Freedom of Choice? Degrees of Freedom?

©

Freedom of Choice? Degrees of Freedom? 3 Will, 3 Agent, 3 Dimensions An abstract contest driven by randomness, mediated with risk trading and constrained under entanglement. Is success determined by ability or probability? Copyright © April 2015 by K. J. L. Percival

Copyright ©2015 by K. J. L. Percival

Introduction 3Will-3Agent-3D is a head-to-head board game that explores concepts of randomness, risk and entanglement and tests the relationship between free choice and degrees of freedom.

Definitions Players: There are two opposing protagonists who ostensibly control moveable entities. Agents: Each player commences with eight stackable, numbered pieces that are manipulated as agents of their objectives. Opposite agents have distinct colours. The Board: Eight rows and eight columns define a domain of prescribed positions in a square grid. The first row, from each player's perspective, represents their base. Dice: Randomness is rendered by a standard six-sided dice. Risk Bank: Trading in risk is implemented by a conceptual banking system that manages risk cards. Cards: Cards are used to facilitate certain processes. Nine numbered cards are involved with endgame agent challenges and eight Double Risk cards are used for special operations. DOUBLE RISK CARDS

CES CARDS

are used for Stacking. They also function as

Covert Entangled State cards are

FREE CARDS

numbered 0 to 8

Double Sided Revision 12:22: 26 June 2015 Page 2

Direction Chart: The direction chart is used to interpret movement directives generated by the dice.

Player point of view

RIGHT

LEFT

LEFT

RIGHT

Note: Refer to the operational rules for details on the application of this chart, the risk bank and the various cards.

Player point of view

Setup Procedure 1. The board is initially configured as represented below. Like coloured agents are randomly distributed into second row positions adjacent to their respective bases.

Base Agents

Risk Bank

Agents

Base

2. Decide the starting player by any conventional method. 3. Shuffle the CES cards and place them face down. The unrevealed top card predetermines whether or not covert entangled states exist. 4. Deal the Double Risk cards. Each player receives an allocation of four cards from the bank. 5. Locate and orient the Direction Chart so that it may be easily viewed and correctly interpreted by both players. Revision 12:22: 26 June 2015 Page 3

Objectives Primary: Reach your opponent's base - a win requires occupation by three legitimate agents. Secondary: Defend your own base. Supplementary: 1. Stack agents if operationally necessary or strategically expedient. 2. A win could be possible by leaving your opponent with no active agents. 3. Recover and reactivate held agents.

Operational Rules General: 1. Directives for Standard Operations must be executed unless it is impossible to do so. This means that if any of your active agents are available for deployment then you must select one and move it. However if all Standard Operations are blocked and any optional Special Operations are either not available, not viable or have failed an attempt then the current opportunity is forfeited (you give up this turn). 2. The current opportunity also terminates after any successful Special Operation. 3. Agents that have attained the primary objective become inactive, locked and secure. They cannot be stacked or moved unless swapped during a successful endgame challenge. 4. There is a stacking limit of two agents. Movement Directives: Each player in succession devises an action by rolling the dice. The directives are interpreted and movements assigned to any one of the current player's active agents. Standard Operation: Allowable movements are one grid position in one of eight possible directions, executed in accordance with the following Direction Chart definitions;

ODD

EVEN

Dice 1 5 4 3 2 6

Movement (one position) Straight FORWARD Diagonal LEFT Diagonal RIGHT Straight LEFT Straight RIGHT Straight or Diagonal BACK

Revision 12:22: 26 June 2015 Page 4

Entanglement: Players have to contend with agents that could be entangled in two possible ways. Overt entangled is a visible state where two agents are stacked. The lower agent is temporarily deactivated and held under the constraint of the higher (active) agent. Covert entangled is a hidden state where opposite agents, bearing the same identification number, may have had their allegiances swapped as a consequence of predetermination. All agents are considered potentially covert entangled until their true states are revealed. Hierarchical Stacking Protocol: The higher (active) source agent stacks over the target active agent and retains the higher status. Any remaining lower source agent becomes active. If the target stack limit is exceeded then the lowest unlocked (moveable) agent is ejected and placed lowest in the source. If an agent is ejected to an empty source then it becomes active. The table below demonstrates the standard possible outcomes (see Interpretation Q.1, Q.2). W

B

W/B

B/W

W/W

B/B

W

0+W/W

0+B/W

B+W/W

W+B/W

W+W/W

B+B/W

B

0+W/B

0+B/B

B+W/B

W+B/B

W+W/B

B+B/B

W/B

B+W/W

B+B/W

B/B+W/W

W/B+B/W

W/B+W/W

B/B+B/W

B/W

W+W/B

W+B/B

B/W+W/B

W/W+B/B

W/W+W/B

B/W+B/B

W/W

W+W/W

W+B/W

B/W+W/W

W/W+B/W

W/W+W/W

B/W+B/W

B/B

B+W/B

B+B/B

B/B+W/B

W/B+B/B

W/B+W/B

B/B/+B/B

Target

Source

PROTOCOL High ⇨ High Low ⇦ Low

White(W) Black(B) Empty(0) outcomes in Source + Target order

Optional Special Operations: 1. Stacking Manoeuvre During the execution of a Standard Operation, if a strategic position within your current scope is occupied you may attempt a Stacking Manoeuvre by taking a Double Risk. If successful, you apply the relevant actions governed by the hierarchical stacking protocol. Tips: Stacking your own agents together could be beneficial e.g. when reaching the primary objective, the pair would qualify as two. A held agent may be recovered and reactivated if it emerges from the stack with the required hierarchical status. 2. Dimensional Leap The Free Standard Operation offers an additional, potentially powerful, capability. Agents located in positions at the extreme right or left edges of the board usually suffer from limited permissible movements. However this restriction is removed during a free operation by extending your scope to the opposite side of the board thus making a dimensional leap possible. Ordinarily the target position must be unoccupied but, if you can afford the risk, you may also attempt a Stacking Manoeuvre as previously described. Double Risk: To accomplish a Stacking Manoeuvre you need to throw a double in the target direction. Before attempting the required extra throw you must expend a Double Risk card to the risk


bank. If you have no allocated Double Risk cards remaining then you can take one from the bank but must immediately hand it to your opponent as a Free card. Free Standard Operation: After performing an ordinary Standard Operation, you can utilise a Free card you possess to undertake an extra Standard Operation. Any number of these cards may be expended seriatim to the risk bank until either you deplete your supply, can’t move or use a Free Standard Operation to initiate a Special Operation. Endgame Agent Challenge: When a player declares victory, the opposite player is allowed to challenge the legitimacy of an agent. The formality is implemented by choosing one agent associated with the context of the win, examining its identification number and revealing the top CES card;  If the CES card is zero (none) then all agents were true and the win remains valid.  If the CES card matches the chosen agent then the opposite agent, with the same number, must be located and the current positions of the two agents, including any existing stacking hierarchies, are swapped. If a winning condition prevails or is newly created then victory is declared and the game ends. Otherwise the game resumes and continues until its ultimate conclusion with no further agent challenges.

 If the CES card is any other number then the challenger failed to expose the covert entangled agents and the win remains valid by default.

Quotes The seemingly something is elegantly reorganized nothing. -- Peter Atkins Prediction is very difficult, especially about the future. -- Neils Bohr Everything should be made as simple as possible, but not simpler. -- Albert Einstein Oh, that I was where I would be! Then would I be where I am not! But where I am, I must be. And where I would be I cannot. -- Unknown All the world’s a stage, and all the men and women merely players. -- William Shakespeare Quantum theory says a lot, but does not bring us closer to the secret of the “old one.” I, at any rate, am convinced that He is not playing at dice. -- Albert Einstein Schrödinger's cat has far more than nine lives, and far fewer. All of us are unknowing cats, alive and dead at once, and of all the might-have-beens in between, we record only one. -- Yoon Ha Lee I don't like it, and I'm sorry I ever had anything to do with it. -- Erwin Schrödinger The human mind may devise many plans, but it is the purpose of the Lord that will be established. -- Proverbs 19:21 (NRSV)


Interpretation Questions & Answers 1. Could you explain the implications of General Operational Rule 3? 2. Can you show me some examples of how to stack? 3. When can I use a Free card to do a Dimensional Leap? 4. Am I automatically allowed to stack during a Dimensional Leap? 5. How do I retrieve a captured agent? 6. Are agent ID numbers supposed to be hidden or visible during the game? 7. What is the meaning of Supplementary Objective 2? 8: How does 3Will work?


Q: Could you explain the implications of General Operational Rule 3? A: When agents reach the primary objective (opposition base) they become inactive, remaining locked in position and secure until the game ends. This means that they cannot be moved or stacked by either you or your opponent. The only exception is where an agent can be moved because it needs to be swapped following a successful Endgame Agent Challenge. There are two interesting predicaments that can arise as a consequence of this rule. The first occurs if you inadvertently drag a stacked opposite agent into your own base. It counts towards a winning total for your opponent and the preceding overt entangled state collapses, allowing you to unstack and move away but leaving the opposite agent fixed in place. The second predicament can occur following a successful agent challenge where an opposite agent is swapped to the higher stack position over an incumbent winning agent. This collapses the entanglement in the same way as mentioned above. If another agent subsequently attempts to stack a disentangled agent that remains on top of a secured agent then a special implementation of the protocol must be adopted. Because the lowest target agent is locked and cannot be moved, the next lowest (middle) agent must be ejected instead. RETURN


Q: Can you show me some examples of how to stack? A: The key to happy stacking is to remember these main principles: Only active agents are allowed to stack other active agents. Highest source agent stacks over highest target agent. Lowest moveable target agent is ejected to the lowest source position if the stacking limit is exceeded. The following diagram illustrates two examples from the table of possible outcomes. RETURN


Q: When can I use a Free card to do a Dimensional Leap? A: Expending a Free card only allows a Free Standard Operation which also happens to add an extra dimensional capability to your scope. You may have noticed that a Dimensional Leap is classified under Optional Special Operations. This is so that, by performing it, your current opportunity (or turn) finishes. However you cannot “spend” a Free card as an entitlement for a Dimensional Leap outside the specific context of a Free Standard Operation. For example if you are performing a regular Standard Operation and see an opportunity that causes you to contemplate using one of your Free cards to make a Leap, then this is not allowed. You must first complete the current Standard Operation and wait for a similar opportunity to arise during a Free Standard Operation. Yes, this may seem tricky to remember but be patient; your opponent won’t be expecting it either - such is luck! RETURN Q: Am I automatically allowed to stack during a Dimensional Leap? A: No. But if your current extended scope, thanks to the Free card (see Q.3), includes a strategically stackable target then you can attempt a stacking manoeuvre in the usual way; by expending a Double Risk card and trying to throw a double in that target direction. Of course if your attempt is unsuccessful then the usual forfeit ensues. RETURN


Q: How do I retrieve a captured agent? A: They prefer to be called “agents in an overt entangled state held under the constraint of a hierarchical stacking protocol”. If you are a single active agent then it’s easy to recover a held agent; just attempt to stack the opposite agent holding it (see Example 2 from Q.2). However if you are currently holding another agent then you need to carefully examine the possible consequences of the hierarchical protocol outcomes before committing to a manoeuvre. RETURN Q: Are agent ID numbers supposed to be hidden or visible during the game? A: It is not specified in the rules that ID numbers be either visible or hidden because the existence of covert entangled states of swapped agents is predetermined by the CES cards alone. So it is left up to the players to decide during setup. Keeping the numbers hidden may afford a sense of mystique and displaying them could strengthen the notion of control. Most players prefer hidden numbers because they offer less distraction. RETURN Q: What is the meaning of Supplementary Objective 2? A: If you are successful in stacking enough of your opponent’s agents to prevent them from winning then you can declare victory. This would be an extremely rare event but not impossible. The number needed to be held depends on the number of your opponent’s agents that have already achieved their primary objective. You must stack all the remaining opposite agents leaving none active. In the ensuing Endgame Agent Challenge the agents in the winning context comprise all of your agents that are holding the challenger’s agents. The convoluted tactics involved in achieving victory in this fashion are particularly tedious. A strategy based on the primary objective is easier to execute and much more likely to succeed. RETURN

Q: How does 3Will work? A: Anything that happens, happens. Anything that, in happening, causes something else to happen, causes something else to happen. Anything that, in happening, causes itself to happen again, happens again. It doesn't necessarily do it in chronological order, though. Douglas Adams. RETURN


1

2

3

4

5

6

7

8

6 7 8

0

NONE

1 2 3 4 5

H L

H L

DOUBLE H L

OPERATION

OPERATION

STANDARD

H L

DOUBLE

OPERATION

STANDARD

RISK DOUBLE DOUBLE

FREE FREE OPERATION

FREE FREE

H L H L

DOUBLE

H L

DOUBLE

H L

STANDARD

RISK DOUBLE

STANDARD

RISK

DOUBLE

RISK

H L

H L

DOUBLE H L

OPERATION

OPERATION

STANDARD

H L

DOUBLE

OPERATION

STANDARD

RISK DOUBLE DOUBLE

FREE FREE OPERATION

FREE FREE

H L H L

DOUBLE

H L

DOUBLE

H L

STANDARD

RISK DOUBLE

STANDARD

RISK

DOUBLE

RISK