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The math of it all

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The Magician
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In the light of the math of

In the light of the math of it all, while math may not be necessary in game design or art or music. Mathematics are vary beautiful. It is the launguage to describe reality and life, not the creator of it, the language. In a great musical composition, we hear math in our ears and it creates order and harmony in our brain. In great art we see it without being aware of it. Same in games, subonsciously the math sings in beautiful mechanics.

Mitchell Allen
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Math, Mazes, Prototypes, Oh My!

Rick-Holzgrafe wrote:
Mitchell Allen wrote:
seo wrote:

A better starting point for automatic generation of perfect mazes would be [this](http://www.mazeworks.com/mazegen/mazetut/index.htm).

Thanks for sharing this resource! Years ago, I wrote a maze game that used an unnecessarily complex algorithm (I didn't have a clue, really). The depth-first search is elegant. I'll finally be able to add mazes to my board game maker software.

I had to smile, Mitch! I also wrote a maze game once, and also had to design my own maze-generation algorithm. Mine is similar to what I saw at the site Seo posted, but not identical. In mine, the first pass only creates the "true path" from the entrance to the exit. A second pass then fills in the forks and dead ends. I found that to get really good mazes, I had to tune it a bit: if the true path wanders and curves too much, it fills the space leaving room for few false paths, making the maze too easy to solve. But if it doesn't wander enough, the solution is too short and obvious. I got the best results by having the true path favor going straight rather than turning a corner.

What else does your program offer? It sounds interesting!

Ah, the joys of reinventing the wheel. My maze game started with a grid of solid blocks, which my algorithm randomly blasted to dust. Then, it picked two random edge blocks, blasted them and tried to make a path.
Finally, my little monster was dropped into the maze and tasked with finding its way out.

I wrote this, not as a game, per se, but as a way to experiment with AI. I created what I thought was a pretty clever Darwinian engine for the monster, who always started out dumb as dirt, bumping into walls. Eventually, it collected enough sense to "see" the walls and go in the direction of an opening. It had no recollection of revisiting, though, so it was always pure luck when it got out. :)

My prototyping program generates grid-based boards, which can contain several layers, such as images, text, fill colors and fill patterns. It can generated different cell sizes and can draw circles or boxes with or without connectors and with or without gaps. Cells can be turned off, creating many different effects, such as a maze :)

If you check out Haulin' Assets, my August GDS entry, you'll see a board that I created and then fleshed out in Photoshop.
(The skeleton is the track, the flesh is the stylized graphics. Sometimes it's easier to import graphics into Photoshop so that you have more control than you do with the prototyping tool.)
I'm going to upload it to my website either today or tomorrow.

Cheers,

Mitch

The Magician
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Prototype program?

@Mitch
Interesting discussion about mazes! You mention that your created a prototyping program for game boards? I am interesting in checking it out. Do you mind sharing? Is there a download?

Mitchell Allen
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Prototyping Program

Hey Magician,

Yep! I have a free version in perpetual beta :)
No mazes in this version, but now that I have the inside scoop :) ...
I just spent the past two days building a setup manager for the program. I have a butt-ugly website that I was hoping would be updated before this latest beta was released.

Let me double-check the setup files and then I'll be happy to post the links.
Hmmm... I thought I edited this last night. Oh Well, here's the link:

http://www.morphoboard.com/downloads.html

Thanks for asking!

Cheers,

Mitch

The Magician
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THanks

Mitch, Thank you for the link. I downloaded the program and checked it out. It looks interesting. I won't know it's full potential though until I learn it. It will take me a bit to get used to. The interface seems a little awkward. I suspect it needs some work am I right? I was looking for a feature like "edit" undo, that I could cancel what I have just done.

The Magician
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math and algorithums

Jon gave a good example about using math to figure game board spaces. Lately I have been interested in two specific points:

*board game math

*board game algorithums

Maybe I'm asking the same question, but is there literature out there about game math for the board game inventor or something like that? Anywhere I can gain knowledge about where in board game design or how to apply math to it? Or, is it something that designers just have to figure out themself using math skills they know? When I look at big games that are pretty complex I wonder that the designer must have had to use a lot of math to handle such a complex system. This is a tool that I want to have with me.

The other thing is algorithums. For example when players are competing against the game cooperatively. I am looking for material about board game algorithums and how to design an enemy in the game that works on it's own through the game mechanics so that players can interact and compete with it. As said by some, I don't want to go the slow route of trial and error if I can make the process simpler for myself.

Mitchell Allen
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Magician

Hi Magician,
Thanks for trying out the software. The interface is tough - I'm studying Photoshop's for help.
Any helpful suggestions will be considered. Which part did you find awkward?

I will consider adding undo - in the meantime, save, save save :) each save is a different file.

Cheers,

Mitch

Mitchell Allen
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Back on Topic ...

The Magician wrote:
Jon gave a good example about using math to figure game board spaces. Lately I have been interested in two specific points:

*board game math

*board game algorithums

Maybe I'm asking the same question, but is there literature out there about game math for the board game inventor or something like that? ... As said by some, I don't want to go the slow route of trial and error if I can make the process simpler for myself.

A college-level text on probability and statistics stays near me at all times. I failed the course, so I NEED that book.

Essentially, a game is about choices. As the designer, you want to abstract or expose the decision-making process.
That can only be done if you've covered your bases - unless you're doing a game like Chess or Go!
Understanding probabilities can make your game interesting; understanding statistics can make it believable.

I used to shake my head whenever I read about the world championship events involving Scrabble or Monopoly. At that level, it wasn't about fun: it was about maximizing your chances. I understand that better now. Think about the various tournaments (chess, again, but also Poker) and notice how much of the strategy is about playing for percentages.

Cheers,

Mitch

The Magician
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Wow, it's snowing outside my

Wow, it's snowing outside my house and is building up. Strange weather here in the pacific northwest for this time of year.

Mitch, that's so funney. You remind me of myself when I was taking college cources. I failed algebra six times. The seventh time, I said "this is not my calling in life" and I haven't looked back at college since. And my life rocks! Anyway, I have a tendency many of the cources I failed in college, I have grown interest in certain subjects and have grown to understand them well through self interest. I guess it really never was about the piece of paper.

I think I am going to dig up my old math books and go through them with a highlighter. I know parts will stick out. I'll say "that could be used in my game designs." Since I started thinking about how card tricks can be linked to game mechanics, I swear, I don't know if it's just me but some games I play that use a lot of cards, the way certain cards play out almost seems like a trick. I want to discover a clear link between how clowns deal out cards in card tricks and making mechanics work for cards to create an interesting system in my games. I have this idea that if you instruct players to deal the cards in a certain way in game setup, some cool experiences could unfold in what cards come up for the player, and have it so they don't know why everything plays out a certain way. That is probably the basis of lots of games. It's so there, if you look at an illusionist performing a trick. What if some event happens early in the game, and magically an event late in the game happens that corresponse to a card played early in the game. And it blows the players mind how that happened. They couldn't see how this played out but it was unique to that particular game experience and was not something that happens over and over. I'm not sure if I'm being clear or making sense. I'm probably drifting a little.

Wow the snow is heavy!

The Magician
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Mitchell Allen wrote:Hi

Mitchell Allen wrote:
Hi Magician,
Thanks for trying out the software. The interface is tough - I'm studying Photoshop's for help.
Any helpful suggestions will be considered. Which part did you find awkward?

I will consider adding undo - in the meantime, save, save save :) each save is a different file.

Cheers,

Mitch


I will answer your questions sometime soon. I want to spend more time with it so I can give you a good list of things I find trouble with.

monica99
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Math sings

I totally agree with the math sings. 12 and the division of that is my language of creation...although 5 definitely throws everyone off guard that sounds observantly sweet to me!

The Magician
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monica99 wrote:I totally

monica99 wrote:
I totally agree with the math sings. 12 and the division of that is my language of creation...although 5 definitely throws everyone off guard that sounds observantly sweet to me!

Lol, I'm pretty illiterate in that language. I can't speak in much more than .0's and 618'es.

coco
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60

I like 60.

It can be divided by 2,3,4,5,6,10,12,15,20 and 30. So it can be used in games for 2,3,4,5,6 players. (for 7 you need 420)

If you take a chess board (8x8) and remove the 4 central squares or the 4 corners you get 60.

If you take an hexagonal board of 5 hexagons per side and remove the central one you get 60.

60 cards can be easily shuffled.

60 cards or tokens don't make a game too expensive.

60 = 6 x 10. 6 and 10 are triangular numbers (1+2+3, 1+2+3+4). Triangular numbers are interesting for scoring (see Ticket to Ride).

etc...

I'm sure you know all this. Just wanted to post it.

Willi B
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I'll second the love of 60

It's one of those things that we all have to consider.... scaling games for number of players... some numbers work well with one another and some do not.

Take a standard hex grid. If you are playing a game like TI:3 where players are fighting for area control in a hex grid, you can easily come up with equal positions for 2, 3, 4, or 6 players.... but that pesky 5-player game!!!! The solution of the hex grid cannot accomplish the feat! Even if you put some players equally distant from one another along the edges of the design, there is not going to be equal distances across from other players.

coco
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The 5th player

You'll need a 11 hexes per side board to accomodate 2,3,4,5 or 6 players.

This makes a perimeter of 60 hexes, and you can distribute players equally. However, this is a 331 hexes board, which can be too large.

Obviously, there is not going to be equal distances across from all of the players, as this is only possible with 3, in a flat board.

The Magician
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lol! fun fun!

lol! fun fun!

Mitchell Allen
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Deal or No Deal?

The Magician wrote:
I swear, I don't know if it's just me but some games I play that use a lot of cards, the way certain cards play out almost seems like a trick. I want to discover a clear link between how clowns deal out cards in card tricks and making mechanics work for cards to create an interesting system in my games. I have this idea that if you instruct players to deal the cards in a certain way in game setup, some cool experiences could unfold in what cards come up for the player, and have it so they don't know why everything plays out a certain way. That is probably the basis of lots of games. It's so there, if you look at an illusionist performing a trick. What if some event happens early in the game, and magically an event late in the game happens that corresponse to a card played early in the game. And it blows the players mind how that happened. They couldn't see how this played out but it was unique to that particular game experience and was not something that happens over and over. I'm not sure if I'm being clear or making sense. I'm probably drifting a little.

Is your description an imaginative way of referring to shuffling the cards? Or do you mean a pre-arrangement like this? I once saw a trick where thirteen cards from a suit were used to spell out every card:
A-C-E- ACE D-E-U-C-E Deuce, etc.
Every time the name of the card was said, the card showing was that rank. What made it cool was that the cards were returned to the bottom of the 13-card deck after each spelling. So all thirteen cards rotated endlessly.

Even though you say "deal the cards in a certain way", I'm thinking that, unless the deck is pre-arranged, you'd have a hard time ensuring that mind-blowing synchronicity.

Cheers,

Mitch

Mitchell Allen
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Triangular Numbers?

coco wrote:

60 = 6 x 10. 6 and 10 are triangular numbers (1+2+3, 1+2+3+4). Triangular numbers are interesting for scoring (see Ticket to Ride).

etc...

I'm sure you know all this. Just wanted to post it.

I sure don't know all this. What are the properties of triangular numbers - the fact that a sequence can be used to add them up?

Is 9 a triangular number? 2+3+4
Shouldn't 1+2+3+4 be a rectangular number? :)

This is interesting. Also, why are they good for scoring?

Cheers,

Mitch

seandavidross
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The Art of Scoring

Mitchell Allen wrote:
This is interesting. Also, why are they good for scoring?
Jonathan Degann, who has written several excellent game design articles, wrote a blog post discussing that topic about 3 years back. I link to it in the designer notes I mentioned in my first post to this thread, since it inspired the scoring system for my game, but, for convenience, here's that link again. As to the properties of triangular numbers... Google is your friend. The Wikipedia link is less technical, but also less complete, than the Wolfram Mathworld link.

The Magician
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Mitchell Allen wrote:The

Mitchell Allen wrote:
The Magician wrote:
I swear, I don't know if it's just me but some games I play that use a lot of cards, the way certain cards play out almost seems like a trick. I want to discover a clear link between how clowns deal out cards in card tricks and making mechanics work for cards to create an interesting system in my games. I have this idea that if you instruct players to deal the cards in a certain way in game setup, some cool experiences could unfold in what cards come up for the player, and have it so they don't know why everything plays out a certain way. That is probably the basis of lots of games. It's so there, if you look at an illusionist performing a trick. What if some event happens early in the game, and magically an event late in the game happens that corresponse to a card played early in the game. And it blows the players mind how that happened. They couldn't see how this played out but it was unique to that particular game experience and was not something that happens over and over. I'm not sure if I'm being clear or making sense. I'm probably drifting a little.

Is your description an imaginative way of referring to shuffling the cards? Or do you mean a pre-arrangement like this? I once saw a trick where thirteen cards from a suit were used to spell out every card:
A-C-E- ACE D-E-U-C-E Deuce, etc.
Every time the name of the card was said, the card showing was that rank. What made it cool was that the cards were returned to the bottom of the 13-card deck after each spelling. So all thirteen cards rotated endlessly.

Even though you say "deal the cards in a certain way", I'm thinking that, unless the deck is pre-arranged, you'd have a hard time ensuring that mind-blowing synchronicity.

Cheers,

Mitch


That is really cool Mitch! Thanks for sharing this! I realy havn't expored this much. My thoughts on this are pure speculation and imagination. What if you set up these tricks in game mechanics? Maybe they wouldn't feel like tricks at all. You wouldn't call them card magic. They would be game mechanics and part of game set up. But, perhaps it wouldn't be one predictable trick if you want to call it that. There could be several for many different circumstances. Maybe players would be instructed to set up the card decks differently in different stages or situations. As an individual, we have the potential to invent any kind of trick really. I would imagine that the mathematics of different card tricks can be custumized and personalized to your own game mechanics. One thought that struck me as an inventor and thinking about this, card tricks use playing cards. Playing cards pretty much look the same. But remember, in designing games, you can make any kind of card you want. It can have symbols, suits, words, names, information, etc. Can you imagine what you can do, useing mathematics of card tricks, and aplying that in a certain way to your custume cards. Okay, stretching my imagination further, lets say you have a card of a certain catigory. We'le call it (A) an event deck. This event deck my have a symbol or suit in the corner. The cards may play out normally in which ever way, but what if you applied the trick aspect to just the suit? Like, some obscure feature of the card that has important value, has a mathematical way in which it is dealt. Maybe it would lead to some interesting results in the game, however the underlying system obscured to the player. Or apply to something else on the card. What if you were applying it to some words or message on the card. These words may not have a strong function in the game, but applying some mathematical tricks, you were able to have certain statements come up that maybe the player felt like those statements were coming up just for them.

The Magician
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Has anyone heard of the

Has anyone heard of the number sequence: (3,3,5)? I heard it has some powerful significance.

seandavidross
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Golden Ratio

I believe you're referring to the Golden Ratio.

The Magician
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seandavidross wrote:I believe

seandavidross wrote:
I believe you're referring to the Golden Ratio.

I refered to the golden ratio several paces up the list, yes. The golden means, the golden section, it's the same thing: .0618 devided by a length.

I'm not sure if 3,3,5 has anything to do with it but I head some interesting things about this. I need to resurch it more before I say anything els about it.

coco
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integer sequences

The Magician wrote:

I'm not sure if 3,3,5 has anything to do with it but I head some interesting things about this. I need to resurch it more before I say anything els about it.

Do you know any more terms of that sequence?

Take a look at this:

http://www.research.att.com/~njas/sequences/

(I added a couple of sequences myself a few years ago)

Néstor

The Magician
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coco wrote:The Magician

coco wrote:
The Magician wrote:

I'm not sure if 3,3,5 has anything to do with it but I head some interesting things about this. I need to resurch it more before I say anything els about it.

Do you know any more terms of that sequence?

Take a look at this:

http://www.research.att.com/~njas/sequences/

(I added a couple of sequences myself a few years ago)

Néstor


In the past, I have mainly concerned myself with the Golden Section as it relates to art and geometry. It has many many applications. Just the last couple days I have started to interest myself with sequences. The Fibinocci sequence comes from the golden means. So, no I am not vary familiar with sequences. I am sure soaking it up though now. Thanks for the link. I will have to bookmark it.

This link here I am finding to be treasure for my mathematical interests in thinking about card mechanics:

http://www.spelman.edu/~colm/cards.html

It's juicy!

Mitchell Allen
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Mathemagical Card Tricks

The Magician wrote:
Okay, stretching my imagination further, lets say you have a card of a certain catigory. We'le call it (A) an event deck. This event deck my have a symbol or suit in the corner. The cards may play out normally in which ever way, but what if you applied the trick aspect to just the suit? Like, some obscure feature of the card that has important value, has a mathematical way in which it is dealt. Maybe it would lead to some interesting results in the game, however the underlying system obscured to the player. Or apply to something else on the card. What if you were applying it to some words or message on the card. These words may not have a strong function in the game, but applying some mathematical tricks, you were able to have certain statements come up that maybe the player felt like those statements were coming up just for them.

This is intriguing. Perhaps the backs of the cards could be numbered and the setup instructions written thus:
"For a Lapis Luzi experience, separate the odd-numbered cards from the even-numbered cards and sort them from lowest to highest, placing the ODD set on top of the EVEN set. For an I Love Lucy experience, simply sort the card from lowest to highest."

It really would be a trick, getting two or more results from the same deck. A Tarot Mechanic, I suppose :)

Cheers,

Mitch

Mitchell Allen
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Triangular Numbers

Thanks, seandavidross!

I thoroughly enjoyed Degann's article, continually thinking of games that I've played that might fit the scoring types.

Pit and Basari are definitely majorities, right?

As for triangular numbers, that's cool stuff, too. I should have googled.

Cheers,

Mitch

The Magician
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seandavidross wrote:I believe

seandavidross wrote:
I believe you're referring to the Golden Ratio.

Well I'll be a munkey's uncle! My memory about the golden means was a bit off. I haven't worked with it in some time and well memories sure can become distorted. I said .618; when it's 1.618. I didn't even think to look on wikkapedia about it.

Anyway thanks!

The Magician
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@ Mitch Yah thanks! Still, I

@ Mitch
Yah thanks! Still, I don't claim anything I said about this because I have a lot of work to see what I can do with the math. I'm working double time with a lack of math skills.

As an aside, I almost can't help bringing this up. You took me by suprise with the "lapis Lazuli" example. This is an unpleasant sincronicity (mispelled I know) for reasons you could never know. First, I love it as a stone and art pigment, but there was this woman I once knew who went by "Lapis Lazuli" as a nick name. Not to tell too much of personal stuff. We weren't "involved", it was nothing like that. All I would say is that I don't like to remember that *****. lol like I said "a weird unpleasant sincrinicity but no fault of yours. And it's not that it bother's me so much as " I would not want another Lapis Lazuli experience" lol.

One thought was, what if you set a number value to non-numerical aspects of a card and applied a mathematical system to make them flow through players hands in a certain way throughout a game.

Mitchell Allen
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Whoa...

Sorry to evoke bad memories.

What are the odds that a throwaway phrase would contain love/hate references to TWO women? I was going to use the old programming stand-in, "foobar"

"To have a foobar experience..."

- in your case, it would have read the same, eh? ;)

As for the synchronicity you encountered in that post, wow! That kind of experience just can't be bought.

But, that's exactly what makes fortune-tellers so compelling...another thread, another day.

Cheers,

Mitch

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