Math of dice/card combos

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chris_mancini
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I'm working on a game in which players roll dice and race to find the matching symbol on a 3x3 grid of cards, in an "I-spy" mechanic:

3d6 - 1 symbol die (6 symbols) / 1 color die (6 colors) / 1 number die (1/2/3, each twice).

The total number of variations based on these 3 dice is 108 (6x6x3).

Each card has a jumble of icons in different color/number combinations. The active play grid is a 3x3 set of cards, with 12 icon targets per card. The reason for this is: 9 cards x 12 icons = 108. Therefore, there is a reasonable chance that the result of the dice roll will be contained within the cards on the table (but not guaranteed).

When a player finds the symbol/color/number combo on a card, they grab it, replace the card with one from the deck to bring the grid back to 9 cards, then reroll the dice to get the next target. Play continues until a grid of 9 cards can no longer be created. The player with the highest number of cards is the winner.

My question is this...what is the ideal distribution of symbol/color/number combinations in order to make the game reasonably reliable to have the matching result present in the 3x3 card grid on any given roll?

My math has thus far concluded this: 9 cards / 12 symbols each / 108 possibilities always present on the table. There are 108 total possible combinations from the dice roll.

If there are 63 cards in the deck (concluded from a 6P game, with the tenth card won concluding the game - 10/9/9/9/9/9 = 55, plus 8 cards remaining on the table as the end condition), then there must be 7 of every possible symbol/color/number combination (63 cards/12 symbols per=756 total symbols; 756 divided by 108=7 of each combo to balance the deck).

Does any of that make sense? I am admittedly do not call math a particularly strong suit, and before I go to the effort of mocking up these cards, I want to be somewhat confident that the numbers make sense in balancing the deck and the grid of 9 cards on a given draw.

Jay103
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I don't know about the exact

I don't know about the exact math you might want to do to get an "ideal spread" of stuff, but it sounds like you have a handle on it.. 7 of each combo, spread on 63 cards.. I'd consider assigning them randomly and then visually tweaking it so that you don't have more than two of a single symbol, number, or color per card. Wouldn't take all that long (compared to the time to come up with something more clever :) )

What do you if there's no exact match on the grid? That's a tough situation, and it's fairly likely.

let-off studios
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Avoid Reinventing

Jay103 wrote:
What do you if there's no exact match on the grid? That's a tough situation, and it's fairly likely.
Can you re-theme a deck of Spot-It to match up with your numbers and such, to test this out? Maybe a multi-symbol re-theme of Set? The permutations on the cards of those games may be similar to what you're trying to do, and they've done a lot of heavy lifting for you already. You may also learn the patterning process they've gone through to most-efficiently arrange the symbols on your cards.

Jay103
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let-off studios wrote:Jay103

let-off studios wrote:
Jay103 wrote:
What do you if there's no exact match on the grid? That's a tough situation, and it's fairly likely.
Can you re-theme a deck of Spot-It to match up with your numbers and such, to test this out? Maybe a multi-symbol re-theme of Set? The permutations on the cards of those games may be similar to what you're trying to do, and they've done a lot of heavy lifting for you already. You may also learn the patterning process they've gone through to most-efficiently arrange the symbols on your cards.

Spot-it is designed specifically to have no failures on any cards.. As soon as you deal out multiple cards at random, that's blown. (and remember, spot-it only requires that you match a single thing, the animal, and you have 6 or whatever choices to try to match...)

With fewer symbols/colors, you could decrease the odds of failure, but you can't eliminate it, and you'd also make spotting the proper thing too easy..

One thing that might help would be to have more than one possible target. Like, if you only needed a match of TWO dice, rather than three.. (maybe the third die becomes 1-6 at that point). Might dramatically reduce (but not eliminate) the odds of the failure point occurring.

pelle
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This game would be easy to

This game would be easy to script to test s few million times for different options and see how often players get stuck (ie no matches seen) for instance.

I think op is slightly off with the card combos since number of combos get a bit weird when you remove cards from a deck and it is rarely as simple as calculations on dice.

The possibility for having a state with no possible match is a bit unfortunate unless you can work that into the game in a god way, like maybe awarding points to the first player that correctly calls out that there is no valid match?

Jay103
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pelle wrote:The possibility

pelle wrote:
The possibility for having a state with no possible match is a bit unfortunate unless you can work that into the game in a god way, like maybe awarding points to the first player that correctly calls out that there is no valid match?

Which you'd have to verify by...

continuing to look at the board, just as you did before?

Jay103
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Looking at your numbers again.. no, this is never going to work.

You have 9 cards visible out of a deck of 63. In that deck, there are 7 cards with the symbol that actually matches. So what is the chance that you have at least one "good" card in your 9?

My probability is a little too weak to be able to compute that easily, but.. it's not high at all. It's on the order of 75% I think... not on the order of 95+%.

You'd need a lot more symbols on the cards to make this work, or more cards in the tableau (or match two of the three things, as I noted above)

X3M
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Probability is also going to

Probability is also going to get lower for a good one, when a certain combination has been rolled more often than another one. Meaning that those symbols are being replaced until non are left.

The most effect you can see of this is in the last few rounds.

***

I'd say, this mechanic is not good at all. Have you tested the idea with a "smaller" deck and less symbols to see what is going on? eg. 2 symbols/2 colours and a 2x2 grid. Perhaps you can discover what is needed?

***

I do not know your target audience. So the following might not be a good suggestion.

The event of "no match". A re-roll does not work well, I am sure of that. There is a chance of rolling over and over again and getting no match at all.

So how about cycling through the options as a game mechanic?
If a player already has noticed that there is no combination possible. Then it may declare this and say what die must cycle to the next items to make the combination. All other players must check if this is the very next possibility.

Maybe even have a reward in order, seeing as how this is more difficult.

eg. Rolled: Symbol C, colour yellow, number 3.

It isn't there.
But it might be Symbol D, or colour green, or number 1.
And if one of those 3 aren't there. Then the next one after that.
The distance has to be as close as possible.
So D-green-1 is 3 removed. While E-yellow-3 is only 2 removed.

chris_mancini
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As Jay and others have

As Jay and others have mentioned, this is the biggest concern; there being no match on the "board" on a given roll. The idea of calling out "no match" is interesting and a simple band-aid for what would otherwise may be considered a broken mechanic...the "Disney Eye Found It" card game does this; if no match is present, flip a new card. Pretty lazy design if you ask me.

Spot It is guaranteed to work every turn as there is only 2 cards ever being matched; it's a brilliant exercise in permutation, but only in its purposefully limited gameplay....as soon as 2 cards becomes 3 or more, the possibility of multiple matches is immediate.

Set is a game I was not aware of, but seems as though some ideas could be gleaned from it. For instance, in Set, if a set cannot be made, the rules say to add cards until a set can be made. In this game idea however, players would just be looking at the one new card, so that exact rule wouldn't work. Replacing a number of cards however, might...something to play with!

As X3M mentioned, there is also the probability of one combo being rolled more than another, thus affecting the likelihood of a smoothly operating game. I was trying to create as much of an equal balance as possible, but with the luck of the roll, it's impossible to control completely...the question is, can it be made to be acceptable?

Anyways this is why I put this idea out to the group! The game is not particularly "gamerly" in its idea, strategy or complexity (it's a super-simple family game idea)...but something I was thinking about.

Jay103
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Yeah, I don't really see a

Yeah, I don't really see a fix for it, unfortunately. Maybe the thing of matching only two of the types would work (giving you three 1/36 chances, for example, rather than a single 1/108 chance), but even then you'd need a bailout rule of some sort just in case..

chris_mancini
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I'd started with only 2 dice,

I'd started with only 2 dice, and using photographs in the seek-and-find - color and category (animals, food, landmarks, objects, etc.) As you mentioned, this would create a total range of 36 core combo possibilities. Much more controllable, while offering a broad range of images within the broad categories. In this case, the likelihood of multiples in the card grid would outweigh the likelihood of no matches, which is a more pleasing situation as multiple players could score, or one player could score multiple points.

It also adds some quirkiness to the game, as you could be searching for a blue dog, or an orange apple. Stock photography along with a little Photoshop manipulation would be all that is needed, so prototyping would be relatively simple.

The bailout rule would still be required, but the frequency in which players would need to invoke it should be greatly reduced.

X3M
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What if you fix the

What if you fix the randomness with something else than dice?

Like a memory? 1 set on the table. Another one in a bag.
And if there is no match at all. It counts as a mismatch. Thus discarded. I am sure that having one of the 2 sets being depleted before the other one isn't a problem for the players. Like a random highest score for each game.

Maybe even WITH dice. Just a bad roll. Going through the combinations systematically will show players from time to time if a combi is posible.

If a couple of rolls failed. Like each player rolled once. Then each player could discard a die for their next roll. With that the chances to find the 2 things instead of 3 is waaaay higher. Eventually 1 die would be the case. And after then. The primary roller can simply select the card that he, she wants.

Just putting out ideas.

wob
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i think your game may work,

i think your game may work, but i agree less dice means less complexity. this is a good thing, both from a design and manufacturing point of view (i am always surprised how large a deck can grow with permetations).

my main reason for posting was the bail out rule. make a feature of it if its unavoidable: 10 points if you spot an impossible item (or rather dont spot) but your opponent can challenge. a successful challenge= +10 points for challenger -10 for spotter. unsuccessful challenge = -10 for challenger. (the amounts were arbitrary but you get the point)

chris_mancini
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I like the bailout rule

I like the bailout rule example; if scores are tallied by number of cards won, then it could be as simple as a +1/-1 card penalty.

So if I go with the color/category combo:
- 36 total permutations
- Grid of 9 cards / 12 images per card = 108 total images
- 108/36=3 / average potential of 3 correct answers per grid

- If only 8 images per card: average potential of 2 correct answers per grid

This feels unlikely that, on the average play, there would not be a match present within the grid. In the event that no match is present, the bailout rule would go into effect.

In addition, a rule could state that, if the same color/category combo is rolled twice in a row, a reroll is required to create a different target combo.

Jay103
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Well, at that point you're

Well, at that point you're Spot It. If there are an average of 3 matches PER CARD, why would you need nine cards in the first place? You could just guarantee that every card had a match to every combination, and you're done. One card on the table.

chris_mancini
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That was my concern and

That was my concern and reason for favoring the 3 dice symbol based design...being further away from Spot It.

The cards in an image based design would average 3 per grid of 9 cards, not 3 per card.

Fri
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Math/Obxervation/variant sugesstions

Math:

If I am understanding the set up correctly each of the 108 rolls dice have 7 matches in the matches in a 63 card deck. When you roll the dice you have to come up with one of these combinations. We know that roll has exactly 7 matches in the deck. So we can simplify our calculations to finding to probability of of one of those cards being on the table. So I am going to rely on wikipedia to explain the concept of the formula to use:

https://en.wikipedia.org/wiki/Hypergeometric_distribution
(Skip to Application and example. Understanding that their example with the marbles is logically equivalent to your example with cards is what is important here.)

Thankfully, we don't actually have to do these calculations some one has graciously set up a website to do it for us:

So you initial set up is this (this will change as the game goes on):
Population size=63
Number of successes in population=7
Sample size=9

You will want to look at Cumulative Probability: P(X < 1). In English this means that there is at least one match but possibly more. The probability of having a match in the initial setup is 74%

Observation:

chris_mancini wrote:
If there are 63 cards in the deck (concluded from a 6P game, with the tenth card won concluding the game - 10/9/9/9/9/9 = 55, plus 8 cards remaining on the table as the end condition)

So without a "bail out rule" this game end condition is impossible because there are only 7 matches.

Variant suggestions:

Could you have players just reroll whenever they wanted? (They would not be allowed to take a card while their dice are rolling)

Could you have three dice that are rolled that are common to all players. Everyone would have equal chances of claiming a card. Also this could accommodate any number of players. Once everyone agrees that there is no match the dice could just be re-rolled.

Jay103
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me wrote: My probability is a

me wrote:

My probability is a little too weak to be able to compute that easily, but.. it's not high at all. It's on the order of 75% I think... not on the order of 95+%.

Fri wrote:
The probability of having a match in the initial setup is 74%

Booya.

Jay103
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Fri wrote:chris_mancini

Fri wrote:
chris_mancini wrote:
If there are 63 cards in the deck (concluded from a 6P game, with the tenth card won concluding the game - 10/9/9/9/9/9 = 55, plus 8 cards remaining on the table as the end condition)

So without a "bail out rule" this game end condition is impossible because there are only 7 matches.

Assuming I'm following you correctly, I should note that you reroll the dice every time there's a match.

Fri
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Oops missed that part - to

Oops missed that part - too busy thinking about the math :)

Edit: fixed grammatical error

chris_mancini
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So I'm modifying the

So I'm modifying the permutations through implementing WILD faces on the symbol and color dice. This reduces the permutations to 75, or 5 symbols x5 colors x3 numbers.

I've reduced the deck to 50 cards, maintaining the 12 symbols per card, for a total of 600 throughout the deck.

Game grid remains at 9 cards/108 symbols in play during a round. Based on the 75 permutations, it works out to require 7 cards at minimum to represent all possible permutations (12x7=84). The deck of 50 is due to balancing the total symbols so that each is represented evenly; 75x8=600; 600/50=12.

What has me flummoxed now is how to define the manner in which I build out each card. I've got the 600 symbols evenly created in Illustrator, but I'm trying to devise a way to most evenly distribute each throughout the deck:

5 symbols: hearts, clubs, diamonds, spades, stars.

5 colors: red, orange, yellow, green, blue.

3 groupings: single symbol, group of 2, group of 3 (groups are always of the same color and symbol).

3 sizes: small, medium, large. Size is only to create visual interest and a more dynamic play area; it does not otherwise matter in play. I do however want to account for this in the distribution of the symbols, so with 12 symbols per card, it would be 4 of each size.

Does anyone have an idea of what formula may create the desired distribution? I'm currently staring at the 600 options in my Illustrator file and completely uncertain as to how to start grouping them on the cards...