About Mu

Play Mu interactively How I invented … Mu

Dialog on a bike at night
The core of Mu's invention came on an eight miles nightly bike ride, from the game club Fanaat to my home, after having been exposed to Atlantis, a monumental but crooked game invented by Martin Medema. You can read more about the context in 'Organic Mechanisms'.

There are things about Mu that I sometimes have difficulty explaining:

• In terms of structure it's a simple game.
• I conceived it within an hour, without touching so much as a checker.
• Its behaviour, though highly 'organic', is predictable enough to allow strategic planning.
• Tactics may indeed backfire in over-the-board play. But the applet allows you to try out moves before submitting one.

The first pillar: the basic mechanism
The basic mechanism goes back to Sid Sackson's Focus and is based on a form of positive feedback: a column - a single included - moves as far as it is high. We're not concerned with bicolored columns, just with single-colored ones like these:



Here are some men arbitrarily divided over 6 columns. Consider it to be one organism. Like an ant colony it answers to a single mind: yours. It can split or merge, go this direction or that, crawl or jump, spread or erect, and it can display efficiency in that there's always a minimum number of step in which it can do things like:

• Spread out completely.
• Raise one stack consisting of all men.
• Get at least one man to A.
• Get say 10 men to B.
• Occupy the area around C completely.

Or reach similar arbitrary objectives. It moves and morphs. For the moment it lacks growth ... but we'll get to that.

The second pillar: the basic terrain
The basic terrain goes back to an obscure seventies game in which a square would 'explode' if it would hold as many men as (or more men than) the number of its adjacent squares, ejecting one man to each of these, leaving the remainder behind, if any. It was called "Explosion" and was featured in issue 55 of Games & Puzzles Magazine. We had experimented with it before at Fanaat, including the hexversion. Martin's segments allowed for boards of different sizes and shapes and a 'one move per segment' protocol, that would enable move combinations without having them get out of hand.

Growth (will kill you)
Under the Atlantis explosion protocol, a column would 'explode' if its height reached the number of a cell's free neighbors, 6 on a centercell, 5 or less along the edge, limiting the columns' ranges accordingly. The game started with each player occupying one cornersegment filled with 7 men, one on each cell. It allowed each player to explode a first (capacity-3) cell on his second move.

Explosions provided the growing mechanism: If a cell exploded, it ejected one man to each of its neighbors, letting any remainder evaporate - that made me raise an eyebrow right away. Next the cell became a 'well', growing one man each turn untill it reaches capacity for a second time. Then it exploded once more and turned into a 'crater' - a solid obstacle. With wells and craters no longer counting as 'free neighbors', that, as it turned out, was a critical growth rate.

The negative side of positive feedback
Here's the thing about positive feedback: you have to keep it controlled or it will spin out of hand. Say you're a cell and your neighbor explodes: you get an extra man and at the same time lose a neighbor. Your capacity decreases while the load increases. That has "chain reaction" written all over it. Add that these chain reactions are most likely to creep inwards from the corners and edges, fueled by the wells, and the picture is clear: you're first and foremost trying to get away from your own wells, with whole sections along the edges eventually turning into craters and 'sinking into the sea' behind you ('cratered segments' were removed entirely). With any luck, you could secure some territory with targeted explosions, at a safe distance from one another, in the remains of what used to be a large board. I thought it was a rather pathetic object and a game that seemed designed with the sole purpose of hampering itself. The segments crumbling off the edges of the board still dominate my recollection of its first impression.
It was a strange night and I finally took my bike and went home in a state of confusion.

I must have been sleep biking on autopilot. I re-entered reality in the monochromatic orange light of an arterial road about half a mile from my home, about two o'clock in the morning. My legs were still peddling. Mu was born. I felt elated.

I had left Fanaat with conflicting impressions. On the one side there was this beautiful organism, versatile, fexible, efficient, volatile and capricious. What did it want? That was the key question. Certainly not the crippled fight to secure some space on a sinking island, fighting another sorry bunch of natives, driven onwards by explosions and crumbling edges of one's own making.
Not only the manner of erecting walls seemed wrong, but the place where they first appeared: in the players' own backyard, at the edge of the board. One needed growth, and low capacity edge cells were the only place to get it, initially. It was like building a wall against a wall. Meanwhile jumping to the center with high stacks to erect walls there, required making high stacks in the first place, without having them explode away accidentally in a chain reaction. In the center you'd need a 6-column for the first explosion, and next you'd need 5-columns for adjacent ones. But the attempted 'wall' would more often than not become an omni-directional 'blob' due to a chain reaction. Some way to build a wall.

Then, somewhere along the way, it occured to me that explosions are used, usually, to clear an area, not to erect something. And then the vision came. Ask any inventor how a game came to be, and it will probably be a more or less rational and evolutionary story along a timeline. But you can't rationalize a vision, or at least not its appearance. So I'll rationalize in retrospect, but it all happened in a fraction of a second.

A vision rationalized
To clear an area there must be something to clear in the first place. What I saw that instant was a board filled with a top-layer of white draughtsmen. Several separate 'holes' appeared in it and they grew bigger and bigger and inevitably encountered one another and ... didn't merge. Instead black draughtsmen appeared to replace any white ones the removal of which would cause a merger otherwise. An organically growing natural separation between different territories that, once completed, would leave a crude 'spiderweb' of boundary lines of black draughtsmen. Those of course would be the territories to conquer and defend.

It was all in one vision, one moment, and it included the growth of a new man on every cell that had its top-layer blown away. It provided fuel for the very same chain reactions, but with the reverse effect: they would actually clear one's territory instead of taking it away. A 'one-man-per-explosion' growth rate would be substantial, but not critical. This, I immediately felt, was what the organism was made for.

By the time I awoke, peddling, I had filled in most of the details: white draughtsmen would only have their own as neighbors, to determine a cell's capacity. The cells of territory they revealed when an explosion occured on them would have their own and those of the top-layer for neighbors. This would guarantee intricate interaction involving both layers. It would also allow any 'overcapacity' of an exploding cell to remain 'in place'. Much of the energy in Atlantis would evaporate as overcapacity, that had bothered me immediately. Of course cells that became part of the wall could also have overcapacity. I realized there would be men on the wall ... men that could travel the whole wall. Oh well, maybe not the whole wall, but they were there, and the question 'what would they want' had an obvious answer: remain involved. It was a detail that would solve itself, I felt. And it did, though it was not at all a 'detail', but rather the crucial key to invading territories! It goes to show once more that if the system is sound, the rule will be there.

Epilogue
The current protocol for laying out a board as part of the game, came later. It makes the game faster (due to a lower average capacity) with more room for opportunism based on local peculiarities. It also makes that 'the wall' does no longer necessarily consist of one connected web because it allows peninsulas to be locked off by seperate sections of the wall that have both sides terminated by the 'out of bounds'.

Rules
The board is the most unusual aspect of Mu. It is made up of connected segments and the lay-out phase is an important part of the game. Moreover, the board is layered. Initially every cell of it is covered with a beige top layer that if removed reveals a green bottom layer. In addition there are hexagons representing 'the wall'.
Pieces can be stacked to several heights. The highest stable height a piece can reach is 5. In the course of a game, pieces may occupy cells of the top layer, the bottom layer and the wall alike.



The opening protocol

• We'll assume two players, Red and Purple, with eight segments each. The lay-out phase establishes a playing area and an initial position. Red moves first.
• Segments are put on the field one by one, to make up a board. When a segment is placed, the player places one of his men in the center. After the first player has laid down his first segment, players take turns. Each new segment must at least have two cells adjacent to the evolving board. After the last segment has been placed it may look something like this:

Note:

• Mu is a 'layered' game in more than one respect, and the actual territory - called "The Commonwealth" for reasons unknown even to the inventor - is as yet buried under a visible top layer called "Virginity".
• The blue area is called "The Out of Bounds" because it is: no piece may move onto or over it. Nevertheless having one's segments bordering on the Out of Bounds as much as possible, is at least one of the considerations in this stage. Therefore lakes and creeks emerge naturally.

Object
The object of each player is to obtain as large a piece of territory as possible, counted as cells of the Commonwealth that are under his control. The end comes after successive passes and the player with the largest territory is the winner. The game may end in a draw.

Movement / Capture

• Players are entitled to one move per segment they occupy at the start of the turn, per turn.
• There is a restriction regarding the 'one move per segment' rule: after the last segment has been placed, the movement phase starts progressively: the first player makes 1 move on his turn, the second player makes 2, then the first player makes 3, and so on till the number equals or surpasses the number of occupied segments.
• The applet will indicate which segments still allow a move to be made by highlighting them thus:
  
• A 'piece' may be a single man or a like colored column. Any number of top men of a piece also make up a piece and may be moved accordingly.
• Players take turns to move pieces. Moving is not compulsory: a player may refrain from moving pieces of any or all segments he occupies at any turn.
• A piece moves in a straight line, exactly the number of cells equal to its height. A piece may move onto or over pieces of any color.
  
  • If a piece lands on a piece of like color, the two merge to one column.
  • If a piece lands on an opponent's piece it captures by replacement, regardless of size.
  
  A player may combine moves any way he likes: it is allowed for instance to move a piece from segment A to merge with a piece of segment B, and, provided no move has originated from B yet, move the whole merged piece or part of it elsewhere, say to a segment C, and do the same there once again.
  Barring intermediate 'explosions', see below, anything goes, so long as pieces are only moved from segments the player occupied at the beginning of his turn, and no more than one piece is moved from any one such segment.
• A piece may not move over or into the Out of Bounds.
• In the course of the game, Virginity disappears gradually to reveal sections of the Commonwealth. These sections appear and expand until they are limited by the encounter with one another, at which point the "Wall" arises as a natural separation: sections never merge.
• The 'one move per segment' rule applies regardless of whether the move originates from Virginity, the Commonwealth or the Wall.
• A piece starting from the Commonwealth or Virginity may never move onto a cell of the Wall!
  A piece starting from the Commonwealth or Virginity may move over a cell of the Wall, if and only if that cell is occupied by a piece of like color.
  If you're wondering how a piece gets onto the Wall, or how the Wall appears in the first place, you're right on the ball.

Capacity and Explosions in Virginity - Growth

• The 'capacity' of a virgin cell is equal to the number of adjacant virgin cells. If a virgin cell gets to hold a piece consisting of a number of men equal to or greater than its capacity, it 'explodes' as part of the same move, thereby distributing its men over the very cells that made up its capacity, one man per cell, while unveiling a cell of the Commonwealth or the Wall. Explosions are compulsory. Pieces landing on other pieces by explosion merge or capture as the case may be.
  The applet indicates hot cells thus: and performs an explosion with a mouseclick on the piece.
  
  
  • If the exploding cell does not border on different sections of the Commonwealth, it either creates or expands such a section. In that case the cell sprouts one new man, immediately after the explosion, that is put onto the cell.
  • If the exploding cell does border on different sections of the Commonwealth, it turns into a cell of the Wall. A cell of the Wall does not sprout a new man.
  
  If the exploding piece consisted of a number of men greater than the cells capacity, the remainder is left behind on the original cell. This is called 'overcapacity' and it is the only way to get a piece on the Wall!
  Overcapacity does not affect the sprouting of a new man.
• Since an exploding virgin cell gives a man to every virgin neighbor, while at the same time taking a neighbor (i.e. itself) from each, these cells in turn may reach or surpass capacity. In that case the original explosion has triggered a chain reaction that is part of the same move. If more such cells exist at the same time, the player is free to proceed with any of them, till the whole chain reaction has come to rest.
• As long as a piece may move, receiving one or more men by explosion does not affect its right to do so. If an explosion causes a piece to land on a segment that the player occupied at the beginning of his turn, but from which no move originated yet, it is included in the move options of the pieces on that segment.

Capacity and Explosions in the Commonwealth

• The capacity of a cell of the Commonwealth is counted in neighbors of both the Commonwealth and Virginity. If a cell of the Commonwealth explodes, nothing special happens except that it distributes its men over the very cells mentioned above, and that overcapacity remains on the original cell. If a cell of the Commonwealth explodes, it does not sprout a new man. Of course the explosion may trigger further explosions and/or make captures in the process.

Capacity and Explosions on the Wall

• Other than in Virginity or the Commonwealth, a cell of the Wall becomes critical but as yet stable if it reaches capacity, and explodes only if the cell has overcapacity, distributing its men over the adjacent cells of the Wall and leaving the overcapacity behind.
• A cell of the Wall only counts adjacent cells of the Wall as its neighbors. The maximum column a cell of the Wall can accomodate is four, on certain intersections.
• A piece on the Wall may move like any piece, but in addition it may move over any cells of the Wall, and may end its move on the Wall, merging, exploding and/or capturing as the case may be. Moving off the Wall is of course a definitive step.

Explosions are immediate

• Explosions are immediate in the sense that they take precedence over making a move. So if a move causes a cell to reach or surpass capacity, the applet requires the player to explode it first and perform subsequent explosions, if any, before proceeding with movement.
  Explosions are not wholly immediate because if there are more cells at or over capacity at the same time, a player is free to choose the order in which to perform them.

An example
It's red's turn. His local opponent green is in sight and has already cleared part of Virginity to make place for the Commonwealth:

	So it's red to move. The two segments he occupies are marked, indicating no red move originated from them yet. He moves the 2-piece (the required two cells) to capture the green man, and the 3-piece (the required three cells) to a virgin cell of capacity three. The markings around the segments disappear.
	He does so in that order, because the 3-cell explodes immediately, distributing its men over the adjacent virgin cells. It should be clear that the order of moves matters.
	After the explosion the uncovered cell of the Commonwealth immediately sprouts a red man, indicated with a small circle. Meanwhile another virgin cell has reached capacity and explodes in a similar fashion, bringing an adjacent cell on one man overcapacity.
	Now this cell borders on different sections of the Commonwealth, so it explodes to transform into a cell of the Wall. The green man on the adjacent virgin cell is captured. The applet leaves the overcapacity in place: subsequent explosions in the same turn may extend the wall and therewith the first cell's capacity, as is the case here. If not, the cell oscillates and is handled accordingly, see below.
	The explosions proceed with the 3-piece on the centercell of the middle segment. Or at least that's wise because the wallcell cannot explode and neither does it have to. If no alternative explosion were available, it would oscillate and be handled accordingly, but now it is spared that fate by the alternative explosion.
	As a result one adjacent cell reaches capacity, another even overcapacity, and Red may choose which one to explode first. Although here the order is irrelevant, it often isn't.
	So a new cell of the Wall appears, with a red man on it. At the same time the first cell now has capacity-1 and is no longer 'hot'. Note: Creating cells of the Wall with overcapacity is crucial, so red did very well here.
	Here goes the other one, sprouting a man and pushing a neighboring virgin cell to overcapacity.
	Which leads to the final explosion, where a new man sprouts on top of the overcapacity that has stayed behind. Note that the piece now threatens the green man on the other side of the Wall, and that this cell is protected twice by the green column of two. Of course this is the result of only two moves of Red - he may have other segments to proceed with in his turn. Now you know how to play Mu velox ... well, almost.

Lost in Oscillation
Sections of the Commonwealth or the Wall may contain arrangements of columns that are in a cyclic chain reaction, even if the combined height of the columns does not equal or surpass the total capacity of the section. This is called 'oscillation'. Here's an example of a section with a stable arrangement of 12 men, next to the same section with an unstable arrangement of 10 men:

• If a section shows oscillation the player is obliged to remove men till the chain reaction comes to rest. Removing men from hot oscillating cells is done by clicking the pieces: the app will notice and remove men till the oscillation stops. Terminating oscillation does not count as a move.

To hold at least one man, a section of the Commonwealth must count at least three cells. Cells of a two-cell section both have capacity one and will oscillate if a man is placed on either, while a one-cell section explodes 'at capacity', which is zero. Thus, a one-cell section of the Commonwealth is 'oscillating on empty'.


Virginity Collapsing & further proceedings
Inevitably Virginity is blown up, sprouting more men, creating and expanding sections of the Commonwealth that run into gridlock with one another, leaving them locked between the Wall and the Out of Bounds.
Players may find themselves alone in such a section, or engaged in a fight with other players. Walls may lock a player in, or out. You may find yourself able to invade an adjacent section without any risk of being invaded, or quite the reverse.
Mutual capture may decide who owns a section or end in a stalemate: a piece cannot threaten to capture a piece of equal height without exposing itself. But it can if it is covered by a like colored piece, adding a Chess like flavor to the fights.
Pieces on the Wall may engage in exchanges till one player wins and owns the section, allowing his armies to use it as a bridge. Even stalemates may occur, where the 'bridges' block one another under the threat of capture. Fights will take place to control key positions on the Wall, because of their strategic importance regarding invasions.

End of the Game
The game ends when all players have passed on successive turns. You pass by double-clicking a mobile piece of your side. The counting is easy:

• If a section of the Commonwealth holds only a piece or pieces of like color, it is the territory of the corresponding player.
• If a section of the Commonwealth holds pieces of different color, it is neutral territory.
• If a section of the Commonwealth holds no pieces at all, but is accessible from the Wall only by a piece or pieces of like color, it counts as territory for the corresponding player. If pieces of different color are able to jump in, or no piece at all, it is neutral territory.
  Note: though not limited to them, this rule typically addresses one- and two-cell sections that will not accept a piece to claim them.
• Fragments of Virginity left inside the Commonwealth at the end of a game count as territory as if they were part of the Commonwealth.

An example game
It will take a few seconds to load due to a multi-move protocol and high mobility.

no Sound
Broken canvas...	to move territory 0 0
Christian Freeling - Ed van Zon (1-0)

Note that in the final position Red can invade the disputed bottom-right territory and win it, while Purple is stuck behind the walls.



Mu velox © MindSports