Cootie - a new FOFC strategy game

[QuikSand]

Okay, so parenthood has given me some new fuel to revisit the once-popular pastime here of launching puzzles and games for the forum to fiddle with. We'll see if this one proves to be worth the effort...

Children's games are loaded with colors and simple directions. In the long-standing tradition of games like chutes and ladders, there's no real need to introduce decision-making at an early age, just have a random event play out over time and have a winner declared at the end (if necessary). Great.

Okay, let's convert one such classic kid's game "Cootie" from harmless bug-building into ruthless probabilistic strategy.


The basics of the game:

Traditionally, the game is played among four players. On your turn you roll a die once, and according to the roll add a piece to your bug, first one to complete the bug wins the game.

The numbers correspond to:

1 - body
2 - head
3 - ears
4 - eyes
5 - mouth
6 - legs (need six legs to complete the bug)

And you must do things in a certain order:

FIRST roll a 1 to get a body
THEN roll a 2 to get a head
THEN you may roll your 3-4-5 and 6s in any order to complete your bug


The challenge:

Okay, to make the game a bit more interesting, here are the rules of how to play FOFC Cootie.

* You may use different dice, and roll one for any given turn. You must establish a clear rule for when to roll each of your dice, that is expressed completely as a function of what pieces of your bug are complete at that time (no reference to the last roll, or any other information is allowed).

* For a price, you may alter the individual sides of your dice, changing an initial side (a number 1-6) to any of the following:

1,2,3,4,5 or 6
R = player re-rolls the die
" = repeats the result of the last roll
+ = the result of the last roll, +1
> = reroll a standard die until its result is greater than the last roll (only the final result counts, no action from the interim re-rolling)
< = reroll a standard die until its result is less than the last roll (only the final result counts, no action from the interim re-rolling)

Note: If the + is rolled on the first roll of thee game, or following a roll of 6 or a roll that has no value, the result is no value and a lost turn. Other special sides that result in bad references, including > following a 6 or < following a 1, or any side referencing the last roll when the lat roll had no value, themselves result in no value.

* The cost to alter any of your dice is a loss of two turns per side altered. In a game, this will be implemented as each player starting after a certain number of rounds. A player with one unaltered die would start rolling in round one, a player with eight total altered sides would wait until round 17 to begin rolling.

* Your dice design are submitted in advance, and you pay in advance for all the alterations they include - there are no re-designs allowable as the game goes on.



Okay... that's the setup, to start with. I have a number of ideas how this could get more complex, but I'll wait to see if there's general interest before we blow it up.

In this thread, please post questions or comments, or just expressions of interest -- but for now, hold your dice designs to yourself. My thinking is that if we get a dozen people interested in taking a few minutes to design their dice, I can run a few games of it as a sort of Cootie Contest, with a little simple work in Excel. We'll keep things simple for now, but if the model holds up, maybe this game can last a while with different alterations.

So, for now... are you interested? Unclear? Post it.

[BrianD]

I'll probably have to read this a few more times to get the options, but I think I'm interested.

[QuikSand]

One more thing -- at least if I'm setting up the "contest," it will be all about *winning* a game involving four competitors. I don't envision any sort of points system for coming in second place, or posting the system that arrives at some long-term lowest average rounds to complete the bug.

It's possible that either of those systems is suitable for another separate game, but for right now, your objective should be to *win a 4-bug game* as often as possible. If that has any bearing on your dice decisions, take notice.

[Vince]

Is there any limit to number of dice?

[QuikSand]

Originally Posted by Vince

Is there any limit to number of dice?

No, though for every side on any die you want in your set, there's a cost of two turns for every altered side.

[Karlifornia]

Quiksand....you are some kind of mad genius, and I would be honored to win your game.

EDIT: By the way, pretty cool blog you have there in your sig.

[QuikSand]

Originally Posted by Karlifornia

EDIT: By the way, pretty cool blog you have there in your sig.

Thanks, I am pretty new to the racket, but am trying to keep with it -- I think it has some potential.

[Vince]

Let's say you have a die with a < or > on it, and while trying to get above or below the previous number, you come up with a value that you can use - do you earn that piece AND continue rolling?

Example:
You've got a Head and Body, and roll a 4 for Eyes.
Next roll, you get a >, so you continue to roll until you roll higher than 4.
While rolling, a 3 comes up.

Do you get the Ears? If you do, do you stop there? Or do you get the Ears and continue rolling until you finish the logical step for the >?

[QuikSand]

Hmmm... what I mean by "continue to roll" is that you take a regular 1-6 die, and you roll it until you come up with a value that suits the condition, and then you take that as your result.

So, if you roll a 2, and then on your next turn you roll a >, that essentially means that you're going to be assigned a random number from 3 to 6. It does not trigger a long sequence of rolls that each trigger an action for you. make sense now?

Do I need to re-word that above?

[QuikSand]

There are no doubt various ideas that might prove to render some of the potential options greater or lesser than meets the eye here. I deleted one public "strategy" sort of post (and responded to the author directly) and would request that you keep your tactical ideas to yourself.

For now, what I want are questions on the rules, and expressions of interest (that you're going to play) so I can gauge how to set up the competition.

[QuikSand]

I guess the easiest setup would be to take 16 entries, and run 5 games -- with the championship game including the four winners from the initial rounds. Anyone have anything more clever than that?

I was thinking it through... tough to keep things in games of 4 players in any alternate setup that includes a sort of loser's bracket.

[QuikSand]

* Your dice design are submitted in advance, and you pay in advance for all the alterations they include - there are no re-designs allowable as the game goes on.

Added to clarify, by request. This game is going to be run centrally -- so all the decisions are made in advance. There's no re-designing dice in the midst of the game.

edit: and corrected, to close yet another angle-shooting loophole already brought ot my attention


Idea here is that you build your dice, send them to me, and I play the game. This isn't going to be a role-playing game where you get to make a ton of decisions as the game goes on. Hope that's clear.

[QuikSand]

Anyone interested in playing, please send me your ENTRY, with that word prominent in the title of your PM. My goal will be to get to 16 entries, but if we fall short of that, I'll fashion some way to make things as fair as possible.

Maybe once we run a few games, we'll have a better sense of whether this is actually interesting. *shurg*

[QuikSand]

Heh, okay maybe I need to reset the goal to 4 entries.

Bump.

[gstelmack]

Ah, I missed the call for entries. I'll throw something together.

FWIW, it would also be interesting to make sure that there is a "control" entry with no unmodified dice, either as a single entry to see how it does, or to fill in the tree.

[BrianD]

If we aren't in a rush, I'd like to turn something in by the end of tonight. I'm not sure I'll get a chance to properly plan during work, but I should have some time this evening.

[QuikSand]

No rush. Just bumping, really.

[QuikSand]

Okay, including one entry that I have already sent to someone who submitted, it looks like we have four entries. Fair enough. I guess I'll run the game, and we'll see if that turns out to make anyone interested.

I like the notion of adding in a "control" player as well, and I might even break my rule of four entries to include that here. I hope it doesn't win.

[QuikSand]

FOFC Cootie Game (#1?)

We have four entries in this initial game of Cootie, and I have decided to include the default one-die competitor as well. Your five players are TredWel, digamma, gstelmack, QuikSand, and cuervo72. For what it’s worth, I sent my entry to TredWel before looking at the others, though it turns out that mine is quite different from any of the others, anyhow.

I’ll give a brief discussion of the dice, and then a little narration for the immersnik crowd.


Default will play with one die, 1-2-3-4-5-6, for the whole game, but gets to start rolling immediately.

TredWel has three dice, starting with a double-one (1-1-3-4-5-6). After he gets piece #1, he will switch to a double-2 die (2-2-3-4-5-6). And finally, after adding piece #2, he will switch to his final die, focusing on the needed 6s (6-6-3-4-5-6). With only a total of five altered sides on this array of dice, he will only be required to skip ten rounds.

digamma has two dice, and follows a similar pattern to TredWel. First die is the 1-2 die (1-2-1-1-2-2) with four altered sides, and the second die (after adding piece #2) he uses the same finishing die (6-6-3-4-5-6). With 6 altered sides, digamma will wait until round 13 to begin rolling.

gstelmack has two dice, and will alternate between them through the entire game. They are built as (1 2 3 4 > 6) and (1 2 R 4 5 6), and with each only having one altered side, he only sits out four turns before starting.

QuikSand pursued a pretty different approach, and his first die will try to take care of the first five numbers in a row, using the + symbol as the key (1 + + + + +). Once piece #5 has been added, he switches to a second die to complete the sequence quickly (6-6-6-6-6-6). This requires a total of ten altered sides, so he won’t start until round 21.

cuervo72 used another fairly complicated approach, somewhat similar to QuikSand’s. First die is designed to start off with a 1, but will get used again (1 1 1 6 6 6). Once the #1 is added, he switches to the second die, all-incremental (+ + + + + +). After the first #6 is added, he shifts back to the original die.


Game play…

Default gets a head start and starts rolling right away, but doesn’t have any pieces in place until placing #1 just as gstelmack starts off in round five. Both have piece #1 in place when TredWel starts up in round 11, but it’s digamma who gets luckier – he starts out in round 13, but strikes for pieces #1 and #2 in rounds 15 and 16 before TredWel has anything on the board at all.

digamma is now switching to his closing die, ahead of everyone except the default who has managed three straight hits to land #2, #4, and #5 in short order. Meanwhile, gstelmack has suffered bad luck and has noly gotten #1 in place and is still waiting for a #2 to get moving, and by design QuikSand and cuervo72 wait patiently to get started.

In round 21, digamma has been on a tear, and now has hit piece #4 and two #6 legs for a clear lead. TredWel finally gets on the board with a #1, and the default player is chugging along having added two #6 legs. Meanwhile, QuikSand and cuervo72 both got lucky to start with a #1, now need to peel off a good run to close it out.

Round 22 – digamma adds a third #6 leg, and now needs pieces #3, #5, and three more legs. Default adds piece #3, and only needs three more legs. QuikSand and cuervo both keep adding with the #2 – cuervo is assured of a run right to his first #6 leg in turn 26 now, but then he will need to get lucky to finish.

digamma keeps the streak going – rolling a 1,3,2, and 1 on his next four turns, which translate (on his current die of 6 6 3 4 5 6) to #6, #6, #3 and #6 – by turn 26, he is only one piece away, needing only to roll a natural 5 to finish his bug. QuikSand got pieces #1, #2, and #3 in sequence, but fell flat after that when he unluckily rolled a 1 and had to start over. cuervo has chugged along, has pieces #1-5 and his first #6 leg, but he is now rolling a (1-1-1-6-6-6) die needing to get five more sixes. The default player has not had much luck, adding only one more leg (its 4th) and still two natural sixes away.

In Turn 27, digamma rolls lucky again, and hits his needed 5. It’s over.

Here is how the field looked when we wrapped up:

AFTER TURN 27                    DONE

Default      1 2 3 4 5 6666       43
TredWel      1 2       666        41
digamma      1 2 3 4 5 666666     27
gstelmack    1 2 3                80
QuikSand     1 2 3                41
cuervo72     1 2 3 4 5 6          33

I will note that gstelmack had a visibly awful run of rotten luck trying to finish up his 6s, which certainly didn’t help things.


Anyway… congratulations to digamma, for the win in the inaugural FOFC Cootie contest.

[QuikSand]

Anyway, building on cuervo's design, I think it might be possible to at least challenge the pretty simple setup that digamma used. Here's what I'd propose there:

A: 111456
B: +++++
C: 126666

It only uses one more changed side than his original, but you have 4 out of 6 sides on the final die, which I think is likely to make up for that.


I confess that my original idea for this involved only using dice with 1-6 on them, but I concluded that the 111222/663456 combo was too obvious, and that we'd have most people submitting the exact same design. So, by adding the various other options, at least we got variety.

*shurg*

[gstelmack]

Originally Posted by QuikSand

I will note that gstelmack had a visibly awful run of rotten luck trying to finish up his 6s, which certainly didn’t help things.

Yeah, but the whole point was to come up with a die to shade that luck in your favor. I'm pretty impressed with what some of the other folks came up with, and wish I'd thought about it a little harder.

[TredWel]

In order to test some designs, I threw together a quick Monte Carlo-style Cootie simulator yesterday. Here are the expected number of rolls that every submitted dice combination takes in order to complete a bug:

TredWel: 8 Penalty Rolls + 20.8 Rolls = 28.8 Total Rolls
digamma: 12 Penalty Rolls + 18.8 Rolls = 30.8 Total Rolls
QuikSand: 20 Penalty Rolls + 14.3 Rolls = 34.3 Total Rolls
cuervo72: 20 Penalty Rolls + 17 Rolls = 37 Total Rolls
gstelmack: 4 Penalty Rolls + 40.2 Rolls = 44.2 Total Rolls
Default: 48.2 Rolls

We all improved on the basic design, but by a much different margins.

Two comments to make:
- I doubt that putting any of the exotic options on your dice is worth two penalty turns. I couldn't find any combination that used exotic values that wouldn't be better served by switching to regular numbers.
- I freely admit that trying to figure out the set of dice that completes a bug in as short a time as possible may not have been the best approach to this game. It is possible that a set of dice exists which has a higher number of expected rolls overall, but in such a way that it gives you smaller results most of the time, and a few stinkers to throw off the final total. I couldn't think of one, though, and given the nature of the game where you want restrictive results early on, my gut says that such a set doesn't exist.

[Passacaglia]

What would have happened if you got a + on your first roll?

[digamma]

What's up now, my bishes?

[QuikSand]

Originally Posted by Passacaglia

What would have happened if you got a + on your first roll?

Originally Posted by First Post

Note: If the + is rolled on the first roll of thee game, or following a roll of 6 or a roll that has no value, the result is no value and a lost turn. Other special sides that result in bad references, including > following a 6 or < following a 1, or any side referencing the last roll when the lat roll had no value, themselves result in no value.

I deliberately wanted to keep the + option from being too powerful -- so I (tried to) clearly state that it couldn't get you your first 1, nor would it get you repeating 6s.

[QuikSand]

So, from your MonteCarlo, TredWel, maybe I set the "price" for an altered die too high. It looks like a price of 1 to 1.5 turns would have made most of these options a bit more competitive, at least in the "average" terms that you describe.

Even though I stated that we were to plan for a game involving 4 entries, I know that my "go for broke" strategy is clearly a better fit for a larger field where a very quick finish would be essential to content (i.e. where 2nd and 25th place are equally useless).

[gstelmack]

But if you reduce the cost, doesn't this quickly become optimal?

111111
++++++
666666

That's a guaranteed 11 rolls to win (on your first one, switch to die 2, on your first 5, switch to die 3), with 16 faces changed, so either a score of 27 or 35 as the top end with your adjusted scoring. The key is to encourage minimal face changes for minimal increases in the 11 rolls to win min case.

[QuikSand]

Originally Posted by gstelmack

But if you reduce the cost, doesn't this quickly become optimal?

111111
++++++
666666

That's a guaranteed 11 rolls to win (on your first one, switch to die 2, on your first 5, switch to die 3), with 16 faces changed, so either a score of 27 or 35 as the top end with your adjusted scoring. The key is to encourage minimal face changes for minimal increases in the 11 rolls to win min case.

Well, if you make side changes free, of course that's optimal. But I was talking about something like 1 - 1.5 turns per change. If TredWel's sim above is correct, then even at the lower end of that range, his setup should average a solution in 24.8, which beats 27.

I guess there are a variety of ways to add parameters here. One would be to have escalating costs for successive changes on a given die. Or perhaps an added cost for each switch of dice during the game.

I mean -- of course, the ideal setup would put all these decisions into play. You'd want to have to consider a whole range of options to find a way to do it. Were there interest, I could construct a new set of custom sides, and basically rebuild the game for another go with different rules and costs... in theory we could tweak this thing to no end.

[QuikSand]

Originally Posted by gstelmack

But if you reduce the cost, doesn't this quickly become optimal?

111111
++++++
666666

That's a guaranteed 11 rolls to win (on your first one, switch to die 2, on your first 5, switch to die 3), with 16 faces changed, so either a score of 27 or 35 as the top end with your adjusted scoring. The key is to encourage minimal face changes for minimal increases in the 11 rolls to win min case.

And, just to springboard from here... if I *know* that you are going to submit this (above) as your entry, then I will just submit:

111116
++++++
666666

I use one fewer altered side than you do, so I get to start at least one turn earlier -- but in 5 out of 6 games we're going to have the exact same sequence and therefore I'll win the sizable majority of the time.

[adubroff]

Originally Posted by QuikSand

And, just to springboard from here... if I *know* that you are going to submit this (above) as your entry, then I will just submit:

111116
++++++
666666

I use one fewer altered side than you do, so I get to start at least one turn earlier -- but in 5 out of 6 games we're going to have the exact same sequence and therefore I'll win the sizable majority of the time.

Wouldn't you stand vulnerable to Price as Right treatment also, i.e.:

111156
++++++
666666

[QuikSand]

Originally Posted by adubroff

Wouldn't you stand vulnerable to Price as Right treatment also, i.e.:

111156
++++++
666666

Oh, absolutely... I'm not saying that my counter proposal was perfect. Just that the previous suggestion didn't completely end the strategy of the game by virtue of it being "optimal" if there's no cost in changing the dice.