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Subject: Mathmatics of Candyland rss

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Call Me Ishmael
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Everyone knows how to play Candyland. You draw one of the 66 cards to move along the 134 space path. Oddly, although "start" is indicated, "finish" or "home," is not. It's just the last space on the path. Others have mentioned the lack of clarity of how to handle the cards running out; one can reshuffle or declare the player in the lead the the winner.

The 134 spaces are divided as follows, and always in the same color order: Red 22, Purple 22, Yellow 21, Blue 21, Orange 21, Green 21. Red and purple each have an extra space at the finish end of the path, and a purple card is always needed to win. Three of these colored spaces are also 'stop' spaces; number 48, a yellow (Gooey Gumdrops); number 86, a blue (Lollipop Woods), and number 121, a red (molasses swamp). A player must stay on that space until the same color is drawn by another player.
The path also contains six character spaces, of which, more later. In addition, there are 2 shortcuts accessed by landing on the lower entrance: Rainbow Bridge, which goes from space 5 to 59, a gain of 54 steps; and Gumdrop Pass, going from space 34 to 47, a gain of 13. A player cannot use both shortcuts on the same trip up the path, for using the Rainbow Bridge makes the Pass inaccessible. The upper exits are one-way; a player landing there does not retreat. These passes always benefit the player, unlike the character cards.
The character spaces add much color, but they are a net loss to the player. Consider the maximum gain vs. the maximum loss of each character:
Plumpy, space #9; +9 -125; net -116
Mr Mint, space #18; +18 -107; net -99
Jolly, space #43; +43 -91; net -45
Grandma Nut, space #75; +75 -59; net +16
Princess Lolly, space #96; +96 -38; net +58
Queen Frostine, space #104; +104 -30; net +74
The characters do not always stand between the same colored spaces; but the sequence of colors is never changed.
Count Licorice and King Kandy have no spaces and no cards.

Therefore, the possible positive move to the player could be +148, but he stands to lose -260.

The colored cards are in 2 suits, single (8 of each color), and double (2 of each color). Drawing a single card will move a player from 1 to 6 spaces; a double moves 7 to 12. Going through the deck one time means the players will move between 216 and 576 spaces; with the character cards included, the deck can move all players anywhere from -44 spaces to 724!

The game can be won in as few as four turns, if a player draws Queen Frostine first, then 3 consecutive doubles, of which the second cannot be red and the third must be purple. Due to the overall negative effect of the characters, it is also possible that a game of Candyland may be started and never end. Or maybe it just seems that way.
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Richard Rowan
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Re:Mathmatics of Candyland
Dude, I think this article and the research behind it qualifies as "way too much time on your hands."
 
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Dirk Chegigo
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Re:Mathmatics of Candyland
The most comprehensive study of Candyland to date. Nice insight. Makes me want to pull out my son's copy, burn it, then play a real game.
 
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Stephen Smith
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Re:Mathmatics of Candyland
Otter wrote:
"finish" or "home," is just the last space on the path.

Red and purple each have an extra space at the finish end of the path, and a purple card is always needed to win.


In the Winnie the Pooh version, this has been fixed in that the last space is striped (replacing the purple space), so any card can get you there. Oddly, it is orange, not red/pink that comes before the purple.
 
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shumyum
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Re:Mathmatics of Candyland
Otter (#36598),

OK, let's say you are playing your kids and on the very first turn you draw a character. This, of course, will illicit cries of "no fair!" and "do-over!". If you are a loving caring parent, you draw again; if you are a strong fair parent you say "rules is rules" and take your move normally; but if you are like me, you want to make your kids PAY for their whining. I would like to be able to say, "OK, I'll draw again, but only if I get to ignore a character card *every* time I draw one." Will this benefit me (probability-wise) for *every* initia-draw character or just the close ones?
 
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Call Me Ishmael
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Re:Mathmatics of Candyland
shumyum (#37492),
Statistically, yes. Over the course of time, two players will each win 50% of the time. However, if one player is using the character cards and one is not, the player not using the characters will always have an advantage. The player using the character cards has a deck runthrough range of -44 to +724; the player not using the characters would have a runthrough of between +216 and +576. He may not have the big leaps forward, but he ALWAYS advances. The noncharacter player is forfeiting a maximum movement of 148 spaces to the other player, but is also dropping a possible loss of 260. That's a +112 space maximum advantage every time the deck is runthrough. That should increase the noncharacter's winning percentage to about 65%.
 
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shumyum
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Re:Mathmatics of Candyland
Otter (#37495),

Yes, but in order to get the deal, I would have to forfeit my first turn early jump. Let's say I draw Queen Frostine as my first card...do I play it normally (getting a HUGE lead) or do I make the deal? Starting so close to the end, it may be more probable for me to finish out the game before I draw another character card.
 
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shumyum
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Re:Mathmatics of Candyland
shumyum (#37500),

In other words, it's clear to me (thanks to your analysis) that being a "noncharacter" player would give me an advantage. But stipulating that I get a "positive" character card at the very beginning changes matters...but it's hard to see by how much.
 
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Call Me Ishmael
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Re:Mathmatics of Candyland
shumyum (#37501),
By drawing Queen Frostine first, you have advanced 104 spaces. However, any other character now becomes a negative for you if drawn. If Queen Frostine were the first card, that leaves 65 cards, five of which are going to hurt you very badly; you can figure that if you draw a character card at this point, it will set you back an average of 58 spaces. On the other hand, it's only 30 spaces to the finish. With 65 cards remaining , you can expect a character card to come up every 13 draws. On average, you are going to take 5 turns to cover that ground. But to since you need a purple to finish, (assuming all cards drawn statistically evenly, and you're not stuck in the Molasses Swamp), once you get in position, you now face trying to draw a purple before you draw a character, a 1 in 6 chance to win, vs a 1 in 6.5 chance of being sent back up to 125 spaces. On average, it could take you 11 turns to win from Queen Frostine, while you could draw a (negative) character card within 6 turns. I'd turn her down, counterintutive as it seems, to be released from the characters' clutches.
 
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shumyum
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Re:Mathmatics of Candyland
Otter (#37511),

OK, I'm convinced. I can't wait to try it out!
 
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Crypt Keeper
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Re:Mathmatics of Candyland
Hmm... Very clever Shumyum, but there's a problem. You've already stated that your kids (like mine) are the type to scream, "no fair!" and "do-over!". Once they figure out that you've played them for suckers the screaming will become unbearable. You could, of course, ignore them and chuckle your way to victory, but I'm not sure if it's worth the damage to your eardrums (not to mention the 'Will you just let them win the damn game and shut them up!' look that your wife will be shooting you.laugh
 
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shumyum
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Re:Mathmatics of Candyland
Gialmere (#37546),

Well, this will definitely turn out to be a one-time gambit that I will remind them about their wholel lives; even when they visit me in the asylum that they will commit me to. "I AM KING KANDY...KOO KOO KaCHOOOOOO!!!!!"
 
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Call Me Ishmael
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Re:Mathmatics of Candyland
shumyum (#37642),
Played three times tonight; even with losing a turn (instead of playing the character) the player not using the characters won all three times by wide margins.
 
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Lilli Bratcher
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Re:Mathmatics of Candyland
Otter (#36598), I haven't seen the game in quite sometime, but I do remember there being a point to it... wait, maybe there isn't?? :shrugs: still one of my favorites of all time.
 
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dave rusin
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I'm embarassed to admit that I have carried out a lengthy analysis of the mathematics of CandyLand, using a possibly faulty understanding of the rules and the board structure, and replacing a properly-shuffled deck with a random-card-generator (the difference being that it is perfectly possible to get e.g. the QF card twice in a row, or never for hundreds of consecutive draws). In particular, I have calculated the exact expected length of a one-person game, which is a ratio of two numbers with hundreds of digits each! A similar analysis computes the expected number of moves for a multi-player game (as a decimal approximation). For details that only a true gearhead would enjoy, see http://www.math.niu.edu/~rusin/uses-math/games/candyland/

As noted there, the game data and a similar analysis were sent to me by someone else who has his own web site for this, so I guess the world is big enough for multiple ubergeeks.
 
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Hector Irizarry
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Candy Land » Forums » General
Re: Mathmatics of Candyland
I always "wash" the deck Casino style before each session. That gives me a fresh new start and the influence of the past is erased from the deck with new shuffles. This is equivalent to re-seed the computer Random Number Generator.
 
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