Reviews


A REVIEW OF: MODULAR GAMES by Eugene William Madison, published by DORRANCE PUBLISHING CO., INC.,

By Alvin W. Logan

 

In his book, the author introduces the term modular games to apply to a particular class of two-player, counting games. Any such game is initiated by fixing natural numbers, say L and M (with 1 < M < L/2 < 50). After PLAYER I is designated, he or she opens by starting with the number 1 and counting up to at most M of the numbers less than L. Then PLAYER II continues the count by enumerating M or fewer numbers. Continuing, the players alternate their moves in this way until a player’s move includes the number L. Of course, this player is declared the winner.

 

The author argues that for any modular game, one of the players has a winning strategy. And, for such games, not surprisingly, PLAYER I will possess a winning strategy a lot more often than PLAYER II; but, as he asserts on page 10, the knowledge of this fact on the part of the players can negate their competitive spirits. Reflecting on this unfortunate fact, the author introduces various enhancements of modular games for the purpose of ameliorating this dilemma… This leads to various game spin-offs from the modular game concept that can be played on computers (or slot machines), cell phones, or as card games or board games.

 

In this review, we will focus on the board-game that the author calls RASCAL’S TRIANGLE. Our reason for this is three-fold: 1. The author devotes approximately one-third of the book to this new board-game; indeed, he describes three different versions of it. 2. A picture of the board for this game appears (unlabeled) on the cover of the book. 3. The progressive version of RASCAL’S TRIANGLE is an interesting and exciting new board-game, we feel, that can be played and enjoyed by people of all age groups and all cultures. Undoubtedly, its popularity will grow exponentially as the publicity is given time to spread.

 

RASCAL’S TRIANGLE BOARD-GAME (the progressive version). This game involves two players, alternately moving their ICONS along a triangular pattern of squares while governed by a fixed set of rules. The players are following the path of a “spiral” to its end and then reversing their direction back to its starting point. Now, we will briefly describe the game board, the rules for playing the game and finally, how the game arises from the concept of modular game.

 

The Board: As the author explains, the board exhibits a triangular array of ten rows of squares which are symmetric about a hidden vertical axis. The first row contains 10 squares and each successive row contains one less square than the previous one. Thus, a total of 55 squares are exhibited. The squares are numbered in a clockwise fashion along something of a spiral with the upper leftmost square designated as SQUARE#1 and the last “middle” square as SQUARE#55. This last square is designated as PREFERRED (or RED). Now, working backwards, every 7th square is labeled RED.

The GREEN ZONE (relevant mainly for the clockwise direction of player’s moves) consists of the 6 squares between the two RED squares #48 and #55, while the BLUE ZONE (relevant mainly for the counterclockwise direction of the player’s moves) consists of the 6 squares numbered 1 through 7, excluding the RED SQUARE#6. See page 23 in Modular Games.

 

Play of the Game: In general terms, the players alternate moves of their ICONS around the spiral and back, each basing his or her moves mainly on tosses of the two dice. The player whose ICON is first to return to SQUARE#1 is declared the winner. To begin, each player tosses a die; the larger value determines PLAYER I. Of course, ties are retossed, immediately. Prior to each move, the “at bat” player gets up to 3 tosses to determine the “length” of the upcoming move. No move has length exceeding 7 squares. If all 3 tosses are larger than 7 (i.e. “too big”, or unacceptable) then the move has length one (square). Otherwise, the “at bat” player fixes the length of the move with an acceptable value of one of the possibly 3 tosses. When a move by a player lands (or ends) on a RED square, this player immediately becomes the “at bat” player for another move (thereby getting two consecutive moves). For this move - since it emanates from a Red square - the "at bat" player may determine it by tossing the dice, as before, or simply advancing the Icon x squares (this players choice of x < 7). When a player’s ICON is in the BLUE or GREEN ZONE, the expression “too big” for a value registered by a toss of the dice takes on a generalized meaning. This is to say, whether or not the value 7 or a smaller value is “too big” depends on the square that the ICON occupies and the direction that it is headed. For example, if a player’s ICON is either in the BLUE ZONE headed in the direction of decreasing numbered squares or in the GREEN ZONE headed in the direction of increasing numbered squares, the value of 7 is “too big” because in either case there are  fewer than 7 squares left in the given direction. Similarly, a value x smaller than 7 is “too big” when there are fewer than x squares left in the direction that the ICON is headed. When Player’s A ICON is in the GREEN ZONE, headed in the direction of SQUARE#55, he or she could be forced to take a  number of one-square moves (because three consecutives tosses registering “too big” always yield a 1-square move) thereby giving the opponent a chance to catch-up or get further ahead.  Of course, if Player A’s ICON re-enters the BLUE ZONE (leading the opponent), his or her adrenalin may begin to elevate because of increased expectations for a win.  However, just as with the GREEN ZONE, Player A may be forced to take several one-square moves which could spoil a potential win.

 

 

Backward Moves: A backward move for say, Player A, is a move in the direction opposite to the direction that the Icon of Player A was headed prior to this move.  The exception is when the ICON lands on Square#55; here, the only direction possible is considered forward. A player may take a one-square backward move (without involving a toss of the dice) whenever his or her ICON is not in one of the zones. This could be done for the purpose of (i) landing on a RED square (which yields another forward move, to more than make up for the 1-square move in the opposite direction) or (ii) forcing the opponent into a penalizing 3-square backward move. Number (ii) occurs whenever the ICON of Player A lands on a square that shares at least one vertical border with the square containing the ICON of the opponent, say Player B. Of course, this case includes the possibility that the ICONS briefly share a square. Our assumptions here, is that the ICON of Player B is not in a zone; i.e. a player whose ICON is in one of the zones is protected from such a penalty move. Also, we assume in (ii) that the backward move of Player B does not lead to back and forth stalling moves between the players.  This is an issue only when the 3-square backward move lands Player B’s ICON on a RED square. Again, we remind the readers that ICONS in either zone are protected from such a penalizing move. This penalizing 3-square move appears natural for, say Player B, whose ICON losses ground (providing that it is not in either zone) anytime that ICON of Player A lands on a square occupied by his or her ICON (when not in a zone). However, this rule, mentioned above, applies more generally,  anytime the ICON of Player A lands on a  square which shares a vertical border with the square occupied by the ICON of Player B (when not in a zone).

 

 Obviously, the aforementioned rule makes good sense for our spiral (numbered) triangular board of squares; however, formulating such a general rule for an arbitrary spiral of squares for some other game does not seem possible or desirable.  In any case, this seems to answer questions concerning our choice of a triangular pattern of square numbered spirally for the RASCAL’S TRIANGLE GAME BOARD.

 

 Now the question is:  how did the RASCAL’S TRIANGLE board-game arise from the modular game concept? The answer is the following: Consider the modular game based on L = 55 & M = 6.  Using the terminology of Modular Games, the matrix below is the analyzing matrix for this game:

 

                                                                                                                       A

1

2

3

4

5

6

7

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9

10

11

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48

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50

51

52

53

54

55

56

 

 

The sixth column of this matrix determines the winning strategy (possessed by PLAYER I) for this modular game. However, for the RASCAL’S TRIANGLE board-game, this column identifies the RED squares. Of course, neither player has a winning strategy in the board-game because of the randomness introduced by basing decisions on the tossing of dice. If one can imagine a scenario where PLAYER I opens by tossing a 6 followed by 14 consecutive 7’s, the result would be a win for PLAYER I with the ICON of PLAYER II left on the X-SQUARE. Fortunately, this scenario, although possible, is highly improbable.  The author gives a short section in the book on probability to help the players formulate their strategy for playing the game.

 

 

REFERENCES

 

E.W. Madison, MODULAR GAMES, Dorrance Publishing Co., Inc.

 

 

The MOD-SLOT (Computer Game) [Revised Jan’ 2012}

Eugene William Madison

 

 

This idea was first introduced in my monograph called MODULAR GAMES, published by Dorrance Publishing Co., Inc. in Sept. 2010. Listed as an exercise called Mod-Slot, this game is a spin-off from the general concept of modular game. Any play of it involves a human player playing against a slot machine (i.e., a computer). The computer randomly selects itself as either Player I (with the human opponent as Player II) or as Player II (with the opponent as Player I). Not surprisingly, a winning strategy exists for Player I a lot more often than for Player II. The computer is programmed so that when the human (i.e. the operator) punches in the necessary preliminary information, its internal calculations allow it to “know” immediately, for which of the two players there is a winning strategy. When the computer has the winning strategy, it will utilize it to score a win. In the Mod-Slot setting – for the human player – the possession of a winning strategy is unbeknownst by him or her. So, the computer will exploit any mistake by the operator & turn it into a win for itself.

 

As with any modular game, Mod-Slot involves counting. In any play of the game the human player (i.e., the operator) and his or her opponent (i.e. the computer) are key-punch counting up to and including a number, say L (to be chosen by the operator, before the counting begins). The counting is done by the players (operator and computer)  alternatively key-punching segments of consecutive natural numbers of length less  than or equal  to a fixed number, say M, with 2 ≤ M  ≤ 10  (where M is picked, at random, by the computer before the counting begins).  The alternating of moves is strict in the sense that no passes are allowed.  Depending on the operator’s objective, initially stated (i.e., either (a) to land on the #L or (b) to miss the #L), the winner is declared as the player (i.e., Player I or Player II) that accomplishes the objective.

 

To describe the external features of the machine, we mention that across its top, there are four buttons: (1) the ON/OFF button, that the operator pushes to begin a play of the game, (2) the L = ? button, that the operator uses to select a value of L, (3) the M = ? button, which is activated by the computer for randomly selecting M and (4) the Objective button, used by the operator to bet on (a) or (b), i.e., to bet that his or her last move in this play of the game will either (a) Land on #L or (b) Miss #L. Along the front of the machine, there are three rows consisting of 25 keys each, numbered (left to right) from 1 – to 75.

 

Play begins when the operator pushes (1) the ON-button. At this moment, the computer randomly lights up a I or II, which designates the operator as Player I or Player II. Then, he or she pushes (2) the “L = ?” button, which allows him or her to choose L  as any number on the third row of keys, i.e., L  is to be a fixed number, so that 50 ≤ L ≤ 75.  Then, the computer activates (3) by choosing M, at random, so that 2 ≤ M ≤ 10. Finally, the operator pushes (4) the Objective button allowing him or her to register a choice of moving so that his or her last move in this play will either (a) Land on #L or (b) Miss #L. At this point, the game must begin within 10 seconds or the computer will shut down.

Also, at most 15 seconds are allowed for the operator to key-punch any move; or again the computer shuts down, amounting to a loss for the operator.

 

When operator is Player I, then beginning with Key I, he or she must push up to k consecutive keys (i.e., in order) where k is the operator’s choice, except he or she must have k ≤ M. Then, the computer following its program, begins with the (k + 1)th key continues the count with a sequence of M or fewer consecutive keys. The assumption is that the pushed keys light-up, whether pushed by the operator or the computer. This alternation of moves continues (no passes allowed) until a move captures L. The winner is the player (operator or machine) that scores the objective initially registered by the operator.

 

When the operator is Player II, then the computer will begin the count by lighting-up the first j keys (in order), with j ≤ M; j being determined by the computer’s program. Then, the operator must, within 15 seconds, begin with the (j + 1)th key, to punch in M or fewer than M consecutives keys. The alternation of moves between the players must continue (no passes allowed) until the number L is key-punched. The keys light-up when they are pushed in sequence with the two players alternating as asserted. Repeating our earlier comment, the winner is the player (operator or machine) that scores the objective registered by the operator.

 

The following example which utilizes the concept of an analyzing matrix referred to in my book on Modular Games will convince the reader that the computer can be programmed to internally utilize this information – regardless of the particular values of L and M that arise and regardless of whether it is designated as Player I or Player II – to win more often than lose.

 

EXAMPLE: If the operator chooses L = 51, while the computer’s random choice for M is 7, then the computer has instant access to the following matrix:

 

        A

 

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

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56

 

 

The case distinction below is useful for giving justification for our conclusion; i.e., the computer wins a lot more often than the human player.

 

 Case 1a. The computer is Player I & the operator has bet (a) Land on #L = 51. The computer, having instant access to the analyzing matrix A, and being Player I has a winning strategy; i.e., simply ending each of its moves on column 3 of A, thereby scoring a win.  This works for Case 1a because, M = 7 and the players are not allowed passes.

 

Case 1b. The computer is Player I & the operator has bet (b) Miss #L = 51.  Again, the computer, having instant access to the analyzing matrix A and being Player I, has a winning strategy; i.e., it can simply end each of its moves on column 2 of the matrix A and score a win. Again, this works because M = 7 and the players are not allowed passes.

 

Case 2a. The operator is Player I and has bet (a) Land on #L = 51.  In this case,  the operator possesses a winning strategy (probably unbeknownst to him or her); but the computer – having instant access to matrix A – pounces  on the operator’s first mistake (i.e., failing to end a move on column 3 of A)  to secure a win.

 

Case 2b. The operator is Player 1 and has bet (b) Miss #L = 51. Again, a winning strategy exists for the operator (probably unbeknownst to him or her). But the computer – having instant access to the matrix A – pounces on the operator’s first mistake (i.e., failing to end a move on column 2) secures a win.

 

In the general cases, once L and M are specified, the computer has immediate internal access to the analyzing matrix, which is an r by s matrix (again, call it A); where s = M + 1 and r is the least natural number so that L ≤ r (M + 1). The general such matrix A is patterned after our EXAMPLE. Of course, when (M + 1) evenly divides L, then r = L/ (M + 1) & L is on column s (i.e. the last column of A).

 

Each of the following remarks generalizes its corresponding case above to the prototype matrix A; of course the assumption that “L = 51” is removed.

 

Remark 1a.  In addition to the assumptions of Case 1a, i.e., (i) the computer is  Player I, (ii) the operator has bet  (a) Land on #L, assume also (iii) L lies in any column j of A, where j ≠ s (i.e., L is not in the last column of A).

In this case, the computer, being designated as Player I and having instant access to the matrix A, has a winning strategy observed by ending each of its moves on column j, to score a win. This works because any move by a player has less than (M + 1) steps and the players are not allowed passes.

When L lies on column s of A (i.e., the last column of A) then a winning strategy exists for the operator (probably unbeknownst to him or her), given that he or she is Player II, in this case; so the computer, utilizing its instant access to the matrix A, simply pounces on the operator’s first mistake (i.e., failing to end a move on the last column of A), to secure a win.

 

Remark 1b. In addition to the assumptions of Case 1b, i.e., (i) the computer is Player I, (ii) the operator has bet (b) Miss #L, assume also (iii) L lies in any column of A, say column k, except column l; i.e., k ≠ l.

In this case, the computer, being Player I and having instant access to the matrix A, has a winning strategy observed by ending each of its moves on the (k + l)th column of A to score a win.

Again, this works because any move by a player has fewer than (M+1) steps and the players are not allowed passes.

When L lies in the 1st column of A, a winning strategy exists for the operator (probably unbeknownst to him or her), given that he or she is Player II. Once again the computer utilizes its access to the matrix A to pounce on the operator’s first mistake (i.e., ending a move on column1 of A), to secure a win.

 

Remark 2a. In addition to the assumptions of Case 2a, i.e., (i) the operator is Player I, (ii) the operator has bet (a) Land on #L, assume also (iii) L is in, say column j, with not(j = s) (i.e., L is not in the last column of A).

In this case, a winning strategy exists for the operator (probably unbeknownst to him or her), given that he or she is Player I and not (j = s), in this case. However, the computer - with instant access to the matrix A - will simply pounce on the operator's 1st mistake (i.e., failing to end a move on column j) to secure a win. When L lies in column s of A, i.e., the last column of A, the computer being Player II, has a winning strategy (given by its access to A) observed by simply ending each of its moves on column s of A (i.e., the last column of A).

 

Remark 2b. In addition to the assumptions of Case 2b, i.e., (i) the operator is Player I, (ii) the operator has bet (b) Miss the #L, assume also (iii) L is in, say column k of A, with k ≠ 1 (i.e., L is not in the first column of A).

Of course, a winning strategy exists for the operator (probably unbeknownst to him or her), since he or she is Player 1 and k ≠ 1, in this case.  However, the computer, having instant access to the analyzing matrix A, simply pounces on the operator’s first mistake (i.e., ending a move on column k) to secure a win.

When k = 1, the computer,  having instant access to the analyzing matrix A, and being Player II can simply score a win resulting from utilizing its winning strategy which involves ending each of its moves on column s (i.e., the last column of A).

 

Perhaps, surprisingly, the computer wins a lot more often than the human whether or not it is designated as Player I or Player II.  Regarding the human player,  he or she is unlikely to know about the role of analyzing matrices in modular games (so, unlike the programmed computer, probably has no access to  analyzing matrices); in those cases where he or she has such knowledge, it’s not likely to be used very effectively in the Mod-Slot setting, because  of the time-constraints that are imposed on moves. The above assumptions that the operator will likely make mistakes are based on the previous observation.

 

 

 

Reference: E. W. Madison, *Modular Games”, Dorrance Pub. Co.,Inc., Pittsb., Pa

 

 

 


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