Battlestar Galactica, Game 2:
On Sunday Morning, I played Battlestar Galactica again. This game there wasn't nearly as much intrigue as the first one. However, at the beginning of the game, we got unlucky draws from the Crisis Deck and drew lots of cylon attack cards, so we were soon mobbed by Cylon ships and boarding parties. Then we got a crisis card that gave the Admiral (Helena Cain) a chance to take the title from the President (Laura Roslin). In this game, Roslin is actually one of the worst Presidents because she has to discard two skill cards every time she activates a location - and the President is going to be activating the "President's Office" location a lot to draw quorum cards. So I thought Cain made the correct move by taking it (there was no need to worry about her being a Cylon - it was a 4 player game, so there was only one cylon, who had already revealed himself). But Roslin wasn't very happy, and used her once-per-game ability to look at the top four quorum cards and play one in order to put Cain in the brig. Then later she spent more cards getting Cain out of the brig, and giving herself the President title back. All this distracted us while the Cylon was able to advance the boarding parties up the "Boarding Party Track" and get to the end to kill us.
Starmada:
This game is a starship battle game. The scenario was that there are two players, the attacker and the defender. The idea is that the defender is defending a "planet" (not actually on the game board, but supposedly off behind his side of the game board) and the attacker is trying to destroy it. The attacker had far more ships (1000 points worth of ships to the defender's 600) but the defender has time on his side - if the attacker doesn't kill the defender within 5 turns, the defender wins. In the first game I was the defender, and soon realized that if I engaged them head on I would be at a large disadvantage, and figured that the best option was to scatter, forcing the enemy to chase me down. The person running the game said that that option wasn't allowed, because the point was to "defend the planet" and the planet would be lost if I moved far away from it.
Him: If you just run away over to the other end of the board, the enemy ships will just ignore you and attack the planet, so you don't win.
Me: So you're saying that if the enemy reaches mu edge of the board, then I lose?
Him: No, it's just that you have to stay and defend, not run away.
Me: I'm still confused. I mean, if the planet is off my edge of the board, can I go sideways to the corner, and I'm still near the planet? Where exactly is the boundary that I have to stay inside?
Despite the fact that he was never explicitly clear on where the boundary was , I was still able to damage the enemy capital ship's engines, then back up each turn, making it so by the time his main capital ship got into range it was too late to make a difference.
We had time for another game so this time I was the attacker. I knew that I would have to strike hard and fast in order to win, so I chose ships that had powerful engines and also ships that had fighter wings (since fighters move fast and so they can close distance). This strategy worked - my fighters surrounded the enemy ships and too one of them out, then I hit his big capital ship with my powerful "spinal mount" weapon, and he conceded. My main comment on this game was that 5 turns seem way too short, because it takes 2-3 turns just to get in range to even shoot most of your weapons.
Miniatures Wargame:
I forget exactly what this game was called, but it was a miniatures game where there were "humans" fighting against "demihumans" like elves and dwarves. The backstory was that humans had invaded the demihumans' lands and cut down their trees, so they were fighting back. I played the humans. The way the game worked was interesting. There were a few kinds of units. Aside from the basic infantry, cavalry, and archers, there were the following special units:
1. Siege weapons, such as cannons. These have a "bowling ball" effect, meaning that during the "missile fire" phase when you fire them, you draw a line out to maximum range in the direction the weapon is pointing, and you made an attack roll against each model it contacts. Also, it has an interesting rule: if the first unit that the line touches is a unit that's engaged in melee combat, the shot has no effect. But if the first unit in the line is a unit not in melee, it will continue through to hit units in melee. I asked the person running the game if it could hit friendly units, and he said yes. The next turn there were lots of enemies lined up perfectly for me to shoot at, but they were all in melee. So I deliberately positioned the very corner of a friendly unit in the line of sight, hoping to sacrifice that one model to allow the shot to go through. He said that it wasn't allowed to do this (I think he interpreted my question as what happens if the friendly unit is one of the later units in the path, not if it's the first unit.) Also another thing is that the sequence of play is missile fire, then movement, then melee, then morale checks, and you aim your cannons during the movement phase. So during the movement phase you have to try to predict what is going to happen in the melee phase and who will be viable targets when the missile phase comes around again. Also sometimes you want your units to fail their morale checks, so they will back up and the enemy will be a legal missile fire target.
2. Spellcasters - each side has a wizard and a cleric. At the end of each turn, each spellcaster can cast one spell from a list, and they can cast it anywhere they can see. Also the cleric can resurrect any one dead model on the battlefield. That again makes movement interesting because you have to try to position your spellcaster so they can cast a spell at the target they want to cast. One of the mage's spells is a "teleportation" spell that will teleport a friendly unit to anywhere within the caster's lne of sight. One thing you have to remember to do is to avoid letting your enemy teleport their unit right next to your caster so they can kill your caster. One time my ally needed help on the other side of the battlefield so I carefully positioned my caster to have a line of sight over to the other end of the battlefield to teleport my unit there, but at the end of the turn I realized that I didn't have teleport, because I controlled the cleric, not the mage.
Anyway this was a fun game. It was 2 on 2 - each player controlled half their side's army. I controlled he right half and my ally controlled the left half. The battle was on and I successfully took control of the right flank, while the enemy pushed my ally back on the left flank, leaving his large archer unit exposed. However, strategic spell use on our part enabled us to block him from attacking that flank for a couple turns, letting me complete the conquest of the right flank and start to wheel around. As I did that, I realized the enemy had left his cleric in the open, and I ran him down. At this point, the enemy counted up the number of units each side had, realized they had no chance for comeback, and conceded.
Thursday, February 4, 2010
Monday, February 1, 2010
Winter War, part 1
Last weekend I went to Winter War, a gaming convention held every year in Champaign. There were games going on from Friday evening through Sunday evening so there were plenty of chances for exciting action. Here are the games I played:
Age of Conan: This game is a 4 player game set in the time of Conan, and each player controls a different kingdom and attempts to get victory points by conquering other provinces, completing special objectives, collecting gold, and taking control of Conan on his adventures in order to collect "adventure tokens" representing the three types of rewards - monsters, treasures, and women. At the beginning of the game, when our armies weren't big enough to conquer anything, I took the initiative to focus on collecting adventure tokens and get ahead in that area. Most of the other players tried instead to send out emissaries to ally with nearby provinces in order to acquire gold. In this game whenever you try to ally with or conquer a province, there is a die roll to see if you succeed, and there are "sorcery tokens" that let you reroll die rolls. One thing that was not clear from the rules is whether or not if you use a sorcery token and don't like the result of the reroll, you can use another sorcery token to reroll it again. The rules say that "only one sorcery token may be used per roll" but other players interpreted that as meaning that once you use a sorcery token to reroll, the reroll is a "different roll" so you can keep using sorcery tokens. This interpretation actually helped me at the beginning, as other players wasted lots of sorcery tokens on rolls that I could clearly calculate had very little chance of success. In the middle of the game, I used my adventure tokens to "bid" for an artifact, the Sword of Atlantis, that significantly improves offensive capability. I used this to conquer several territories and get ahead. Unfortunately near the end of the game, I overextended myself and got myself caught between two opponents, and got attacked from both sides and lost lots of territory. With time running out, I had an army stranded in enemy territory, and I had to strike out and make a desperation move to attack the nearest enemy territory hoping to conquer it. First I had to get rid of the enemy that was there to turn it neutral, then do a "campaign" to conquer it. I expected the initial siege to be a pushover because I had a bigger army and the Sword of Atlantis, but it didn't turn out that way. During the siege, there were three times where my opponent had a 1 in 27 chance of rolling well enough to kill one of my guys - and he succeeded two of those times. Fortunately I managed to get him, and with just two out of five soldiers left, completed the campaign on my last turn of the game, which gave me just enough victory points for the win.
Heroscape: This is a hex-based collectible miniature game where you build your army with points, and unfortunately this did not turn out as favorably. The mission was a 3-on-3 battle where there was a central castle in the middle that we were both competing for. There were six premade armies that we drew from, and the army I drew was the only army with no ranged attackers. The opposing team managed to get inside the castle and shut all four doors in the first couple turns, and then they could get on the castle walls and attack us with large height bonuses. We had no flying creatures, so the only way it would even be possible for us to retake the castle is to break down the doors, which have extremely high defenses (and of course to break down the doors you have to stand in front of them, out in the open, which is a very vulnerable position). Also my army started in the corner opposite from where all the action was, so by the time I was even able to get my army over to the action (remember, I had no ranged attackers, so I had to close to hand-to-hand distance) the battle was basically over (we lost).
Battlestar Galactica: This board game, based on the television series, features players as crew of the Battlestar Galactica trying to get to Kobol. However some players are secretly Cylons, robots disguised as humans that are programmed to destroy humanity. At the beginning of the game each player is secretly dealt a "loyalty card" indicating if they are human or Cylon, and then halfway through the game there is a "sleeper agent phase" where everyone gets another loyalty card, so you can think you are human and then turn into a Cylon. Each player chooses a different character from the show, and each character has three special abilities - one which can be used every turn, one which can be used only once per game, and one which limits them in some way. I chose Gaius Baltar, whose once-per-game ability is "Cylon Detector" which lets him look at any other player's loyalty cards. Also in this game there are two special titles - President and Admiral. The President gets control of the "quorum cards" which can help humanity deal with problems (or which can be used by a clever Cylon to sabotage them), and the Admiral gets control of Galactica's nuclear arsenal as well as choosing which of two destinations they jump to at each hyperspace jump. Also, some "crisis cards" come up that force the President or Admiral to make a decision. I started out as the President and Helena Cain started out as Admiral. Near the beginning a crisis card came up that forced both me and the Admiral to discard most of our "skill cards" and draw "treachery cards," a type of skill card that is useful mainly to Cylons. A couple turns later a crisis card came up that forced the Admiral to make a choice: either the Admiral and President each discard two skill cards, or the President has to give up his title to the Admiral. I argued that Cain should choose the first option: it's too dangerous to have all the power in the hands of one person if he turns out to be a Cylon, and discarding cards would give him a chance to prove his loyalty to humanity by discarding the treachery cards. I threatened to check him using my Cylon Detector:
Baltar: If you make me give up my Presidency, I'm going to check you. Normally I wouldn't use my power until the sleeper phase (because that way I get to see both loyalty cards rather than just one) but this seems really suspicious.
Cain: I have be best interests of humanity at heart. It's always a good idea to have the titles for yourself.
Baltar: Actually, let's see. I think I'll give you another chance. We're about to jump, so if you pick a 1-distance then I'll check you, otherwise maybe not. (Destination cards have "distance" values from 1 to 3 that indicate how far you've jumped. In general you want to jump as far as possible, so picking a low distance destination helps the Cylons.)
Another player: See, Baltar, you're not very decisive. I think it's good that we don't have you as President.
For the rest of the first half of the game, me and Cain want back and forth accusing each other of being Cylons. After the sleeper phase I checked her - "nothing personal," just that she could have gotten a cylon card and I want to make sure. Guess what, she was human all along. The game continued with a race to the finish line, and we eked out a victory.
The game is so fun that it makes me want to watch the show. Unfortunately, the DVD box set costs $250, so I'm not sure if that will happen...
(Part 2 coming soon!)
Age of Conan: This game is a 4 player game set in the time of Conan, and each player controls a different kingdom and attempts to get victory points by conquering other provinces, completing special objectives, collecting gold, and taking control of Conan on his adventures in order to collect "adventure tokens" representing the three types of rewards - monsters, treasures, and women. At the beginning of the game, when our armies weren't big enough to conquer anything, I took the initiative to focus on collecting adventure tokens and get ahead in that area. Most of the other players tried instead to send out emissaries to ally with nearby provinces in order to acquire gold. In this game whenever you try to ally with or conquer a province, there is a die roll to see if you succeed, and there are "sorcery tokens" that let you reroll die rolls. One thing that was not clear from the rules is whether or not if you use a sorcery token and don't like the result of the reroll, you can use another sorcery token to reroll it again. The rules say that "only one sorcery token may be used per roll" but other players interpreted that as meaning that once you use a sorcery token to reroll, the reroll is a "different roll" so you can keep using sorcery tokens. This interpretation actually helped me at the beginning, as other players wasted lots of sorcery tokens on rolls that I could clearly calculate had very little chance of success. In the middle of the game, I used my adventure tokens to "bid" for an artifact, the Sword of Atlantis, that significantly improves offensive capability. I used this to conquer several territories and get ahead. Unfortunately near the end of the game, I overextended myself and got myself caught between two opponents, and got attacked from both sides and lost lots of territory. With time running out, I had an army stranded in enemy territory, and I had to strike out and make a desperation move to attack the nearest enemy territory hoping to conquer it. First I had to get rid of the enemy that was there to turn it neutral, then do a "campaign" to conquer it. I expected the initial siege to be a pushover because I had a bigger army and the Sword of Atlantis, but it didn't turn out that way. During the siege, there were three times where my opponent had a 1 in 27 chance of rolling well enough to kill one of my guys - and he succeeded two of those times. Fortunately I managed to get him, and with just two out of five soldiers left, completed the campaign on my last turn of the game, which gave me just enough victory points for the win.
Heroscape: This is a hex-based collectible miniature game where you build your army with points, and unfortunately this did not turn out as favorably. The mission was a 3-on-3 battle where there was a central castle in the middle that we were both competing for. There were six premade armies that we drew from, and the army I drew was the only army with no ranged attackers. The opposing team managed to get inside the castle and shut all four doors in the first couple turns, and then they could get on the castle walls and attack us with large height bonuses. We had no flying creatures, so the only way it would even be possible for us to retake the castle is to break down the doors, which have extremely high defenses (and of course to break down the doors you have to stand in front of them, out in the open, which is a very vulnerable position). Also my army started in the corner opposite from where all the action was, so by the time I was even able to get my army over to the action (remember, I had no ranged attackers, so I had to close to hand-to-hand distance) the battle was basically over (we lost).
Battlestar Galactica: This board game, based on the television series, features players as crew of the Battlestar Galactica trying to get to Kobol. However some players are secretly Cylons, robots disguised as humans that are programmed to destroy humanity. At the beginning of the game each player is secretly dealt a "loyalty card" indicating if they are human or Cylon, and then halfway through the game there is a "sleeper agent phase" where everyone gets another loyalty card, so you can think you are human and then turn into a Cylon. Each player chooses a different character from the show, and each character has three special abilities - one which can be used every turn, one which can be used only once per game, and one which limits them in some way. I chose Gaius Baltar, whose once-per-game ability is "Cylon Detector" which lets him look at any other player's loyalty cards. Also in this game there are two special titles - President and Admiral. The President gets control of the "quorum cards" which can help humanity deal with problems (or which can be used by a clever Cylon to sabotage them), and the Admiral gets control of Galactica's nuclear arsenal as well as choosing which of two destinations they jump to at each hyperspace jump. Also, some "crisis cards" come up that force the President or Admiral to make a decision. I started out as the President and Helena Cain started out as Admiral. Near the beginning a crisis card came up that forced both me and the Admiral to discard most of our "skill cards" and draw "treachery cards," a type of skill card that is useful mainly to Cylons. A couple turns later a crisis card came up that forced the Admiral to make a choice: either the Admiral and President each discard two skill cards, or the President has to give up his title to the Admiral. I argued that Cain should choose the first option: it's too dangerous to have all the power in the hands of one person if he turns out to be a Cylon, and discarding cards would give him a chance to prove his loyalty to humanity by discarding the treachery cards. I threatened to check him using my Cylon Detector:
Baltar: If you make me give up my Presidency, I'm going to check you. Normally I wouldn't use my power until the sleeper phase (because that way I get to see both loyalty cards rather than just one) but this seems really suspicious.
Cain: I have be best interests of humanity at heart. It's always a good idea to have the titles for yourself.
Baltar: Actually, let's see. I think I'll give you another chance. We're about to jump, so if you pick a 1-distance then I'll check you, otherwise maybe not. (Destination cards have "distance" values from 1 to 3 that indicate how far you've jumped. In general you want to jump as far as possible, so picking a low distance destination helps the Cylons.)
Another player: See, Baltar, you're not very decisive. I think it's good that we don't have you as President.
For the rest of the first half of the game, me and Cain want back and forth accusing each other of being Cylons. After the sleeper phase I checked her - "nothing personal," just that she could have gotten a cylon card and I want to make sure. Guess what, she was human all along. The game continued with a race to the finish line, and we eked out a victory.
The game is so fun that it makes me want to watch the show. Unfortunately, the DVD box set costs $250, so I'm not sure if that will happen...
(Part 2 coming soon!)
Saturday, January 23, 2010
Superheroes, classes, and math, part 2
So after the first week of classes, and the character creation session for the superhero game, here is what has happened:
- I don't have anything particularly interesting to say about Stephen Bond yet.
- Steve LaValle is kind of a funny professor. The first day in class he said that "normally I try not to stay too close to the book, but in this case I am kind of confused as to whether I should do that, because I wrote the book." Also he had us look through the book and vote (by email) on which chapters we want him to focus on for the class. Not everyone turned their vote in on time, so he said that "just like real elections, we have low voter turnout here" and joked that "actually, I'll just ignore the voting and do whatever I want, just like the real government."
- Jeff Erickson called me a "great graduate student" on his web page.
- At Armored Gopher Games we had our character creation session for the superhero game. I had to take a cab there because my bike got a flat tire. But it was really cool and people had lots of funny superhero ideas. One of the superhero ideas was an alien who was given superpowers and sent to Earth to live among the humans as part of a "reality show." One of the character's disadvantages is that occassionally the "producers" will teleport him to a "more interesting" location, or teleport him back to the mothership to alter his powers. My idea was a "gadgeteer" type character who has an army of robotic UAVs (unmanned aerial vehicles) that will help him - like a "combat" one that has powerful attacks and a "scout" one that has stealth and detection powers. Dave said that this was a good idea for a character, but warned me that the focus of the game was on the heroes, and if the robots took up too much of the spotlight, he would find ways of disabling them.
- I don't have anything particularly interesting to say about Stephen Bond yet.
- Steve LaValle is kind of a funny professor. The first day in class he said that "normally I try not to stay too close to the book, but in this case I am kind of confused as to whether I should do that, because I wrote the book." Also he had us look through the book and vote (by email) on which chapters we want him to focus on for the class. Not everyone turned their vote in on time, so he said that "just like real elections, we have low voter turnout here" and joked that "actually, I'll just ignore the voting and do whatever I want, just like the real government."
- Jeff Erickson called me a "great graduate student" on his web page.
- At Armored Gopher Games we had our character creation session for the superhero game. I had to take a cab there because my bike got a flat tire. But it was really cool and people had lots of funny superhero ideas. One of the superhero ideas was an alien who was given superpowers and sent to Earth to live among the humans as part of a "reality show." One of the character's disadvantages is that occassionally the "producers" will teleport him to a "more interesting" location, or teleport him back to the mothership to alter his powers. My idea was a "gadgeteer" type character who has an army of robotic UAVs (unmanned aerial vehicles) that will help him - like a "combat" one that has powerful attacks and a "scout" one that has stealth and detection powers. Dave said that this was a good idea for a character, but warned me that the focus of the game was on the heroes, and if the robots took up too much of the spotlight, he would find ways of disabling them.
Tuesday, January 19, 2010
Superheroes, classes, and math
Today is the first day of classes at UIUC. I didn't have any classes today because all my classes are on Mondays, Wednesdays, and Fridays. In addition to working as a research assistant for Jeff Erickson, I am also taking two courses: Planning Algorithms with Steven LaValle and Numerical Analysis with Stephen Bond.
Additionally, on the gaming front, I'm about to start playing in a new role playing game campaign run by Dave, the owner of Armored Gopher Games. The game is going to be a superhero game, based on a rule set called the HERO System. The unique part about the HERO System is that unlike games such as Dungeons and Dragons where you have to pick from a given list of powers, in the HERO System there is a point system where you spend "character points" to create your own powers by picking from a list of effects (like "ranged killing attack", "energy blast," "entangle," and "transform,") and add in "advantages" and "limitations" to improve or restrict your powers (such as "armor piercing," "limited number of charges," or "reduced endurance cost.") As you can probably guess, there is a lot of math involved in this, which is why I'm excited about it. For example here is an actual conversation about one of the powers in the game, which allows you to have a group of "followers."
Me: "So, if you build your follower as an N-point character, you can have (2^k) of those followers for a point cost of (N/5)+5k, right?" (I am writing this formula on the whiteboard)
Dave: "That could be right, I don't know. You're talking in a foreign language to me. That's math. I don't speak math."
Me: (Pointing to a page in the rule book) "It says in the rule book that the follower costs 1 point for every 5 points it's built on, and you can double the number of followers by spending 5 points extra. Like it says in this example, if the follower is a 200-point character, he costs 40 points, and you can get two of him for 45, or four of him for 50, etc."
Dave: "Yes that sounds about right."
Anyway, we're going to be finalizing our characters on January 23rd, and then the actual game is going to start the second Saturday in February. I will post more on my blog about any more superhero action!
Additionally, on the gaming front, I'm about to start playing in a new role playing game campaign run by Dave, the owner of Armored Gopher Games. The game is going to be a superhero game, based on a rule set called the HERO System. The unique part about the HERO System is that unlike games such as Dungeons and Dragons where you have to pick from a given list of powers, in the HERO System there is a point system where you spend "character points" to create your own powers by picking from a list of effects (like "ranged killing attack", "energy blast," "entangle," and "transform,") and add in "advantages" and "limitations" to improve or restrict your powers (such as "armor piercing," "limited number of charges," or "reduced endurance cost.") As you can probably guess, there is a lot of math involved in this, which is why I'm excited about it. For example here is an actual conversation about one of the powers in the game, which allows you to have a group of "followers."
Me: "So, if you build your follower as an N-point character, you can have (2^k) of those followers for a point cost of (N/5)+5k, right?" (I am writing this formula on the whiteboard)
Dave: "That could be right, I don't know. You're talking in a foreign language to me. That's math. I don't speak math."
Me: (Pointing to a page in the rule book) "It says in the rule book that the follower costs 1 point for every 5 points it's built on, and you can double the number of followers by spending 5 points extra. Like it says in this example, if the follower is a 200-point character, he costs 40 points, and you can get two of him for 45, or four of him for 50, etc."
Dave: "Yes that sounds about right."
Anyway, we're going to be finalizing our characters on January 23rd, and then the actual game is going to start the second Saturday in February. I will post more on my blog about any more superhero action!
Tuesday, December 8, 2009
Gaming Math - Problem 12
Problem 12: Energy Crisis
In the collectible card game "Magic", players get "land" cards which they can play to generate "mana," which represents magical energy that can be used to play other cards which represent magical spells. Each land card can be "tapped" for one mana per turn, and different cards cost different amounts of mana to play. If you get lots of cards but not enough mana to play any of them that is called being "mana screwed." Unhinged, a "joke" Magic card set, contains a card called "Mana Screw", which allows the user to (as many times as he wants to) spend one mana to flip a coin, and if he wins the coin flip he gets two mana (for a net gain of one.)
Problem 12.1:
Suppose that a player has X mana, and he has a spell that he wants to play that costs Y mana (where Y is greater than X). He uses Mana Screw's power repeatedly until he either (a) runs out of mana, or (b) gets enough mana to cast his spell. What is the probability that he will get enough mana to cast the spell.
Problem 12.2:
There exists another Magic card called Krark's Thumb (this is not a joke card) that allows the user to, whenever he is called upon to flip a coin, flip two coins and ignore one. Thus if a player used Krark's Thumb he would have a 3/4 chance of winning each coin flip, rather than 1/2. Suppose that such a player had one mana, and wanted to play Gleemax, another "joke" card that costs 1,000,000 mana. He again continually used Mana Screw's power until he either ran out of mana or got enough to play Gleemax. What is the probability of success? (You can take the limit as Gleemax's cost approaches infinity.)
(Hint for both problems: Let f(x) be the probability of success starting from x mana, and find a recurrence relation for x.)
The solution is here.
In the collectible card game "Magic", players get "land" cards which they can play to generate "mana," which represents magical energy that can be used to play other cards which represent magical spells. Each land card can be "tapped" for one mana per turn, and different cards cost different amounts of mana to play. If you get lots of cards but not enough mana to play any of them that is called being "mana screwed." Unhinged, a "joke" Magic card set, contains a card called "Mana Screw", which allows the user to (as many times as he wants to) spend one mana to flip a coin, and if he wins the coin flip he gets two mana (for a net gain of one.)
Problem 12.1:
Suppose that a player has X mana, and he has a spell that he wants to play that costs Y mana (where Y is greater than X). He uses Mana Screw's power repeatedly until he either (a) runs out of mana, or (b) gets enough mana to cast his spell. What is the probability that he will get enough mana to cast the spell.
Problem 12.2:
There exists another Magic card called Krark's Thumb (this is not a joke card) that allows the user to, whenever he is called upon to flip a coin, flip two coins and ignore one. Thus if a player used Krark's Thumb he would have a 3/4 chance of winning each coin flip, rather than 1/2. Suppose that such a player had one mana, and wanted to play Gleemax, another "joke" card that costs 1,000,000 mana. He again continually used Mana Screw's power until he either ran out of mana or got enough to play Gleemax. What is the probability of success? (You can take the limit as Gleemax's cost approaches infinity.)
(Hint for both problems: Let f(x) be the probability of success starting from x mana, and find a recurrence relation for x.)
The solution is here.
Monday, November 30, 2009
Gaming Math - Problem 11
At the board game night no Sundays at the local game store there is guy who has been coming in to playtest a new card game he invented. The game is based off of computer fighting games like Mortal Kombat. The way it works is as follows: Each player chooses a character, and each character has a deck of "attack cards" and "defense cards." The players take turns attacking and defending (so the first turn player 1 attacks and player 2 defends, then the second turn player 2 attacks and player 1 defends, etc.) The sequence of play in each turn goes as follows:
(1) The attacker draws attack cards until he has 5 attack cards in his hand; the defender draws defense cards until he has 3 defense cards in his hand.
(2) The attacker plays attack cards. Each attack card has three attributes: a "range" (high, medium, or low), a numerical "attack value" indicating how much damage the attack does, and a "level" (1, 2, or 3). The attacker can either play any one attack card as a basic attack, or he can play a combo. To play a combo the first card in the combo has to be level 1, then the second level 2, then the third (if it's there) level 3 (you can play either a 2 or 3 attack combo)
(3) The defender tries to block the attack. Each defense card is either a "High Block", "Medium Block", or "Low Block." In order to block the attack he has to play a block card that matches the range of the attack. If there is a combo attacks must be blocked in order. For example if it's a 3 hit combo, and he blocks the first and second attacks, the third attack still does its damage. If he doesn't block the first attack because he doesn't have a matching block card, he takes damage from all the attacks, and he can't block the second or third attacks even if he had a matching block card.
(4) The attacker can discard attack cards he doesn't want, and the defender can discard defense cards he doesn't want.
(In the actual game there are more complicated elements, like "counters" that let you turn your enemy's attack back on himself, "special moves" that let you power up your regular attack cards with special powers, and certain characters have special rules, etc. But for the purpose of this problem we are just using the elements above.)
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Problem 11: You've Been Blocked
Consider the game above. Assume that the defense deck has an equal quantity of high, medium, and low block cards, and ignore finite deck size effects (i.e. assume that each time you draw a card from the defense deck, you have a 1/3 chance of getting each of the types of cards, regardless of what cards have already been drawn).
Problem 11a. If the defender has just drawn a new hand of three cards, what is the probability that he will successfully be able to block a 3 hit combo where each of the attacks has a different range? How about if all the attacks have the same range (e.g. 3 medium attacks)?
Problem 11b. Suppose the defender draws a new hand of three cards. What is the probability that if I attack with a single attack (e.g. a medium attack) he will be able to block it?
Problem 11c. Suppose I've just attacked with a single medium attack, he blocks it (thus using up the medium block card). If I attack him with another medium attack on his next turn (don't forget he gets to draw a new card to replace the block card he used up) what's the probability he will be able to block it? (Assume that the defender will always block an attack if he has a block card available.)
The solutions are here.
(1) The attacker draws attack cards until he has 5 attack cards in his hand; the defender draws defense cards until he has 3 defense cards in his hand.
(2) The attacker plays attack cards. Each attack card has three attributes: a "range" (high, medium, or low), a numerical "attack value" indicating how much damage the attack does, and a "level" (1, 2, or 3). The attacker can either play any one attack card as a basic attack, or he can play a combo. To play a combo the first card in the combo has to be level 1, then the second level 2, then the third (if it's there) level 3 (you can play either a 2 or 3 attack combo)
(3) The defender tries to block the attack. Each defense card is either a "High Block", "Medium Block", or "Low Block." In order to block the attack he has to play a block card that matches the range of the attack. If there is a combo attacks must be blocked in order. For example if it's a 3 hit combo, and he blocks the first and second attacks, the third attack still does its damage. If he doesn't block the first attack because he doesn't have a matching block card, he takes damage from all the attacks, and he can't block the second or third attacks even if he had a matching block card.
(4) The attacker can discard attack cards he doesn't want, and the defender can discard defense cards he doesn't want.
(In the actual game there are more complicated elements, like "counters" that let you turn your enemy's attack back on himself, "special moves" that let you power up your regular attack cards with special powers, and certain characters have special rules, etc. But for the purpose of this problem we are just using the elements above.)
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Problem 11: You've Been Blocked
Consider the game above. Assume that the defense deck has an equal quantity of high, medium, and low block cards, and ignore finite deck size effects (i.e. assume that each time you draw a card from the defense deck, you have a 1/3 chance of getting each of the types of cards, regardless of what cards have already been drawn).
Problem 11a. If the defender has just drawn a new hand of three cards, what is the probability that he will successfully be able to block a 3 hit combo where each of the attacks has a different range? How about if all the attacks have the same range (e.g. 3 medium attacks)?
Problem 11b. Suppose the defender draws a new hand of three cards. What is the probability that if I attack with a single attack (e.g. a medium attack) he will be able to block it?
Problem 11c. Suppose I've just attacked with a single medium attack, he blocks it (thus using up the medium block card). If I attack him with another medium attack on his next turn (don't forget he gets to draw a new card to replace the block card he used up) what's the probability he will be able to block it? (Assume that the defender will always block an attack if he has a block card available.)
The solutions are here.
Wednesday, November 25, 2009
Trip to Chicago
Since there are no classes this week since it is Thanksgiving, I decided to go up to Chicago to see some sights there.
I got to the Illinois bus terminal on Monday morning and bought a ticket for the 11:00 Greyhound bus up to Chicago. Unfortunately there wasn't enough space on the bus for all of us so they had to bump me to the 12:00 bus. I knew planes could be overbooked but I had never heard of a bus being overbooked. Anyway on the way up there the bus broke down. Fortunately there was another Greyhound bus there to rescue us. By the time I had gotten up there and checked into the hotel it was already 4:00, and most of the attractions were closed. Instead I took the subway over to downtown here I could have dinner at the ESPN Zone and explore the rest of downtown where there's lots of shopping. I got a new winter coat (my old one was really worn out) and got a cool looking miniature RC helicopter from Brookstone. Also in Chicago there was a whole store that sold nothing but Legos. Here is a picture of the display outside:
All those things are made out of Legos. On Tuesday I went to the Museum of Science and Industry. There were some cool things there - in one of the activities we got the chance to use a "Human Patient Simulator" medical training dummy of the same kind as those actually used in medical schools. The way it worked was they showed us how to check for different vital signs (pulse, heart rate, breathing etc.) and then we were assigned to go up there, check the vital signs, and diagnoes the patient by matching the results to the results on a diagnosis chart we were given. Also while I was there I ran into Courtney Fontaine, one of the people that I took the comedy class from back at ComedySportz in DC. She now works at the science museum, and is also in another comedy group (not ComedySportz, although there is a ComedySportz in Chicago). She was having a performance that night and I went to see it. There were several comedy groups performing that night, and there were lots of funny parts. Courtney's group got the suggestion "gift-wrapping", and I thought it would be a boring suggestion but they went off on lots of funny tangents with it. The last group I gave the suggestion "Dungeons and Dragons" to, and I was actually quite impressed with their knowledge of D+D themes - they incorporated lots of classic D+D elements including traps, castles, wizards, and even "elemental powers" into their routine. The climax of the scene was funny - someone was talking about going to the "center" of a castle to find treasure, but someone else misheard it as "senator", and the scene then segued into a senator on a talk show whose shtick was "gratuitous profanity" - but since this is a family friendly blog I won't go into it any further.
Then on Wednesday, I first went to the Travisa office to drop off the materials for my trip to India, then went back to get my stuff from the hotel. There wasn't really enough time to go to another attraction so I just went to the train station (I figured I might have better luck on the train rather than the bus) and got there about an hour and a half before the train was scheduled to leave. It was a good thing I got there early because I got the last available ticket to Urbana-Champaign. The train left on time and only got back to Urbana-Champaign about 20 minutes after the scheduled time.
I got to the Illinois bus terminal on Monday morning and bought a ticket for the 11:00 Greyhound bus up to Chicago. Unfortunately there wasn't enough space on the bus for all of us so they had to bump me to the 12:00 bus. I knew planes could be overbooked but I had never heard of a bus being overbooked. Anyway on the way up there the bus broke down. Fortunately there was another Greyhound bus there to rescue us. By the time I had gotten up there and checked into the hotel it was already 4:00, and most of the attractions were closed. Instead I took the subway over to downtown here I could have dinner at the ESPN Zone and explore the rest of downtown where there's lots of shopping. I got a new winter coat (my old one was really worn out) and got a cool looking miniature RC helicopter from Brookstone. Also in Chicago there was a whole store that sold nothing but Legos. Here is a picture of the display outside:
All those things are made out of Legos. On Tuesday I went to the Museum of Science and Industry. There were some cool things there - in one of the activities we got the chance to use a "Human Patient Simulator" medical training dummy of the same kind as those actually used in medical schools. The way it worked was they showed us how to check for different vital signs (pulse, heart rate, breathing etc.) and then we were assigned to go up there, check the vital signs, and diagnoes the patient by matching the results to the results on a diagnosis chart we were given. Also while I was there I ran into Courtney Fontaine, one of the people that I took the comedy class from back at ComedySportz in DC. She now works at the science museum, and is also in another comedy group (not ComedySportz, although there is a ComedySportz in Chicago). She was having a performance that night and I went to see it. There were several comedy groups performing that night, and there were lots of funny parts. Courtney's group got the suggestion "gift-wrapping", and I thought it would be a boring suggestion but they went off on lots of funny tangents with it. The last group I gave the suggestion "Dungeons and Dragons" to, and I was actually quite impressed with their knowledge of D+D themes - they incorporated lots of classic D+D elements including traps, castles, wizards, and even "elemental powers" into their routine. The climax of the scene was funny - someone was talking about going to the "center" of a castle to find treasure, but someone else misheard it as "senator", and the scene then segued into a senator on a talk show whose shtick was "gratuitous profanity" - but since this is a family friendly blog I won't go into it any further.Then on Wednesday, I first went to the Travisa office to drop off the materials for my trip to India, then went back to get my stuff from the hotel. There wasn't really enough time to go to another attraction so I just went to the train station (I figured I might have better luck on the train rather than the bus) and got there about an hour and a half before the train was scheduled to leave. It was a good thing I got there early because I got the last available ticket to Urbana-Champaign. The train left on time and only got back to Urbana-Champaign about 20 minutes after the scheduled time.
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