The warrior monks of Huro colony (fiction)

Huro was once a Space Navy outpost, but the frontier has gone far beyond the area, and for years it remained a very quiet colony with failing infrastructure. But then a group of warrior monks arrived on the planet, and it has since been transformed into one of the most esteemed destinations for training of the mind and body.

The order of fighting monks has no name. The vast majority of the planet’s inhabitants are part of the order, or on there way off-world. Thousands come to test their mettle, and many leave, unable to overcome their induction trials. Even those who fail to get into the order report being transformed by the challenges. For those who are not particularly serious about the trials, its a chance to revitalize your body and reconfigure your perspective.

I was a bailiff on New Boston Chicago for eight years. I told them that Boston and Chicago were two separate Old Cities, but they didn’t comprehend why combining them into one name was so silly. Eventually I got fired, for reasons I have no desire to get into, and I ended up working as a bouncer. I was a tough guy in a small community, but that doesn’t require being all that tough, after all. Eventually, I fought a drunk trainee from Huro who roughed me up when I tried to throw him out. He took pity on me, and told me I needed to step up my game. I’m pretty small for a bouncer, only 5’10 and 190 pounds. I decided he was right, and starting saving up, so I could afford to get there, right away.

It took me over a year to hitchhike, work and connive my way towards Huro. But I made it

The spaceship that got me here, The Black Claw, was piloted by ex-trainees. I spent the last of my savings to pay my way with them. Luckily, the monks don’t charge anything to try out. They are looking for new members of the order, and if you fail to make it through induction, you don’t get another chance. Several Black Claw crew members told me that.I asked them incessantly about the training process. They didn’t have much to tell me.

Twenty of us hopefuls arrived at the monk’s enrollment building that morning. The night before we had arrived at the spaceport, and found lodging. Only a small Japanese sign identified the building’s purpose. Unfortunately, I didn’t know Japanese at the time.

Inside, the hallway was only big enough for five, so the rest of us waited outside. After about twenty minutes, and ten of our number processed, it was my turn to go inside. There I waited some more, as a monk wearing a big hat took the others down the hall, one by one, until at last I followed him down that hall, past many doors, until finally he found the door he was looking for, and lead me into it.

“Now, I will seek from Chance your training name.”

The monk tossed three many-sided, golf-ball sized dice, and they skittered across the floor of the small room. It seemed  eerily empty.

The dice was scattered across the room, and I noticed each face had a different word on it. The monk picked up the three huge dice, and made sure to keep the correct faces on top, so he could read off the words.

“Mamushi, the venomous snake who ambushes it’s prey.” the monk said, interpreting the first die.

“Ash. Your motivations are uncertain. Your past is behind you.”

“Palm. The open palm strike shall be your technique.”

“Together these Gifts of Chance reveal much to me. I see that you prefer the closed fist. That way will not succeed here. Ash Palm Mamushi shall be your name.”

“Thank you,” I said to the monk. “I will train with ferocity,” I added, wondering if I should have said such after the words escaped my lips.

“Strike me then, Mr. Mamushi. Your training begins now.”

This monk was fairly young, he looked to be in his early twenties. But in my overactive imagination, he was a powerful old master. I feared him.

“Do not restrain yourself. Strike me now!” the monk yelled.

I hesitated for a second, and then clumsily approached the monk, trying to bury my fist into his face, like I would have attacked a violent drunk.

My fist was hitting empty air, as the monk leaned slightly to the side. Before I could follow up the whiffed attack, the monk hit me hard in the sternum with his open palm.

I tumbled backwards, breathless, and my head hit the far wall first.

“Your prior ‘training’ is of no use here. That is the first lesson of Sa’are.”

I blacked out.

Revising a Novel

I’ve been busy plotting and planning, and rewriting “Valeraine” for the last few weeks. The first draft was wild, scattered, and didn’t have the kind of portrayal of Valeraine and Roland that I wanted.

In the first draft, the story started with war which still is part of the 2nd draft. But in the second draft, I wanted to pull back. Instead of Valeraine being a commander and a powerful mage, I made her more focussed on being a mage, and made her a novice to the  military sort of path. I rewrote about 10 chapters during this process, and with help from the Traverse City Nanowrimo group, I added the idea that she was actually adopted, and not Sapphiron by blood. I added a whole conspiracy / story behind covering up the adoption.

The early parts of the story, which were revised in the 2nd draft, go all over the place. I decided a few weeks ago, that the story was getting too scattered and unfocused, and so I decided to place the story in the eyes of Valeraine and Roland. I also thought about what the motivations of all of the characters.

The whole adoption scheme was for the purpose of continuing the line of Sapphiron Kings.  King Deocius would never let Valeraine be put into danger.

Instead, I decided to start the 3rd draft with the King forcing Roland and Valeraine to be married, and to head north to the safety of Roland’s keep. I just started writing that draft.

This creates conflict, which wasn’t really there early on in the 2nd draft, but we also see more depth to Roland and Valeraine and the King because of it.

I would like to record here how the process of restructuring, reenvisioning and revising “Valeraine” goes. I have gradually changed my thoughts on who these people are, and what the story should be about.

In order for conflict to occur, Valeraine had to be stripped of the one thing she wanted most: the chance to research and practice magic.

Roland is put in the awkward position of getting what he wanted: marrying Valeraine, but objecting to the forceful method the King pursued, and finding himself hated by Valeraine. He now had to convince her that it wasn’t his plan. This is a classic case of miscommunication. They both would have been happier if the whole thing had come about voluntarily, but they were both on paths taking them further from each other, so in the end, there’s the question of whether the King’s harsh actions in confining Valeraine’s powers and forcing them to marry were actually justified in the end.

Valeraine has to struggle, because she cares for her students, who are now lost without any instructor in fire magic. She has to scheme to establish a school on the sly in Baird duchy, and this creates the possibility of political intrigue.

The central question I have now, is how to prevent all of this tension from unwinding as Sapphiron is conquered. What new twists can keep the conflict and tension high in Casposea and onward until the climax.

I’m starting to think getting a little inspiration from Shakespeare might be interesting.

Railroad Tycoon 2′s North American Scenario

When I went to college, I got my first Windows PC, and one of the first games I played on it was Railroad Tycoon II.

I tried the campaign, which was a collection of scenarios basically, but most of the scenarios were time limited, with small goals and a local area where you were building your railroad network up. No, after playing the early campaign scenarios a few times, I spent vast hours playing one scenario in particular. I call it a sandbox scenario, because it was a scenario that shows what the main campaign of the game should have been like.

The scenario was called North America, and it covered Mexico, Cuba, the US and Canada. It was as if they had all been unified into one country as you didn’t have to worry about gaining the right to build railroads and stations in each of these countries. The European Scenarios seemed to focus much more on negotiating with, say France, to have the right to extend your german railroad company’s line into French territory.

But in the North American scenario, you could run your own railroad company over this huge landscape, all the way from 1830 to shortly after 2000. I never made it further than about 1920. This sandbox scenario was basically the nirvana for someone interested in operating a rapidly growing business sprawling out of control.

Railroad Tycoon II Platinum became available on Steam a few years ago. I had bought Railroad Tycoon III when it came out, and promptly stopped playing it when I discovered you couldn’t play on a big map with a long scenario length. I think the scenarios were limited to something like 20 years. It just lacked the pure thrill of the North America map in Tycoon II.

The thing I liked about North America scenario is it was a wide open sandbox, but with financial constraints, and also performance evaluation. It really did a great job in simulating “What have you done for me lately” shareholders and financial analysts. You could build up a monstrously wealthy corporation that was basically printing money by the bushel, but eventually it became hard to keep up the growth rate when you already had so many lines laced throughout the continent.

Immediately after I purchased Railroad Tycoon II on Steam, I begin building the most grandiose railroad corporation I had ever built in the game. I started my corporation on the basis of an Jacksonville to Tampa, Florida railroad line.

Before long I had connected to Miami, and then into Louisiana, and so on. Previously, when I had played this scenario, I tended to start in the midwest, and I never really built anything in Mexico or in Cuba. This time, I became a Cuban, Mexican and Southern united states powerhouse.

I always used to keep all of my lines connected to each other. But this time, I kept looking for opportunities elsewhere, and building independent networks there. This helped to keep congestion on my lines down, and let to great profits.

I also would never build a line to a city that was already occupied by one of my competitors. But this time, I decided that big cities were lucrative enough to share. Having a large population center as an origin and destination for passenger service was hugely helpful. The main problem with a small rail network is that passengers can’t get to popular destinations, and short routes don’t pay  much.

Eventually, I expanded all over the map in a spidery way, sharing rail service with my competitors in many ways, and setting up San Francisco to Miami routes, or even Mexico City (all mine) to New York City (competitive).

Making money from passenger service, and providing restaurants and bars at stations, as well as eventually putting dining cars on your passenger trains was the easy part. The difficult, even baroque part of making money in the North American scenario is the only way real railroad companies make money now: hauling freight.

There’s a lot of considerations when hauling freight. You have to study and understand the entire industrial model in the game, and try to figure out how much demand there is for ingredients, and how degraded your rail network will be from traffic.

You will end up with main interstate rail routes which get bogged down. After all, you can’t really make a highway for trains, and your freight routes will be competing with and slowing  down your time critical passenger routes.

Milk can’t sit around in a train car for very long, but steel can, and eventually, making stuff gets vastly complicated as steel, canning facilities, bauxite mines and many more enterprises have to be connected.

A steel mill needs coal and iron, and often the nearest coal is in the appalachians while the mill is in mexico city and the oil is in texas. This can take a lot of time and coordination, and after steel mills show up, the complexity begins to ramp up even more (automobile plants, rubber, aluminum, etc.

All of this complexity gives this game a lot of sandboxy realism, but it lead to the collapse in my ability to optimize or even maintain my railroad empire. At one point, I had over 160 trains to keep track of, repair, maintain and replace, and in addition I had to determine where else to expand, and what freight or passenger areas were underserved, and where line congestion was leading to big problems.

I would find out that a station was not sending out its raw materials in a timely fashion, and then order two more trains to make deliveries, only to find out that the original problem was that the trains I had purchased to make deliveries in the past were getting stuck in the middle of massive traffic snarls. Added more trains just added to the problem.

Eventually I sat down, paused the game for thirty minutes and managed to pare down the number of trains to around 120. I began to act like a modern manager, getting rid of trains who showed a negative lifetime profit. But this had knock on effects, in that often the unprofitable trains were supplying intermediate goods for more profitable manufactured goods transporters.

It did help overall though, as the huge snarls around my main hub in mexico city started to easy. I had so many steel mills in mexico, and so much need for coal and oil from elsewhere, as well as a steady thrum of produce, coffee and passenger travel, that the whole Mexican region was in a state of lockdown, as more trains kept pouring into the region, while those that needed to get out struggled in the traffic jam.

One thing my massive corporate empire did show me is that Mexico City can be an excellent hub for all kinds of railroad traffic. The main problem is that a lot of the routes through mexico have to deal with steep grades. Steep grades can cripple travel speed for all of the 1800s trains, especially the earliest ones.

One of my classic tricks, which I used in this case as well, is to take all of the money your player has collected from CEO pay, sell all of your shares in the corporation you’ve been running, and with that jumbo pile of money, start over with a new corporation which you own outright.

I started a new corporation, but this time, I had already exploited most of the easy opportunities on the map. I really didn’t have enough opportunities for the capital I had invested into the new company, so I doubt I would get a good return. On the other hand, as soon as the computer took over my old corporation, they started running routes like computers always do, and the bad AI shined through. They were losing money hand over fist by sticking to the basic routes that the AI used.

I think the lesson that the N/A scenario has is that as any corporation grows, the amount of  information that people at the top need in order to make wise decisions increases. But no one can handle an unlimited amount of information, and you can only delegate to a certain extent. Obviously, as a single person playing the game, you hit the limit fairly quick. I often set up a few stations and some new train routes, and then let them be, moving onto the next opportunity. I was the growth guy, but I couldn’t also pay attention to how well the current lines were working. As you delegate, you gain a more and more filtered view of actual events, but you have to delegate because there is simply too much information to consider.

It makes me wonder if an AI could be built to design and maintain these railroad networks in the artificial, simplified case of a video game. I tend to think it would take a highly evolutionary, experimental sort of AI to take on the challenge. But maybe something in graph theory would shed light on the trouble. Also, I’m sure today’s actual railroad companies have insight on this issue.

Anyway, I started thinking today of giving the N/A scenario another shot, but this time thinking about the optimal way to build and maintain a continent wide passenger-only railroad corporation. By constraining the design to just passengers and postal mail, I feel like I can test things more thorougly, and I believe with Railroad Tycoon’s design, avoiding low profit freight in pursuit of 19th century high profit passenger travel will give a network that is both less congested and more profitable than a freight and passenger combined network. And I think I can give the corporation a faster rate of return by not holding onto so much cash, like I did last time. I will borrow aggressively as well, because of the great rate of return on passenger line investments.

One of the things I want to discover on this blog in future posts, in the optimal way to design a passenger network, and the optimal set of train routes and station stops. How do you balance short distance trips with long distance trips. I’d like to find some information on the Railroad Tycoon II model to see if it seems at least semi-realistic in terms of where passengers want to go.

Some of the corporate management bits of the game are rusty. When my last corporation was making money hand over fist, the game wouldn’t let me provide a special dividend to get rid of some of my excess cash, like Microsoft has done. I was holding onto about 10 million in cash that I hadn’t found an investment opportunity for. 10 million dollars back in 1895 was a lot of money to be holding onto. I was making more money than I could plough into reinvestments or provide as dividends because of the too strict rules on dividend payouts.

So when the network gets to that stable point, I won’t be able to actually provide the kind of dividends you would expect a railroad to provide. I’ll just have to calclulate them afterwards based on the level of cash accumulation I obtain.

There is also a whole stock market side to the game that I’d like to play around with sometime, and if I master the passenger rail corporation, I’d like to tackle the much tougher freight only side of things. Maybe once I get a perfect passenger rail network, I’ll start a new freight only corp with a massive bundle of cash to see what I can do. The capital you get at the beginning of a new game isn’t enough to build a proper freight network.

 

MWC Pawn strength adjustment

Giving pawns 2 * neighboring pawns in attack strength gives them enough power to take on anything if clumped together. Pawns should be important, but no individual pawn should have 6 attack strength. It totally changed the test game I was playing with the new rules.

I will put up that game for later, but my new rule for pawn attack strength is: neighboring pawns + 1. Defense strength will still be neighbors + 2. This still gives two pawns with two neighbors each a combined attack strength of 6.

Medieval War Chess attack/defense strengths update

I decided to adjust the attack and defense strengths for medieval war chess. Previously, I had the general rule that the defense strength would be double the attack strength for a piece. Unfortunately, I think that makes the knights require either a queen and knight to defeat, or three pieces to defeat.

Since there are three actions in a turn, its reasonable to expect two pieces attacking together to defeat an enemy, but making a three piece attack requires the enemy sliding up to you ahead of time.

I don’t want to make defense too weak, because then, neither player would be able to be the first one to approach without getting wiped out.I also want to contrast the pieces more from one another. Since they don’t have their own movement styles, they kinda blend together right now.

Piece Attack Defense Reason
King 2 7 Defensive specialist piece.
Queen 6 6 Powerful but endangered by the other queen.
Knight 3 5 More defensive minded than the bishop
Bishop 5 3 The glass cannon
Pawn Neighboring pawns * 2 Neighboring pawns + 2 This is something of a radical redesign that gives pawns some real punch if they can group up. This also makes pawns slightly more sturdy.
Rook 0 2 I wanted to make sure rooks can’t be taken out by a King alone, as that would result in Kings going all over the map chasing down rooks and being unstoppable.

Medieval War Chess: game 1

Medieval War Chess is a chess variant that uses the same pieces and the same board but has entirely different rules. The idea is to capture the essence of medieval melees. The long range movement of chess pieces doesn’t really capture this. But the game moves slowly, because pieces can only move to one adjacent square per move, so each player gets 3 actions a turn, which can include moves, attacks or cannon attacks.

The notation for recording games should be pretty clear from my games. I’m using the chess position editor at http://www.apronus.com/chess/wbeditor.php  to visualize the games because that system allows moves that aren’t legal in standard chess.

I would like to test this game and tweak it, but for that to happen, I need to find someone else interested in playing. For now I’m just playing solitaire to explore the possibilities of the game.

The rules are listed at the bottom of this post.

1. c2-c3, d2-d3, b1-c2   — A new opening I haven’t tried before. Cannon attacks aren’t much of a worry yet because the rooks have little range to manuever with all of the pieces still on the back row. So why not develop the queenside knight and/or the queen itself.

1. …g7-g6, f7-f6, g8-g7 — Open up an attack on white’s a2 point. I think there are broadly two types of cannon attacks to consider. There’s the protected attack, where the rook takes one action to get along the open rank, file or diagonal, one action to destroy the targetted piece, and the last action to move back to where he was hiding before.

Alternatively, there is the unprotected attack, where the rook moves twice to get in position, and then attacks. Early in the game, these unprotected attacks often expose the rook to too much danger of being either destroyed by a counter-battery from the other sides rooks, or to being melee attacked. The rooks should be protected like the queen is in normal chess. They are extremely valuable and fragile.

In this case, black’s threat isn’t likely to work out, but it forces white to do something on the queen side again to defend against it.

2. d1-d2, d2-e3, c3-c4 — Defend the a2 pawn and advance the queen into a more aggressive position. Be careful with your queen, but black hasn’t really presented any melee power yet, so the white queen may be able to do something.

2. … b7-b6, c7-c6, b8-b7 — Black does the same thing on the queen side that he did on move one. He is slowly advancing the black knights, but very slowly.

3. g2-g3, g1-g2, g2-f3 — Protect the h2 pawn and move the king’s knight forward. White is going for a melee assault plan.

3. …b7-a6, a6-b5, b5 attack c4 — The c4 pawn was exposed without a strong counterattack threat. Easy pickings. 2 c3-c4 was a blunder perhaps.

4. e3-f4, f4-g5, g5 attack g6 — White leaves the a2 pawn open to cannon attack in order to advance the queen and attack a black pawn. This is risky, but it doesn’t seem like black can take the white queen without preparation.

4. …h8-g8, g8 cannon a2, g8-h8 — White’s queen is threatening Black’s kingside rook, so he has to hide it back away in the corner after attacking, but this situation is dangerous for black as well as white because of that threat.

5. a1-b1, h1-g1, h2-g2 —  Both of white’s rooks as well as the h2 pawn were under the threat of cannon attack. So white has to play defensively.

5. … a8-b8, b8 cannon g3, c8-c7 — White anti-cannon defense didn’t quite work. White now has only 5 pawns left to black’s 7. Cannons become very powerful as the board empties.

6. g5-g6, g6 attack h7, g6-g5 — White has to avoid the h row, which is a free firing lane for the h rook, and the threat of bishop and knight together which can overpower the queen. Black’s rook is now in the hot seat. Black has to drive off the black queen.

6. .. d8-d7, c6-c5, h8-g8 — Black offers the trade of rook for queen, but protects his rook from cannon attack, while starting to get his queen in the game.

7. b1-a2, a2 cannon g8, a2-b1

7. …d7-e6, e6-f5, [f5, f6] attack g6.

8. c2-c3, d3-d3, b1 cannon f5 — This whole sequence surprised me. Every turn I thought I knew what was going on. I’m pretty much Mr. blunder at this game right now. White now has the lead with an extra rook.

8. … c8-c7, c7-c6, b8-c7 — Gets the queenside bishop and rook closer to the fight.

9. g2-h2, g1 cannon g7, h2-g2 — As pieces start to fall, a bad formation like black’s will fall apart. White’s formation is very defensible against cannons in contrast.

9. … f8-f7, f7-e6, d6-d5 — Black is pretty much lost, but trying to get pieces together for a final melee assault attempt.

10. b1-a1, a1 cannon a7, a1-b1

10. …c5-c4, d5-d4, c6-c5 — Preparing for a melee attack.

11. c3 attack c4, c3 attack d4, f3-d4 — This is strange, it makes me wonder if there should be some rule about how many melee attacks can happen in a turn.

11. … c7-d7, d7 cannon d2, d7-c8 — The loss of black’s pawns gave him range on the white pawn. Finding a safe place for the last black rook is getting harder thought. Finally black makes an aggressive action again.

12. c1-d2, d2-e3, f3-e4 — Knights feel a little too indestructible currently. Every other piece feels fragile, but I don’t know how anyone is supposed to lose their knights once the queens are off the board. Maybe I’ll adjust the knight defense to 7, so 2 knights can defeat 1 knight.

12. …c5-d6, c8 cannon c3, b5-c4 — good move on black’s part, but not enough material left, especially after his own knight falls

13. b1-c2, c2-d3, d3 cannon c4 — trading rook for knight.

13. … c8-d8, d6-c5, d8 cannon d3 — taking the rook with a bishop results in losing the bishop.

14. f2-g3, g1-f2, f2 cannon f6 — trading down pawns.

14. …d8-c7, c7 cannon g3, c7-d8

15. e2-f3, f1-e2, e1-f1

15. … e8-f7, c5-d6, b6-c7 .. defending against the threat of a cannon attack on his rook and trying to link up his 2 remaining pawns eventually.

Rook sight lines are a huge factor in restraining movement by both sides here.

16. e3-f4, e2-e3, e3-d4 — preparing for a melee assault, the queenside bishop is exposed, but the black rook would be lost if it was cannoned.

16. … d6-c6, d7-c8, d8 cannon d4

17. f2-e2, e2-d1, d1 cannon d8

17. … c7-d8, c6-d7, e6-d6

Its over, black doesn’t have the power to do anything, and the knights and bishops no longer are restrained since black’s rook is gone. They can march on, finishing off the rest of the black’s army with help from the white rook.

The Final position: http://www.apronus.com/chess/computer.htm?fen=02kp4X3bp3X4b3X8X4NB2X5P2X1P4P1X3R1K2_w_-_-_0_1

RULES OF THE GAME:

Chess Piece Summary
Piece Off. Strength Def. Strength Note
Queen 6 6 The Queen is the only unit without doubled defense strength
Rook 0 0 The Rook has the special cannon attack but no melee strength
Bishop 3 6 Bishops are a moderately powerful piece.
Knight 4 8 Knights are the best melee combatants.
Pawn Varies Varies Pawns have 1 off., 2 def. strength for every pawn surrounding them.
King 2 4 In version 1 of the rules, the King has special powers. He is mundane now.

Detailed Rules:

  1. Initial Setup:
    1. The game is played with a standard chess board and pieces
    2. Pieces are placed in the same positions are normal chess at the start of the game
    3. (Alternative) Players may instead agree to set up random positions according to the style of Fisher Random Chess or any other Chess position randomization scheme. Some starting positions may, however, be excessively imbalanced in favor of White
  2. Winning the Game:
    1. The game is won, if a player’s opponent has no pieces left on the board.
    2. The game is drawn, if neither player has enough strength on the board to take out any of their oppositions’ pieces, and there are no rooks left on the board.
  3. Turn Sequence:
    1. Players alternate taking turns, with white starting first, as in normal chess.
    2. Unlike in normal chess, during each turn, a player must make three legal actions
    3. An action consists of either: moving a piece, initiating a melee attack, or initiating a cannon attack.
  4. Movement:
    1. A movement action consists of moving any piece to an adjacent square that is not occupied by any other piece. Vertically, horizontally, and diagonally adjacent moves are all allowed.
    2. Another way of stating it, is that a piece’s (including pawns) movement action is like the King’s movement in normal chess.
    3. Enemy pieces can not be captured through movement. You can not move any piece onto a square that is already occupied by another friendly or enemy piece.
    4. You can use two or even three of your turn’s actions in order to move a single piece further, but in each action, the piece can only move like a king.
  5. Melee Attacks:
    1. Unlike movement, a melee attack is a collective action that will usually involve multiple pieces of yours.
    2. To start a melee attack, declare an enemy piece as your target.
    3. You must have at least one of your own pieces adjacent to the targetted enemy piece, in order to declare a melee attack on the target.
    4. All of your pieces adjacent to the target enemy join the offense.
    5. The target defends alone.
    6. In order to determine what happens during the attack, see the next section
  6. Resolving a Melee Attack:
    1. If an attack succeeds during resolution, the target piece is removed from the board.
    2. If an attack fails, nothing happens, which is a waste of an action.
    3. To determine whether the attack succeeds or fails, total up the power of the offense and compare it to the target’s power
    4. If the attackers have more power than the target, then the attack wins.
    5. Otherwise, if they have less or the same power, the attack fails.
    6. The power of the offense is the combined strength of every offensive piece involved in the attack. Piece strengths are listed in the next subsection.
    7. The power of the defense is the double the strength of the target, with one exception.
    8. Queens contribute their normal strength on defense, rather than double their strength.
  7. Piece Strengths:
    1. Queen: 6 offense, 6 defense. The queen is the only piece that doesn’t have doubled defensive strength
    2. Knight: 4 offense, 8 defense. The knight is the best piece on defense
    3. Bishop: 3 offense, 6 defense
    4. Rook: 0 strength. The rook is used for cannon attacks as covered in a later section. It has no inherent strength.
    5. Pawn: varies. Pawn strengths are covered in the next subsection.
  8. Pawn Strength:
    1. The strength of a pawn is based on the number of friendly pawns adjacent to it.
    2. If there are no pawns adjacent, then the pawn has 0 offensive, and 0 defensive strength.
    3. But if there are three pawns next to it, it has the same strength as a bishop, with 3 offense, 6 defense.
    4. In general, if there are x pawns adjacent to it, it has x offense, and 2x defense.
  9. Cannon Attacks
    1. A rook is the only piece that can execute a cannon attack.
    2. Like movement, and unlike melee attacks, Cannon attacks are individual piece actions.
    3. To start a cannon attack, a target enemy must be selected. The enemy must be on a vertical, horizontal, or diagonal line coming from the rook.
    4. This line between the rook and its target is called a ray. Only the part of the line between the Rook and its target is considered the ray.
    5. A cannon attack will fail if:
      1. There is another piece (friend or foe) between the rook and its target.
      2. There is another piece (friend or foe) after the target along the same ray.
      3. Essentially, if you start at the rook and follow the row, column or diagonal until there are no more squares on that ray, there shouldn’t be any pieces other than the rook and target. If there are any other pieces, the attack fails.
    6. If none of the conditions for failure occur, the attack has succeeded, and the target is removed from the board.
    7. A rook may make multiple cannon attacks per turn, as long as the player has a sufficient number of actions remaining in the turn.

Extensions of boolean operators and their closure

A topic I’ve been pondering lately is the extension of the And, Or, Not and Implication operators from the binary set 01 to the reals, and how that extension might be carried out.

This is something which is done in fuzzy logic, but the Fuzzy Logic extension, while following all of the axioms of Boolean operators, is useless for real-valued mathematics.

The fuzzy extensions:
a \wedge b = Max(a,b)
a \vee b = Min(a,b)
\neg a = 1-a

The Max and Min functions destroy any analytical power this system would have. There are some neat things in fuzzy logic, but for my purposes I have moved on to a different set of extensions which breaks a lot of the axioms, and does not confine values to [0, 1].

I don’t believe there is any alternative to the above which keeps all of the axioms of the Boolean system. Any continuous non-constant function for and or or will break some of the consequences of the axioms. So I decided to break them anyway and explore my alternate system for its properties.

My System of Extensions
a \wedge b = a * b
a \vee b = a + b - a * b
\neg a = 1- a

The and has its most basic continuous definition with multiplication, and the negation keeps the definition from fuzzy logic. To define the Or extension I use De Morgan’s laws.

De Morgan derivation of Or extension

\neg (\neg a \wedge \neg b) = 1 - ((1-a) * (1-b)) = 1 - (1 - a - b + ab) = a + b - ab

Using these operators you can create some polynomials but not all. In fact, it is impossible to express the function f(x, y) = x + y using these operators. So the next thing I was interested in is what set of polynomials can be expressed using this system.

And and Or both create polynomials of a larger degree than their operands. And Not is its own inverse, so it can only be applied once to a given polynomial to create another of the same degree.

Because Or is defined in terms of And and Not, we don’t need to consider it for the algebraic closure over these extensions.

Therefore the algebraic closure can be calculated by the following infinite process:

1. x is an element of the closure.
2. if m is an element and n are an element then m * n is an element.
3. if m is an element then 1 – m is an element

At first I thought it would be closed over the following set of polynomials:

(1-x)^m * x^n  or 1 – (1-x)^m * x^n

But the following polynomial construction shows why that set does not cover all of the elements in the closure.

-x^2 +1 = 1 - x^2 = 1 - x * x
Its constructible using  ( x is an element, x * x is an element, 1 – x * x is an element.).

-x^3 + x = x * (-x^2 + 1)
Its constructible from the above element, but it does not fit the form. Therefore the set formed above is not the closure.

I’m not sure what tools are used to find out things like this, so I haven’t made much progress, but it is an interesting question.