piece, provided it is not resting on the throne or an conditions: adjacent square. If the king is on or adjacent to the throne, the 1) Attackers win if the king is captured king is only captured if surrounded by hostile squares. Each 2) Defenders win if the king reaches a corner square valid capture scenario has been considered and is explained in 3) Defenders win if all attackers are captured the following sections. Additionally, if the same board state is repeated three times 1) On Throne: When the king is on the throne, four attacker in a row, the game is a . pieces are required for capture. V. CALCULATIONS A. Na¨ıve State-space Complexity A na¨ıve approach to calculating the state-space complex- ity of Brandubh would be to consider every possible state stemming from the initial state. We call the rough estimation UBnaive, which is described as: 5 44 Figure 5. On Throne Capture UBnaive =2 × 4 (1) ≈ 9.91 × 1027 2) Adjacent to Throne: When the king is in a cell adjacent to the throne, three opposing pieces are required for capture. The purpose of the calculations within this paper are to cal- 3) Hostile Throne: When the throne is not occupied by the culate the most accurate upper bound possible by considering King, the throne is considered hostile, and therefore can be as few illegal moves as possible. The simplifications made to used to help capture pieces by either attackers or defenders. UBnaive are explained in the proceeding sections. B. Rotations and Mirroring D. Upper Bound of Non-End Sates

The game board used by Brandubh is a perfect square, and D is populated by the values of ar and dr. Each column of as such can be divided into four equally sized quadrants. This the matrix represents the number of defenders being consid- equality can be used to simplify UBnaive into what we will ered from 0 to 4, and each column the number of attackers in refer to as UBtight, by treating mirrors or rotations of each the same range. To make the process used for all non-end state quadrant as one state. That is, if a piece occupies the square calculations clear, the process for the first case is explicitly 3 × 2; through mirroring it is the same as occupying square shown. However, the function specified in (4) will be used to 3 × 6 or 5 × 2, or through rotation it is the same as occupying represent the remainder of the cases. square 5 × 6. • On Throne (OT) OT is the case where the king has not left its initial C. Definitions position. The throne is shown as the red tile in Figure 3. In order to simplify the calculation of states, the board has The calculations for all states given this assertion are as been broken down into sections. The sections to be considered follows: are the throne, the corners, the cell the king occupies, and the 11211 cells adjacent to the king. The various types of game pieces 1 2 2 1 will be referenced by the following variables: D = 1 2 1  A = [1, 1, 2, 1, 0] (5) • Attackers: a 1 1    • Defenders: d 0    The variable k will represent all the tiles on the board minus   the four corners, the throne, the tile the king is on, and the Let i be the count of attackers adjacent to the king. king’s adjacent tiles. The following equation will be used to Due to the rule of capturing a king in the center tile calculate the set of possible state variations, given the variables requiring 4 attackers surrounding him, i has a maximum a, d, and k. value of 3. Ci will denote the possible cases where the king is surrounded. In all sub-cases where the king is on the throne and not captured, n = 4 and k = 40. The k! P (a,d) = (2) components of on throne are: k a!d!(k − a − d)! We will use a single variable Kronecker delta, δ , to denote 8 4 8 3 n (a,d) (a,d) if there is at least one attacker adjacent to the King. The C0 = D0,0 P40 + D0,1 P40 (6) a=1 d a=1 d necessity of δn becomes clear when accounting for capture =0 =0 X8 X2 X8 X1 states. The function is as follows: (a,d) (a,d) + D0,2 P40 + D0,3 P40 1 if n =0 a=1 d=0 a=1 d=0 δn = δ0,n = (3) X8 X X X (0 if n =06 (a,d) + D0,4 P40 Next, let ar be the number of hostiles adjacent to the a=1 X 12 king and dr the number of defenders. Note that ar and dr ≈ 4.99 × 10 can individually range from 0 to n, where n is the total number of open spaces adjacent to the king. Similarly ar + dr can range from 0 to n. Now, consider the case where the 7 4 7 3 (a,d) (a,d) king is surrounded, ar = n. Let A be a single row vector C1 = D1,0 P40 + D1,1 P40 (7) where A represents the number of unique arrangements a=0 d=0 a=0 d=0 n X X X X that make this possible. A single row vector is needed for 7 2 7 1 (a,d) (a,d) simplifying equation (4). With ar surrounding attackers and + D1,2 P40 + D1,3 P40 a=0 d=0 a=0 d=0 dr total defenders. Let D be a 2 dimensional vector where X X X X ≈ 1.44 × 1012 (8) Dar,dr represents the number of unique arrangements of the surrounding attackers and defenders. Now, given n spaces around the king and k spaces non-adjacent to the king that are occupiable by either attackers or defenders, let f(n,k,A,D) 6 4 6 3 (a,d) (a,d) calculate the number of unique states that can arise: C2 = D2,0 P40 + D2,1 P40 (9) a=0 d=0 a=0 d=0 X6 X2 X X n−1 n−ar 8−ar 4−dr (a,d) (a,d) + D2,2 P40 f(n,k,A,D)= Aa Da ,d P r r r k a=0 d=0 a =0 d a δ d X X Xr Xr=0 X= ar X=0 ≈ 3.14 × 1011 (10) (4) COA is represented by the orange tiles in Figure 3 and 5 4 5 3 can be calculated with the following summations: (a,d) (a,d) C3 = D3,0 P40 + D3,1 P40 a=0 d a=0 d COA = f(4, 39, ACOA,D) (18) X X=0 X X=0 (11) ≈ 9.63 × 1012 ≈ 5.16 × 1010 (12) Using the above calculations gives:

OT = A0C0 + A1C1 + A2C2 + A3C3 (13) OA = DOA + ADA + COA (19) 12 ≈ 7.11 × 10 ≈ 3.78 × 1013 • Adjacent To Throne (ATT) • On Edge Not Adjacent to Corner (OE) ATT refers to all states where the king is adjacent to the OE defines the state where the king is along the edge of throne, and is not being captured. These positions are the board, but not adjacent to the corners or in a captured represented as blue tiles in Figure 3. The calculations for state. This section is broken up into 2 subsections to take all possible states for ATT are as follows: advantage of mirroring along the center-line. These will 1 33 1 be called On Edge Not Adjacent to Corner Center 1 2 1 (OEC) and On Edge Not Adjacent to Corner Not D = A = [1, 2, 2, 1] (14) 1 1  Center (ENA). Each of these sub-cases share the same 1  matrix D with differing A vectors:     1 3 3 1 AT T = f(4, 40,A,D) (15) 1 2 1 AOEC = [1, 2, 2, 1] 12 D =   ≈ 9.63 × 10 1 1 AENA = [1, 3, 3, 1] 1    • Open Area (OA)   OA refers to all states where the king is not along the OEC is represented by the pink tiles in Figure 3. The edge, adjacent to the corner, adjacent to the throne, and calculations are: is not being captured. The calculations for OA have been broken up into 3 unique parts; Diagonal Open Area OEC = f(3, 40, AOEC,D) (20) (DOA), Adjacent to Diagonal Open Area (ADA) and ≈ 9.63 × 1013 Center Open Area (COA). The breakdown into three sections reduces redundancy by taking advantage of the ENA is represented by the army green tiles in Figure 3 symmetry of the game board, and hence makes the overall and can be calculated by: upper bound calculation more accurate. The three state space calculations will be summed together and referred ENA = f(3, 40, AENA,D) (21) to as OA for the remainder of the paper. As seen below ≈ 1.14 × 1013 each of these sub-cases share the same D matrix with different A vectors. Using the calculations above gives: 14641 1 33 1 ADOA = [1, 2, 4, 2, 0] OE = OEC + ENA (22)   13 D = 1 2 1 AADA = [1, 4, 6, 4, 0] ≈ 2.10 × 10 1 1    ACOA = [1, 3, 4, 3, 0] 0  • Adjacent To Corner (ATC)     ATC refers to when the king is adjacent to a corner, but DOA is represented by the purple tiles in Figure 3 and not in a captive state. ATC is shown as a brown tile in can be calculated with the following summations: Figure 3, and its calculation is as follows:

DOA =2 × f(4, 39, ADOA,D) (16) ≈ 1.68 × 1013 1 2 1 D = 1 1 A = [1, 2, 1] ADA is represented by the yellow tiles in Figure 3 and 1  can be calculated with the following summations:  

ADA = f(4, 39, AADA,D) (17) AT C = f(3, 41,A,D) (23) 13 ≈ 1.14 × 10 ≈ 1.14 × 1013 E. Upper Bound of End States Final attacker being captured with one defender or In addition to all possible positions of the king where the the king, and a hostile square (CE): game does not end, those which lead to a win condition CE = x(2, 4, 42) (30) must be considered. Unlike in the non-end state calculations, matrices do not simplify calculations for end state conditions, = 1241244 and therefore will not be used or shown. End state condi- tions are more complex than non-end state conditions and as Final attacker being captured with two defenders or a such require more than one function to represent them. The defender and the king (CNE): following will be used to reduce the redundancy CNE = x(11, 4, 41) (31) which would otherwise be included in to the Upper Bound End = 506495 States. The variable dr is still used to represent the number of defenders adjacent to the king, and u represents how many Using the calculations above gives: defenders could be adjacent to the king, with a maximum value of 4. NAL = CE + CNE (32) = 1747739 u u! = (24) d d !(u − d )! • King Surrounded by Two Pieces (KS)  r r r KS refers to the scenario where the king is captured The variables y and t will be used to represent the maxi- by two attackers on either of its sides as depicted in mum number of attacker and defender pieces which are not Figure 8. This case is completely different from the others adjacent to the king. within this section, as it takes place in the open area, and therefore has a unique calculation as shown below: y t (a,d) −d g(y,t,k)= Pk (25) 2 6 4 r 2 (a,d) a=0 d=0 KS =4 P39 X X dr d a=0 d Variable j represents possible transformations:  Xr=0   X X=0 1 5 4−dr 1 (a,d) − − + P39 t t 1 t 1 dr (0,d) (0,d) (0,d) dr=0   a=0 d=0 x(j, t, k)= j P + P + k P (26) X X X k k k−1 4 4 "d=0 d=0 d=0 # (a,d) X X X + P39 (33) a=0 d=0 8−q −d  1 4 r X X 12 1 (a,d) ≈ 2.59 × 10 h(q, k)= Pk dr d a δ d  r=0   = q =0 • King Captured On Edge (KCE) −X− X X 8 q 1 4 KCE finds all situations in which the king is captured on P (a,d) + k (27) the edge of the board, but not in the corner or adjacent a=0 d X X=0  to corner. • King Surrounded On Throne (KOT) This case covers all possible combinations of valid cap- KCE =2 × h(2, 40) ture states while the King is on the throne. ≈ 6.53 × 1011 (34) KOT g , , = (4 4 40) • King Adjacent to Corner (KAC) ≈ 6.91 × 109 (28) KAC captures every instance of the king being captured while adjacent to the corner tile, as depicted in Figure 9. • King Surrounded Adjacent To Throne (KAT) This case covers all cases where the king is captured KAC = h(1, 41) while adjacent to the throne. ≈ 2.03 × 1012 (35) KAT =4 × g(5, 4, 40) • King Corner (KC) ≈ 4.61 × 1010 (29) KC is the win condition in which the king enters any 4 • No Attackers Left (NAL) of the board corners. This is represented as a black tile NAL is the win scenario where all attackers have and the following math: been captured. For simplification, this condition has been split into the possible scenarios leading to this state. KC = h(0, 42) ≈ 1.14 × 1013 (36) F. Report Findings follow a similar process to the Brandubh analysis presented here. UBNE is the summation of all potential game states which do not result in the end of the game. The summation is as Based on this work showing Brandubh’s tractability, the follows: authors are working towards a weak and strong solving of the game to establish the rules’ gameplay balance. This work UBNE = OT + AT T + OA + OE + AT C should also provide resources to improve AI agent develop- ≈ 8.68 × 1013 (37) ment for the Tafl family of games.

UBE is the summation all possible end states, and is as VIII. VERSION AMENDMENT follows: In a previous version of this work published in the 2020 IEEE Conference on Games (CoG) [21], the related works UB = KOT + KAT + CE E citation of Andrea Galassi’s work done on Tablut was acci- + CNE + KS + KCE dentally left out during the final stages of the drafting process. + KAC + KC (38) The authors apologize to Galassi for this oversight. 13 ≈ 1.67 × 10 REFERENCES

UBtight will be calculated by the addition of all possible [1] T. Harding, “’A fenian pastime’? Early Irish board games and their states the board can exist in. identification with chess,” Irish Historical Studies, vol. 37, no. 145, pp. 1–22, 2010. [2] H. J. R. Murray, A history of board-games other than chess. Clarendon UBtight = UBNE + UBE (39) press, 1952. ≈ 1.04 × 1014 [3] C. Linnaeus, Lachesis lapponica, or a tour in Lapland. White and Cochrane, 1811. 27 [4] J. C. Ashton, “Linnaeus’s game of tablut and its relationship to the UBnaive was found to be approximately 9.91 × 10 as ancient viking game hnefatafl,” The Heroic Age: A Journal of Early shown in equation (1). Our calculations for UBtight came to Medieval Northwestern Europe, vol. 13, pp. 1526–1867, 2010. be approximately 1.04 × 1014. The vast difference between [5] D. Walker, Reconstructing Hnefatafl. Lulu Com, 2014. [6] N. Davidson, “Hnefatafl: an experimental reconstruction.” Online, 2011. UBnaive and UBtight highlights the effects of redundancy on Accessed: 2020.05.24. the number of states. [7] Pentagram, Pentagames. Fireside, Simon & Schuster Inc, 1990. [8] L. Dantas, “Brandubh board.” Online, Jan. 2007. Released to public VI. CONCLUSION domain. [9] K. Tuyls, J. P´erolat, M. Lanctot, G. Ostrovski, R. Savani, J. Z. Leibo, The summations presented within this paper represent a T. Ord, T. Graepel, and S. Legg, “Symmetric decomposition of asym- careful estimation of upper bound state-space calculations for metric games,” Scientific Reports, vol. 8, p. 1015, Jan. 2018. the Brandubh variant among the family of Tafl games. A com- [10] R. Cressman and Y. Tao, “The replicator equation and other game 27 dynamics,” Proceedings of the National Academy of Sciences, vol. 111, binatorial upper bound using a na¨ıve approach of 9.91 × 10 no. Supplement 3, pp. 10810–10817, 2014. was initially found. After applying state-space reductions when [11] R. Cressman, C. Ansell, and K. Binmore, Evolutionary dynamics and accounting for mirrored states, board edges, special king states, extensive form games, vol. 5. MIT Press, 2003. [12] P. Kalavathi, M. Senthamilselvi, and V. B. S. Prasath, “Review of and edge cases by applying the game’s rules, a tighter bound computational methods on brain symmetric and asymmetric analysis of approximately 1.04 × 1014 was achieved. from neuroimaging techniques,” Technologies, vol. 5, no. 2, p. 16, 2017. From the results shown, it can be determined the com- [13] P. Feldman, A. Dant, and A. Massey, “Integrating artificial intelligence into weapon systems,” 2019. plexity is roughly between that of connect four [20] and [14] C. E. Shannon, “Programming a computer for playing chess,” The 8x8 domineering [16]. As both connect four and domineering London, Edinburgh, and Dublin Philosophical Magazine and Journal are considered solved games, the conclusion can be drawn of Science, vol. 41, no. 314, pp. 256–275, 1950. [15] S. Edelkamp, “Bdds for minimal perfect hashing: Merging two state- that Brandubh is tractable for both weak and strong solving. space compression techniques.” Online, 2017. Checkers was solved for 1020 positions using hundreds of [16] L. V. Allis et al., Searching for solutions in games and artificial computers over two decades. Newer computers, faster storage, intelligence. Ponsen & Looijen Wageningen, 1994. [17] J. Schaeffer, N. Burch, Y. Bj¨ornsson, A. Kishimoto, M. M¨uller, R. Lake, and being six orders of magnitude smaller than checkers P. Lu, and S. Sutphen, “Checkers is solved,” Science, vol. 317, no. 5844, should allow Brandubh to be solvable on a much quicker pp. 1518–1522, 2007. timeline. [18] J. Slater, “On tafl: state space complexity.” Online, 2015. Accessed: 2020.05.24. VII. FUTURE WORK [19] A. Galassi, “An upper bound on the complexity of tablut,” 2021. [20] H. [van den Herik], J. W. Uiterwijk, and J. [van Rijswijck], “Games This initial state-space evaluation should allow for updates solved: Now and in the future,” Artificial Intelligence, vol. 134, no. 1, and variants of the game to be evaluated more quickly. All Tafl pp. 277–311, 2002. [21] K. Compy, A. Evey, H. McCullough, L. Allen, and A. S. Crandall, “An games are being changed and updated to find game balance, upper bound on the state-space complexity of brandubh,” in 2020 IEEE so this state-space is expected to change in the future. Conference on Games (CoG), pp. 519–525, IEEE, 2020. These techniques and approaches to address various game state cases should be applicable to other Tafl games. Similar games such as Hnefatafl, Tablut, Ard Ri, Tawlbwrdd, and Alea Evangelii are all open to continued analysis, which should