Button Men Strategy

On January 19, 2022, the beatpeopleup.com website went down alongside all Cheapass Games content at cheapass.com, including the print-and-play files for Button Men and all other Cheapass games. Presented here is an archive of the strategy articles posted at cheapass.com and beatpeopleup.com. Please note that these are from the early 2000's and may not reflect current strategy and theory in the Button Men community.

## The Highest Lowest Number (James Ernest and Glen Barnett)

Here’s the situation. It’s your turn. You’re playing against a larger character, let’s call him Hammer, who’s got two dice left: (3/4, 17/20). Your dice lie (5/6, 10/10). You have already figured out that Hammer needs to keep his 20 to win. How do you get your best chance of taking it?

If you’re playing to keep the highest number on board, as you usually do, you’d do 5-takes-4, and reroll the 6-sider. But that’s going to give you an expected value of 3.5 on your lowest die (an even chance of 1 through 6, averaged). Hammer is obviously going to take your highest number next turn, to have the best shot at protecting his 20. So you want to come out of this turn with the best possible value on the lower of your two dice.

With the hand above, rolling the 10-sider is the better choice. Half the time, the 10 will roll higher than the 5/6, leaving 5 as your lower number. In fact, the average expected value of the lower number in this roll is 4.0.

Unfortunately, it’s not as simple as always rolling the larger die. If your 6 were showing a 3 rather than a 5, you’d be better off rolling the 6-sider. Rolling the 10 in this case only gives you an average lower number of 2.7.

And now, before you read further, figure out which die you’d roll if you were showing (4/6, 5/10.)

The Best Lower Number

Glenn Barnett

There’s an easy way to calculate the expected value of the second highest number when you have a choice of which of two dice to roll.

First, some preliminaries. Triangular numbers are the number of dots you get when you lay dots out in triangles, like so:

A quick way to work out these numbers is to write down consecutive numbers from 1 on up (to 12 will do), and then underneath write a 1 under the 1, and then under each other number write the sum of the number to its left and the number above it, so the number under 2 is 1+2 = 3, and so on:

1 2 3 4 5 6 7 8 9 10 11 12

1 3 6 10 15…

The next few triangular numbers are 21, 28, 36, 45, 55 and 66. These numbers are important; if you know them already, great. If not, they are easy to work out.

Let’s say you have a 4 on a d6 (six-sided die) and a 5 on a d10, and your opponent has a 3 on a d4 and a 17 on a d20 – you want to know what to take the d4 with to maximize your chances of getting that d20. As James points out, you need to maximize the average value of the lower of your two numbers after the roll (since your opponent will take the higher of the two). Maximizing the lowest number’s average is the same as maximizing the probability that your rival will roll under your lower die.

Here’s how to find the average value of the lower number when rerolling a given die:

Take the face-up number on the \*other\* die. Now subtract the previous triangular number divided by the number of sides on this die. For example, if you reroll the d10, then the number on the other die is 4. Subtract the previous (ie third) triangular number (which is 6) divided by 10: 4 – 6/10 = 3.4. That’s all there is to it! (That’s not quite true – sometimes you do something that’s even easier.\*)

Now what if we reroll the d6? The average lower number is 5 – 10/6 (i.e. 5 on other die – 4th triangular number divided by 6 for the d6), which gives 3 1/3. So you’d better take with your d10.

* That formula works when the number on the other die is smaller than the number of sides on the die you’re rerolling. If it is bigger, just take the average for the die you’re rerolling (since it will always be smaller than the other die, right?).

You don’t generally have to calculate the values to this accuracy -typically you only need to go far enough to figure out which one is bigger. Let’s look at another case: 3 on a d6 and 7 on a d10. Rerolling the d6: 3.5 (since 7 is bigger than 6) d10: 7 – 21/10

You don’t need to work out 7 – 21/10 if you can see it is bigger than 3.5 (which it is). You can say “21/10 is just over 2, so 7 minus something less than 3 is more than 4”. Neat, huh?

“Help, what’s the 8th triangular number?” If you don’t have the numbers to hand, and you don’t want to work the whole list out to get just one of them, here’s how to get a particular triangular number. You take that number, multiply it by the next number, and halve. So the 8th triangular number is 8 x 9 / 2 = 36. If you don’t know your multiplication tables, it only takes a moment to write down those two rows of numbers I mentioned before (if you can’t /add/, pick a different game to Button Men).

Here’s the same information in a table. The column headed “f” is the face value on the other die. The column headed “V” is the formula for the values in the table, unless we have gone past the number of sides on the die already.

Table: Average value on lowest die. (4 through 12 represent sides on rolled dice)

Note that we don’t need to continue the table to 20, since all but the last column don’t change, the rest of the values in the d20 column are bigger than all the values in the table, and if they are both d20’s you obviously just reroll the lower one.

You can notice some interesting stuff right away. For example, one rule you can see is that if the die with the fewest number of sides is showing 8 or more, reroll the other die. Another pretty obvious rule is that if you work out a lowest number value for one die that’s already above the other die’s ordinary average, it must be above the other die’s reroll value. Also, if the die with the larger number of sides is showing the lower number, reroll it. There are lots of other things to notice that I’ll leave up to you.

## Button Men Theory (*Simon McGregor)*

Button Men Theory is tough. Leaving aside Swing Die theory and character selection (and I have some ideas of my own on that one), ordinary play strategy isn’t simple.

I’m currently working on “endgame analysis” (i.e. play strategy when there are only a few dice left), and it’s evident that rules of thumb aren’t infallible by any means. Nor is it always obvious when they don’t apply. And it gets worse! Not knowing what “best play” is, it’s really hard to tell either from theory or experience how likely the various endgame scenarios are in the “perfect play” case.

Yesterday, I whipped up a small Button Men computer game for personal use. I know you’ve got your own one in development, but if you want a look at mine, you’re welcome to it.

Anyway, here are some (unsolved by me) questions about Button Men strategy:-

1) How much of an edge, in general, does optimal play have (vs. reasonable but sub-optimal play) in a single round? And in a three-round match?

2) Are there situations where sub-optimal play has a greater edge over sub-optimal play than optimal play does? (i.e. are there ever times when, knowing the “best” strategy versus “best play”, you can choose another strategy and better your chances by assuming your opponent won’t spot “best play”?

3) Assuming “best” swing die selection, is there a “best” character? Or are we playing scissors-paper-stone?

I am looking at some brute-force computational approaches right now. Button Men Theory is amenable to brute-force solution in two areas:-

1) First-go odds. A complete (60 \* 60) table of first-go odds would be useful, and is straightforward to calculate.

2) Play strategy. With five dice, Button Men is simple enough to exhaustively calculate the best play strategy from a given starting position, and the chances of winning.

I’ll let you know when I’ve calculated the first-go table.

If Button Men takes off like it should, I guess we’ll see a whole lot of mathematical analysis. And I don’t think it will be solved quickly. Or maybe ever.

I’ve come up with some simple (and potentially conflicting) heuristics for the game in general. I’m sure I’ve missed some, so feel free to point them out.

i) Keep the highest number you can for your next turn. Sometimes your highest number is going to be taken in your opponent’s turn, so this means maximising your next highest number.

ii) Force your opponent to re-roll their best numbers to lower ones.This usually makes them easier to take later.

iii) Lock your opponent (i.e. leave them without a legal move) whenever possible.The bigger characters usually depend on a lock to win. Locking requires you to have the highest, though not necessarily the heaviest, die on the board.

iv) Take your opponent’s heaviest dice (i.e. the ones with most faces). This increases your chances of locking later on, and decreases your opponent’s chances of locking.

v) Maximise your spread (i.e. the range of different numbers you can take) and minimise your opponent’s. A large spread protects against locking.

In general, smaller numbers give better skill plays, because you can add lots of them together without going over 20. But if your numbers don’t add to more than the weight (i.e. number of sides) of your opponent’s heaviest die, you are better off with larger numbers. The chance of getting a skill play is the same, and your power plays are better.

## So You Wanna Play Iago (*James Ernest and Glen Barnett)*

Players love the art and characters of Button Men. Players also love the strategy. In fact, some of them even engage in long conversations about the ideal ways to play. I’ve constructed this article from a recent e-conversation between myself and Glen Barnett, a PHD in statistics working in Australia. Glen’s comments are in blue... (Glen's comments are prefaced with his name in this wiki archive)

I was disappointed that no one took up your challenge of discussing Button Men strategy on rec.games.abstract. But I’m always up for it.

Glen: Glad to hear it! Anyway, I was interested in anyone’s thoughts on good heuristic strategies for the game. Some things are pretty obvious, but the obvious simple strategies are clearly suboptimal often enough to worry me.

I think the basic play strategy evolves something like this:

* 1: I must take more points
* 2: I must therefore lose fewer points
* 3: In a given situation, I should choose an attack which simultaneously lets me roll up my most vulnerable dice, and scores the most points for me.

Glen: Yes. I actually had this much by the time I started my first game.

My current comfort level is with what I’m calling 2nd-level strategy. This introduces two new factors: Score Differential, and Change Killing.

Score Differential says this: With any set of characters, there is probably a “bigger” character. That player must keep some number of sides to win. For example, in Avis(10) vs. Hammer(10), Hammer must keep more than 18.7 sides (i.e., his 20) to win the game.

The equation for sides-you-keep is 2(you-them)/3. This equation is instrumental in choosing appropriate Swing Dice versus a particular opponent. For example, if Avis is playing vs. Hammer(10), she’s better off choosing a 6. Now Hammer must keep 21.3 sides to win.

Glen: Yes, we figured this one out a few weeks ago. It makes a big difference to how quickly you can figure out the best rolls, \*and\* it helps with choosing swing dice, as you note. I mentioned this approach to some of your Demo Monkeys who I am friends with.

Change Killing is how Hammer will protect his 20’s. Suppose the situation is this:

Hammer: 3/6, 3/12, 7/20, and 10 points.

Avis: 3/4, 4/4, 2/10, 6/12, and 30 points.

“Basic” strategy might tell Hammer to take the 6/12, either with his 7/20 or by adding the other two. However, that’s going to hurt him. Hammer’s best move is to take Avis’ 3/4, since that prevents her from making 7, and therefore protects the 7/20.

Taking the 6/12 would leave his 20-sider vulnerable, even if he rolled it. If he rolls the 20-sider to take the 6/12, he runs about a 40% chance of losing the 20-sider on the next turn, and with it the game, since it’s already understood that Hammer must keep 20 points to win.

Glen: Yes, we make similar calculations ourselves. We quickly figured out the importance of leaving your rival with multiples of the same number (like 3,3,6), and of killing combinations that can take your good dice. Often there are two combinations that will get you, and then you need to figure out what dice they have in common and try to kill those.

Glen: e.g. you have 17 on a d20 (which I write as 17(20) myself), and your rival can get that with a 7,6,4 or with a 10,7 (and he only has one 7!). Then if you kill that 7, you have zapped both combinations.

Right. I call that die the “keystone,” if it’s there. Sometimes the opponent has two independent ways to take your big die, so there’s no keystone. Then you have to roll the target die to have a chance at protecting it.

I’m curious if there are situations in which that’s so risky that there’s a smarter thing to do. In other words, leave the die that’s probably lost in favor of trying to protect something else, or make a more devastating attack.

Glen: Of course there are – there are situations where you have to give up a neat attack in order to reroll your forked dice (two independent attacks that add up to your die is like a fork in chess, right?). If that reroll is going to be into nasty territory, you will be better going for the good attack and giving up your die. for example, consider (these may not be real characters, but I could make one up that was):

Glen: 2/4 3/6 12/12 11/20  vs  3/4 5/6 6/6 8/8 12/20

Glen: The first player listed is to move. The d20 is gone unless he rerolls it. Say he takes the d8. The values in the shade are 13 16 19. Erk. Much better to go for an exchange of d20s. Even if that last die of his opponent’s was a d12 he still might be slightly better off taking it in this case.

Glen: This work led me to an early realisation that if you had highish odd values, your opponent couldn’t get them if he only had lower even numbers. Of course that then led straight to what you call change killing – which you \*can\* do in a game, since it isn’t so calculation-heavy.

On the subject of Swing Dice, I’ve made a first draft of a very clunky chart of Avis’ ideal swing dice against all opponents.A point on the chart dictates what Avis should pick against a certain character with a certain Swing Die. “X!Y” means that Avis can either go big or small, and these are the ideal ways of doing either.

“Ideal” is pretty unscientific right now, though. Here’s how I did it:

For smaller characters: If Avis is against a large character, she knows he must keep a certain number of sides to win. Avis will choose the largest die she can without reducing that number of sides. For example, in Avis4 vs. Bauer20, Bauer must keep more than 24 sides to win. This means keeping a 20 and one other die. If Avis upgrades to an 8, Bauer needs only 21.3 sides , but this is equivalent to 24 in terms of his actual dice. Avis gains attack power with no compromise in points. However, if she chooses a 10, the break point is 20 exactly, so Bauer can now tie by keeping a 20. So, vs. Bauer20, Avis chooses 8.

For larger characters, the goal is to choose the largest die possible without being forced to keep more dice. This is essentially the opposite of small characters, though you’re still going as large as possible within a given zone.

For roughly equivalent characters, the goal is to choose the smallest die possible. This is the argument: Let’s say Avis is fighting Karl4. Between the range of Avis4 to Avis12, the game is the same: either player must keep one die to win. Within this range, I propose that it’s preferable to go small, with the intent of going first more often. Players who go first are more likely to retain a die, and in this case that’s all it takes. You’ll notice that this is the only zone where I go low instead of high.

Glen: That top-left-corner item strikes me as a bit odd; if 4 is “ideal” against Avis4, that means the best you’re expecting to do with a free choice of swing die against Avis4 is break even!

In Blackjack, hitting 16 vs dealer 10 is the best move, too. Even though it loses a majority of the time, it still beats the alternative (barely). I won’t try to prove that 4!10 is the right solution to that set, because my math is admittedly mushy. But it -is- true that sometimes the ideal response to a given situation still loses more than half the time.

I’m just starting to get into Shade Theory, which deals with true probability of rolling a number the opponent can’t catch, sort of the refined equivalent of “rolling high.” If a 20-sided die is rolling into (2,3,4), as described above, then it is in the shade on an 8, or a 10 through 20. That’s 12 spots out of 20, or 60 percent safe.

If a 20-sider is rolling into (7,7), then it’s 60% safe. Rolling into (5, 4, 3) is 55%. Rolling into (9,3) it’s 50%. Here’s a neat one: (7,7,7,7) is still 60%.

Glen: I looked at explicit probabilities long before we worked out the 2/3 rule – for me that probabilities was second stage – but the problem is that it’s hard to do the calculations during a game.

Glen: (One thing I did get from playing around with the probabilities is just how important Skill attacks can be in the early game.)

Glen: With the probabilities, it isn’t just rolling what your opponent can’t catch but also setting up rolls that can catch more of your opponent next time around – and rolling a nice low 1 or 2 can often be very important in leaving you with options even after your opponent has done his worst.

## Basic Swing Dice Theory, May 1999 (*James Ernest)*

Here’s the basis of Swing Die theory, as I first wrote it up for a colleague in Berkeley, CA. I’ve made some adjustments to the discussion, but it’s the basic theory responsible for the Avis chart above. If you really hate math, you’ll really hate this.

This discussion pertains to the ideal choice of Swing Die versus a particular opponent. It assumes that you know not only who you are fighting, but also what their Swing Die is. In tournament play, both players choose their initial Swing in secret, knowing their opponent’s fighter. After the first round, only the loser is allowed to re-tune, which makes this chart the most useful.

It’s understood that the bigger of two characters will have to keep a die to win. More specifically, he will have to keep a certain number of -sides- to tie the game, which is equal to two thirds of the difference between the characters’ side totals, or (big-little)\*2/3. Actually, the general case is (A-B)\*2/3 where A isn’t necessarily the bigger character, and when the result is negative it means that B must keep that number of sides.

The exception to this rule is when the two characters are “roughly equivalent,” i.e., either character will win by keeping his smallest die. In this case, either character -must- keep one die to win, because there is no existing way for both players to take all of each others’ dice, and no existing way for both characters to have dice remaining at the end.

Here’s an example. S(Avis6)=36. S(Hammer4)=62. Hammer must keep (62-36)\*2/3, or 17.3 sides to tie, more to win. This means either keeping a 12 and a 6 (18 sides, the closest possible thing above 17.3) or, more likely, a single 20.

(Notations above: Avis12 means Avis with a 12 as her Swing Die. S=Sum of all sides.)

Avis can go for more power in this case and not compromise her position. In fact, she should choose a 12, because Hammer must still keep either 12+6 or 20 sides to win: (62-42)\*2/3=13.3, which is essentially equivalent to the 17.3 she got with her 6. Avis gets more attack power with no loss of scoring position by choosing a 12 over a 6. If, however, Avis chose a 20, Hammer could tie by keeping only 8 sides.

It’s a matter for further debate whether the number of initiative rolls Avis will lose makes up for her added attack power, but I don’t think it does. The primary liability of larger dice is that they are worth more points, and this is irrelevant in the range of 6 to 12 in this case.

So, when playing a small character, the ideal choice is to pick the largest possible die, while still forcing your opponent to keep the most possible points.

When playing a larger character, the goal is to choose the largest possible die without being forced to keep more sides. When Avis faces an even smaller character, Stark4, she must keep some sides to win. S(Stark4)=22. Avis is best off choosing either an 8, 10, or 12 in this case. Versus Stark4, these dice come up as -8, -9.3, and -10.7 for Avis. At Avis8 (-8), she can win by keeping any die larger than 8, and ties if she keeps her 8. At Avis10 (-9.3) she can win by keeping any die 10 or greater, which is three of her dice. At Avis12 (-10.7) she must keep one of her two 12’s to win. Of the three, I prefer Avis10, where she wins by keeping any one of her three largest dice.

When two characters are roughly equivalent, meaning that either must keep one die to win, I prefer to take the smallest possible die that won’t change that. Vs. Karl4, Avis can choose any die between 6 and 12, and the result is the same: Either character must keep one die to win. In this range, we’re assuming that Avis should try to go as fast as she can, and choose the 6. If she goes first more often, and the two characters are the same size, she’ll keep a die more often. That’s the theory, anyway.

All of this is mushy, and the Avis chart above is the result of squinting at the numbers, and the dice each character has, and taking the best shot. But it’s at least a starting point.

## **Article 5: More Swing Die Theory(**Ethan Glasser-Camp)

First, I want to say that that ("himself@cheapass.com") is probably THE coolest email address I have ever seen.

One of the great things about Button Men theory is that anything you "discover" is likely to be noticed because of how little has been done with Button Men as yet. Anyhow, the other day en route to a gaming place in Manhattan, I got kind of bored and considered:

Let's say I'm playing as a character, X, who has a sides total without Swing Die of 60. I'm playing against character W, who has a sides total with Swing Die of 54. What would be the Swing Die that I could choose that would be the only one I would have to keep? Or, in other words, since I have to keep 2/3 \* (X-W), what die could I choose that would equal 2/3 \* (X-W)? Since whatever die I choose will add to X, I found myself facing a problem similar to the rocket equation that engineers find themselves facing. More weight requires more thrust. More thrust requires more fuel. More fuel means more weight.

Anyhow, after algebrizing the equation, I found that the swing die I sought was equal to double the difference of the two sides totals. Thus 2 \* (X-W) or 2 \* (60-54) or 2 \* 6 or 12 was the number of sides I was looking for. I decided to call this the round swing die, and realized how rarely it was going to be used, since double the difference of sides totals is rarely between 4 and 20. But anyhow I noticed that since the sides total is always a whole number, the difference would also always be a whole number, and so would the round swing die. The other thing I noticed was that a round swing die would always be a multiple of 2.

As I said, I haven't found a use for this yet, but hey, Rome wasn't burned in a day. And when in Rome..

## More Swing Dice Thoughts (*Ryan Keane)*

Here’s a general model for determining the “optimal” swing die each game. This model assumes that each character being used has 1 swing die and no Option, Poison, Null, Turbo Swing, Mood Swing, or Winslow dice, as these affect one’s hand size for scoring purposes. Also note that the model is based on a simplified game ending where the larger player is left with 1 die; it does not predict the outcome of games where the larger player is left with 2 or more dice (I’ve left this out for simplification) or where there is a “stalemate” (both players still having dice at the end).

Player A is the character with the larger initial hand size (i.e. hand total w/o swing die). Note that Auxiliary and Reserve dice, once added to the hand, should be included in the initial hand size total.

Player B is the character with the smaller initial hand size.

Player A’s formula for swing die choice:

Ax<=Bx-Di+3/2\*(An)-1/2

Player B’s formula for swing die choice:

Bx<=Ax+Di-3/2\*(An)-1/2

Ax: Player A’s swing die size

Amax: A’s maximum swing size

Bx: Player B’s swing die size

Di: Player A’s initial hand size minus Player B’s initial hand size (greater than -1 by definition).

An: The size of one of Player A’s dice which must be kept for him to win. This value can vary from Player A’s smallest to largest die, including his swing die. Optimally, Player A wants An to equal his smallest die, while Player B wants An to equal Player A’s largest die. However, in practice one must vary An to find the optimal swing die that fits in one’s swing range

(eg. 4-20).

Let’s do an example:

You, playing Avis, just lost against Hammer(4) (i.e. with a d4 swing die).

Avis: 4,4,10,12,X

Hammer: 6,12,20,20,X

What swing die should you use next round?

I’m player B because my initial hand size is 30, whereas Hammer’s is 58; so we use the second formula above. Ax=4 and Di=28, so…

For An=4:

Bx<=4+28-3/2(4)-1/2=25.5 With a Swing d20 (since Avis can’t use a d24 swing), Player A will lose with just a d4 left.

For An=6

Bx<=22.5 With a Swing d20 (since Avis can’t use a d21 swing), Player A will lose with just a d6 left.

For An=12

Bx<=13.5 With a Swing d13, Player A will lose with just a d12 left.

For An=20

Bx<=1.5 Since Avis can’t use a d1 swing, no possible swing will result in Player A losing with just a d20.

So now it comes down to a choice between a d20 (Hammer(4) can win with 3 out of his 5 dice) and a d13 (can win with 2 of his dice). I’m not sure what the optimal choice is – you’ll need another model :).

Another example:

You’re Hammer and just lost to Avis(20).

In addition to the An values included in Hammer’s initial hand size, we’ll also include the case where you win with just your swing die left

(An=Ax=2(Di-Bx)+1).

For An=6

Ax<=20-28+3/2(6)-1/2=0.5 You can’t win with just a d6, no matter what swing you use.

For An=12

Ax<=9.5 With a Swing d9, you can win with a d12 (3 of your 5 dice).

For An=17

Ax=17 With a Swing d17, you can win with a d17 (3 of your 5 dice).

For An=20

Ax<=21.5 With a Swing d20, you can win with a d20 (3 of your 5 dice).

So a d20 gives you the most bang without giving up anything in terms of hand scoring.

How about against Avis(13):

For An=6

Ax<=13-28+3/2(6)-1/2=-6.5 You can’t win with just a d6, no matter what swing you use.

For An=12

Ax<=2.5 You can’t win with just a d12, no matter what swing you use.

For An=20

Ax<=14.5 With a Swing d14, you can win with a d20 (2 of your 5 dice).

So d14 is your best swing.

On a side note, it would be interesting to know which pair-offs reach swing equilibrium, where both players are using their optimal swing dice and do not change them when they lose, and which pair-offs oscillate, each player changing their swing die each time they lose to counter their opponent’s swing choice. If equilibrating pair-offs exist, then for those pair-offs you would always know the optimal swing die to start with. Maybe I’ll work on that next…

## Speed Dice Counter Strategies (*Rick Healey)*

First, the subject that stressed me before – Speed Dice, and the level of power that they grant a player in Button Men. I’ve been doing rough experiments (involving both the buttons I have and the BM 1999 poster for the recepies that I don’t), and I’ve got some concrete strategies in place.

Now, first, it’s obvious that the Speed Dice were made to compensate for powerful dice of small values, such as the Trip Dice of the Lunch Money Button Men. That’s what you said, at least.

Thus, the best shot at taking down the Speed Dice is to have as many disparate dice as possible, to minimize the impact of Speed Dice. This is of vital importance against Bennett, with his two Speed d20s. However, on the flip side, you’re not going to worry much about Hale’s solitary Speed d8.

However, if you just go for sheer overpowering (read: you pull out Iago and take a d20 as a Swing Die), you’ll still discover that you’re being munched. So far, it seems that the best strategy against them is to have as wide a range of die sizes as possible. Obviously, against Bennett, you’ll want to favor the heavier sizes. But if you are already playing a heavy die user, like Iago, Hammer, or Crusher, it’s actually best to use a d4 as a Swing die. Logic being that all of the other dice are already relatively safe (of course, nothing is going to save you from watching all of your dice coming up between 2 and 5 and Bennett getting a 1 and a large Speed roll), you need the speed edge to take care of your opponent. Since very few of the Brawl Buttons even have the chance to go first (Darwin being the only one that has a d4 in his recipie) without some luck, just one tiny die amidst a horde of large dice can mean the difference between a close game and pummeling the Brawlers.

This generally means one thing – trying to equal the opposition’s recepies when possible. This isn’t always that vital – again, odds are against Hale doing a massive Speed attack, while I’ve seen multiple situations while testing that have Bennett taking four dice at once. I’m bad at math, but so far as I can tell, you need to have a number of sides no fewer than 1.5 times the total of “Speed sides” (number of sides on combined Speed Dice) to put yourself at an even pace. While this means that you can pretty much treat Hale as a normal Button that can occasionally do a decent die nab, Bennett (40 Speed sides) and Morgan (minimum 38 Speed sides) can only be taken on by some of the more hefty characters. If your character can equal the target number I’ve estimated above (again, you’ll probably want a math person to give you a more precise number) before swing dice, then shoot low with the Swing. If you need to bulk up to get there (Stark players take note), then grab your d20 and start rolling.

Actually, another interesting phenomenon I’ve run into concerning Speed Dice. They do have one serious flaw – they don’t stand up well to Focus Dice. Reason being this: they can “minimize” themselves on the first turn, taking away a possible Speed Attack. Since the rules do not prohibit reducing a Focus die that has nothing to do with initiative (such as reducing a 9 on a Focus Die to a 7 when your opponent has a 1 showing, you can nullify any potential Speed attacks that your opponent could try.

Now, to make a mad rush to get the Lo5R Button Men…

Actually, I could see another theoretical sort of die, which could be reduced at any time, not just on the first turn, to help against this sort of thing. I shall call this theoretical die a “Fading Die,” and it also couldn’t attack the same turn as it was reduced.

## Standard Versus Twin Dice? (*Ryan McGuire)*

I just wanted to go into a little more detail about Twin Dice being roughly equal to a regular die with the same mean and number of sides.

Spoiler: Twin Dice are better than normal. Normal dice are better than ‘Sidekicks’.

[Definition: “Sidekicks” are coins which show either their face value (heads) or one (tails). For example, a 5-point Sidekick will always show either 5 or 1 in a 50/50 distribution. They haven’t been used yet, but they are interesting. This essay also uses modified Triple Dice, such as (2,2,2)-1. These are also theoretical, and a little cumbersome. The (2,2,2)-1 is equal to the sum of three 2-sided dice minus one, and generates the results (2, 3, 3, 3, 4, 4, 4, 5).]

Power Attacks:

Case 1: Compare (2,2,2)-1 to d6 to a 6-cent sidekick (1 or 6)

——-

Table 1: I have a (2,2,2)-1 and you have a d6. Who would win? (Y=you, M=me, E=either, depending on who goes first)

Result: 20 M’s, 20Y’s, 8E’s.

Table 2: I have a d6 you have a 6-cent Sidekick

Result: 5M’s, 5Y’s, and 2E’s.

Conclusion: I guess it just doesn’t matter.

Case 2: You have a d4, I have either a (2,2,2)-1, d6, or a 6cent Sidekick (either 1 or 6).

——-

Table 3: My (2,2,2)-1 vs. your d4:

20M’s = .62500; 5 Y’s = .15625; 7E’s = .21875

Table 4: My d6 vs. your d4:

Result: 14 M’s = .58333; 6 Y’s = .25000; 4E’s = .16666

Table 5: My 6-cent sidekick vs. your d4:

Result: 4M’s = .500; 3 Y’s = .375; 1 E = .125

Conclusion: Versus a d4, I’d much rather have the (2,2,2)-1

Case 3: You have a d8, I have either a (2,2,2)-1, d6, or a 6-cent sidekick.

(tables omitted)

Results:

(2,2,2)-1: 20 M’s = .3125; 36 Y’s = .5625; 8 E’s = .125

d6: 15 M’s = .3125; 27 Y’s = .5625; 6 E’s = .125

Sidekick: 5 M’s = .3125; 9 Y’s = .5625; 2 E’s = .125

Conclusion: Well THAT was surprising. Considering these three Power Attack results, I’d rather have Twin Dice rather than a single die with the same number of sides and same average roll. Against equal or larger dice there’s no real difference. Against smaller dice the Twin Dice do slightly better.

Skill Attacks:

That’s all fine, but what about comparing the dice for Skill Attacks?

Case 4: You have a d12 left. I have a d10 and a d6.

——-

What are the chances that I can make a successful Power Attack? There is a 365/720 chance of at least one of my dice being greater than or equal to yours.

What are the chances that I can make a successful Skill Attack? There is a 50/720 chance of a good Skill Attack. Together that’s 415/720 or about 58% chance that I can take your last die this turn.

Case 5: Same as Case 4 but I have a d6 and a dime sidekick (1 or 10).

——-

The chances of a successful Power Attack are still 365/720. However the chances of a successful Skill Attack drop to only 7/720. The total of 372/720 is only about 52%.

In this case, you only have one die so Skill Attacks are out of the question for you. Plus I’ve looked at the chances of a this-turn kill without worrying about retaliation.

## Even More Button Men Theory (*James Ernest and Rick Healey)*

James’s comments are ~~in blue~~ notated with his name.

I’ve been pouring over the site over the essays given out on Button Men so far. I picked up the game when it first came out, and I now have at least two packs from each CAG expansion. Here’s a couple of things I have noted with the addition of the new die types.

With Shadow Dice, you can achieve something initially impossible – a stalemate. Let’s say you take Shore from Button Men: Soldiers and face him off against Buddy from Vampyres. We will wind the match down by assuming that the following values are left on the dice:

Shore(6): 2/4, 1/4, 1/20 points = 34

Buddy(4): Sh5/6, Sh18/20 points = 26

(Sh next to a die indicates a Shadow Die)

Shore can’t get at Buddy’s dice by any sort of attack, and Buddy’s values are too high to perform a Shadow attack. this leaves both sides with adding half of their remaining dice. Technically, the game rules don’t detail what happens in the event of a draw, but I play with both sides get half of what remains. This would give the following tally:

Shore = 48 pts.

Buddy = 39 pts.

Obviously, Shore won this round. It also helped that he did capture more dice in this example, but here is the strategy that someone is Shore’s position could use: capture only those dice which are a threat. Clearly, if all your rolls are under a Shadow Die’s purview, you can ignore it. of course, this can result in very humorous matches, such as when anyone from the first series goes against Peace or Wastenott – you could theoretically have a draw before any dice were taken (although it would take some incredible chance to pull that off).

James: Actually, while the rules in the core set don’t cover this case, the rules in Vampyres (and afterwards) actually do. The game ends no when one player is out of dice, but when both players pass. Thus, the situation you described above isn’t actually a stalemate, but just a game end.

James: I think you are scoring the game the way it’s intended, we just don’t call it a “draw.” It’s simply the end of the game. And it’s true: The existence of Shadow Dice makes the strategy of always trying to roll as high as possible no longer valid. Especially when those Shadow Dice are also Poisoned.

I personally love Shadow dice, and my current favorite to use of the buttons I have is Dunkirk. Even non-Poison Shadow Dice are excellent – the psychological advantage that they give over the opponent can allow you to rip through their dice as they ignore a normal d12 to eliminate a Shadow d6 with a 1 showing.

As for the scoring, since I think of it as a fight of sorts, I guess that’s why I see both opponents with dice remaining as a “draw”. So long as I score it correctly, though. Coincidentally, I have gotten one legitimate draw. I was playing Wastenott, while my opponent played Shore. I forget the exact Swing Dice values, but we ended up with an endgame scenario and both of us had the same points. We weren’t playing tournament style, so we just went at it again with the same Swing Dice (I sadly lost the rematch).

Poison Dice are rather fun, and can easily foul your opponent’s day. Poison Dice are hard to judge at the moment, especially for Reaver, with the Poison Swing Die. In my experience, players are a bit shy about taking the Poison Die in general, so a low Poison Die such as a d4 gives you two advantages – you can go first, and you also have a better chance to pull of a skill attack. However, if your opponent thinks that munching a Poison d4 is no problem, it may be time to shift over to the Poison d20, which has lots of stopping power and can make your opponent think twice before eliminating it. So far, I favor the low values, but the fact that they only lose two points with a Poison d4 doesn’t bother them as much anymore.

James: The way to force away your singular large Poison Dice is to constantly make attacks with them, until they are too small to ignore. If your p20 is showing a 12, roll it again and try for a 3 or 4. You will eventually force it away.

So far, this has rarely been an issue, as the only Button Man that I own with a Poison d20 is Jellybean (I’m a college student – I can only afford to buy half of CAG’s catalogue 😉 ).

James: The strategy of rolling low on your biggest Poison Dice is never so important as when you are playing Max, a Freak with eight Poison dice, including: p12, p12, p20, p20, p30, p30, pX, pX. In a game with Max, the object is always to wind up with the higher negative score!

So I see. Given Max’s absurd values, I’d go for another pair of d20’s for Swing dice. Of course, I’m the only guy I know of that could handle rolling 4d20 simultaneously (although when I do pick up Max, I’m going to need to buy one more d30. Just one.)

Finally, the Speed Die. You opened up a can of worms with this one. Already, I get dirty looks from the other Button Men fans aong my friends when I pull out the Brawl expansion. If any die ever invoked irrational fear in an opponent, it’s that. Actually, it may be the best psych-out tool ever. While they concentrate on the Speed Die, fearful of its effects, you can mop up with the other dice. The only saving grace is that most of the Brawl buttons have either small Speed Dice or the ‘S’ swing to prevent them from taking a low swing and going first.

James: It’s true: Speed Dice look more powerful than they really are, especially in the particular recipes you find in the Brawl Button Men set. The only truly overpowered Brawler is Sydney, the rare, who we knew would be too powerful the moment we designer her. Some rares just need to be strong, to compete with, you know, the other strong rares. 😉

Again, since I don’t have any rares outside of Gordo, I can’t really judge Sydney there. Of course, I don’t think that the rares are super powerful. Echo, for example, is just like a normal Button (literally). Giant is rather horking, but the fact that there is no way for him to go first takes some of the meat from him. Changeling is rather absurd, of course, but Zeppo isn’t that bad at all – he depends on good rolls and a good choice of Swing die each round. All in all, even the rares don’t hold a very large advantage over the opposition, with the exception of Changeling.

And then there’s Bennett. With a d6, he has a reasonable shot at going first against a lot of foes. Not only that, but unless you’re playing another Brawl character and Bennett’s unlucky with the roll, you’re going to have a decent shot at facing at least one Speed d20 on your turn. One friend of mine is still bitter when he watched Bennett take four of Stark’s dice at once with a Speed Attack. Granted, you don’t get that every day, but I had the largest score I ever had, including the time I endplayed Iago with a d20 swing.

Right now, the Brawl characters are absurdly too powerful. While it does add another dimension to the game (by rolling just one die, you could ruin a couple of Speed Attacks), it has too much power. It just munches the normal characters, and even the Shadow Dice have a hard time with them. Personally, I’m going to be wary of anyone walking in with Bennett pinned to their shirt/bag/Ass-kicker 2000.

I’m hoping that Something in Freaks will balance them out (personally, I’m hoping Max’s 8 Poison Dice will do the trick). Until then, I’m going to try to find strategies for facing down Speed Dice.

James: You’ll find them. Speed Dice were designed specifically to destroy the previous dominant Button Men set, the oft-overlooked Lunch Money girls. In the hands of a decent player, the six Lunch Money characters with their tiny dice and Trip Dice were by far the strongest in the set. The Speed Die characters take out anything with smaller dice, but fare rather poorly against heavy hitters like Bauer, Hammer, Crusher, and even Giant.

Perhaps that’s my catch. I like finesse moves more – I like to take small characters and use smaller Swing Dice to be able to have several options in terms of taking dice. Plus, if I go first, I have a good chance of endplaying my opponent, making him lose more sides than he can take from me. Of course, I mix up this strategy to keep my opponents from being able to read me like a book.

As for the Lunch Money girls, they aren’t sold in this area. I had to play at the local games shop owner (Pandemonium Books & Games of Cambridge, MA) to provide more Button Men. He didn’t start carrying the Brawl expansion until I asked him about it. I better get to work on asking about the girls (of course, since he is making an effort to carry them now, I really ought to by the Sailor Moon ones too).

James: So, when you see Bennett coming, grab your big hitter. Tune him high, take out Bennett’s big die, and protect the points you need. Bennett’s got so many points himself that you won’t need to protect much.

James: If that doesn’t work, call Nick Sauer. He wrote the damned thing. 😉

I may ring him up… oh wait, I don’t have any way to contact him. I’m sure he will eventually see this anyways, so maybe I’ll talk to him eventually.

For Bennett… I guess it’s going to have to be a case of munch quickly at the Speed Dice and hope I don’t roll what he needs to Speed attack me. Either that, or I’m going to have to pull out Wastenott and hope for some good Shadow d20 rolls.

## Trip Dice Odds (*Joe Kisenwether)*

*(Joe Kisenwether is the co-designer of the Freaks and Fantasy Button Men expansions)*

Odds of Successful Tripping

Suppose you’re trying to trip a d10 with d6. How likely are you to get away with it? The most straightforward way to figure this out is making a table of the possible outcomes of the trip roll:

The columns indicate the result on the d10, the rows the result on the d6 and X’s appear in those boxes where the trip was successful. We can see that the trip works 21 out of 60 times, which simplifies to 7/20 = 35%.

The layout of the X’s in the table above should suggest something to you if you’ve read the article on “The Highest Lower Number”. Triangular numbers. They show up again in the formula for how likely a trip is to succeed. If you have a trip die with N sides and your opponent has a regular die with X sides, and his die is at least as large as yours, then the probability that you can successfully trip is: (The Nth triangular number)/(N\*X). Since the formula for the Nth triangular number is N\*(N+1)/2, we can simplify the previous formula to (N+1)/(2X). Considering the previous example, d6t vs. d10, we get a probability of (6+1)/(2\*10) = 7/20 = 35%.

If the trip die N is larger than the target die X, then the formula is 1-(the (X-1)th triangular number)/(N\*X). Which simplifies to (2N-X+1)/(2N). So a trip six trying to take a d4 would win (2\*6-4+1)/(2\*6) = 9/12 = 75% of the time.

One more time. The probability of an N sided trip die successfully tripping an X sided die is:

(N+1)/(2X) when X>=N

and

(2N-X+1)/(2N) when X<N

OK, Now lets do a practical example.

You’ve got a d10 showing a 10, he’s got a trip d6 showing a 4 and a d8 showing a 6 (10/10) vs. (4/6t , 6/8) You need to keep your 10 to win. Do you take the trip die, or the die showing the larger number?

If you take the trip die, you will have to re-roll the 10, and must deal with a 60% chance of rolling a 6 or less and losing. On the other hand, if you capture the d8, you will only have a 40% chance of rolling a 4 or less, but your opponent will then get a chance to trip you. The probability that you would then lose the game =

(Probability of rolling <=4) + (Probability of rolling >4)\*(Probability of being tripped) = .4 + .6 \*(7/20) = 61%

So in this case, the (slightly) wiser thing to do is take the trip die.

OK, for extra credit, here’s a tough one:

(6/6t , 8/12) vs. (5/6 , 20/20)

Your opponent will win if he keeps his 20. Do you:

* a) Take the d6 with the trip die
* b) Trip the d6
* c) Take the d6 with the d12
* d) Trip the d20

Good Luck!

## Why are Shadow Dice powerful? (*Joe Kisenwether)*

Shadow dice just make power attacks upward instead of downward. This doesn’t increase their field of attack (the set of numbers they could capture) at all, just changes it. A roll of a 4 on a regular six-sider corresponds to a roll of a 2 on a shadow six. In each case four of the six numbers in the die’s range can be hit. No net increase in attacking ability. On average, a randomly rolled N-sider will be able to attack (N+1)/2 different values. This doesn’t change when the die is shadow.

So why are they so effective? Well, the first reason is that it is USUALLY the dice showing the larger numbers that our opponent doesn’t want taken, and these are more likely to fall to a shadow die than a regular because the shadow’s field of attack consists of higher numbers. But that’s only “usually”. If our opponent has shadow dice as well, he wants to hold on to the shadow dice that show low numbers. The numbers that would not typically be in our shadow die’s range. This would seem to indicate that regular dice are more effective against shadow dice than other shadow dice are. And yet, I’d much rather face an all shadow character like Wastenott < (4) (8) (10) (20) (X) > with Starchylde < (6) 8 (10) 12 X > than with Niles < 6 10 10 12 X >. In fact, I would much rather face any character with Starchylde than with Niles. Why does making a few of the dice shadow make such a difference between characters that have almost the same die sizes? And if Shadow dice are so Powerful, why aren’t Wastenott and Peace < (10) (12) (20) (X) (X) > more potent than they are?

A pair of regular six-siders showing 2 and 4 could capture a 1,2,3,4, or 6. Five different values. The formula is simply MAX(die1,die2)+1. The plus one is the skill attack using both dice. On average, a pair of randomly rolled regular N-siders will be able to attack (4N^3 + 9N^2 – N) / (6N^2) different values. (I’m not going to go into detail as to how to derive these formulas. Just generate the first few values by hand then solve for the rational function that passes through them.)

A pair of shadow six-siders’s showing 2 and 4 could attack 2,3,4,5,6. Five different values. For two shadow dice, the formula is N-MIN(die1,die2) (plus 1 if die1+die2>N). For a pair of shadow N’s, the expected field of attack is (4N^3 + 6N^2 + 2N)/(6N^2). Calculating (or just referring to the chart below) we see that this means a pair of shadows actually has a WORSE average field of attack than a pair of regulars! Between 3% and 9% worse. What’s going on here?

Let’s look at what happens when you mix shadow and regular together. A regular die showing a 2 and a shadow showing a 4 could capture 1,2,4,5 or 6. Five different values. The formula is MIN[(regular die &endash; shadow die + N +1),N] ( plus 1 if die1+die2 >N) One regular N-sider and one shadow N-sider will have a field of attack of: (5N^3 + 6N^2 + N) / 6N^2. Considerably better than either type of unmixed pair. Between 6% and 19% better.

The question is why? The answer is that a regular die and a shadow die are less likely to have their fields of attack overlap one another. Less redundant attacks leads to a larger combined field of attack even though each individual die of the mismatched pair is no stronger than each individual die of matched one.

Conclusion: The strength of shadow dice comes largely from combining them with regular dice because when you do, your fields of attack are less likely to overlap.

OK, now what about Queer dice? They are shadow if showing an odd number and regular if showing an even. Individually, they are slightly stronger than either regular or shadow, having an average field of attack of (N+2)/2 rather than (N+1)/2.

A pair of queer six-siders showing 3 and 4 would be able to capture 3,4,5,6 (with the 3) 1,2,3,4 (with the 4) or 7 (with the skill attack). Seven different values. Taken in pairs, they have an average field of attack of (6N^3 + 5N^2 &endash; 6N) / (8N^2) when N is an even number and (6N^3 + 9N^2 + 2N – 1) / (8N^2) when N is an odd number. Not surprisingly, we find that their average field of attack lies between that of the matched pairs and the unmatched.

Here are a few things that I highly suspect, but have yet to “do the math” for:

Conjecture 1: The difference between mixed sets and all regular / all shadow sets will become more pronounced as more dice are used, with the optimal mix being 50% shadow, 50% regular.

Conjecture 2: Larger sets of Queer dice will behave more like mixed sets and less like unmixed ones i.e. queer dice work better when you have a bunch of them. (Which is good, since at present the only character who has them has nothing but Simon <4q 6q 12q 20q Xq> )

OK, one last question. Does any of this have any bearing on playing the game? Well, it can help with character selection, now that you know mixed sets are more powerful than unmixed. It can also come up during the game. If you have a choice between taking a shadow die and a regular, all other things being equal, you will want to take the one that leaves your opponents shadow/regular ratio furthest from balanced.

Die Size vs. Expected Size of Field of Attack

## More Thoughts on Shadow Dice (*Michael Martin)*

I really like the new Button Men website, though I notice that the strategy section didn’t have anything new on it. I cut my Button Men teeth on Grist and Wastenott, and Shadow Dice have been near to my heart ever since. Here’s some thoughts my game group and I have had about Shadows:

Shadow Dice are interesting on a number of levels. At first glance, they appear strong — they can wipe out high-valued dice. At second glance, they appear weak — since they can’t be both safe and useful at the same time, except against other Shadow dice. After a few games against Peace or Wastenott, you then realize that playing as or against Shadow Dice completely change the game.

I classify my Shadow Dice into three types:

– “Blind” dice are showing values higher than all my opponent’s dice. They’re vulnerable to Skill attacks, but otherwise are completely out of play. The existence of blind dice is what lets both people have dice left at the end.

– “Active” dice can make captures. Unless that’s against another Shadow Die, that means it’s vulnerable, too.

– “Hosed” dice can’t make captures, but that’s because the opponent’s dice are showing values above the Shadow Die’s maximum value. They can’t act (except as part of a Skill Attack), but they can be taken.

When playing with a mix of Shadow and Normal Dice, you can use your Normals to attack the dice your Shadows are blind to. Things get more interesting when dealing with characters like Wastenott or Peace, or once your normal dice have all been captured.

If you’re Peace, an endgame that benefits you is one where you’ve hung on to your heavies, they’re showing high values, thus locking you out, and your opponent is locked out with a bunch of lighter dice. Yes, that’s an advantage even if you’re not using Shadows, but with Shadows, you can hit their heavies a bit more easily, and then proceed to blind all your dice. Round ends, you get lots of points for retaining dice, they don’t get nearly as many.

Hence, Peace and Wastenott, right? Conflicts that end early, without too much destruction. Very appropriate.

So, some basic strategies against each type of die would be:

– Against normal dice, as above. Let them keep the dice that are lighter than yours. Force a stalemate and net points on defence. Or, just wipe them out, if you can.

– Against Shadow dice, this is essentially the same as normal vs. normal.

– Against poison dice, don’t take them. Unlike players with normal dice, you can retain lots of dice left and still leave them the poison at the round’s end. Your only real danger is poison d12s and d20s, as Shadows are \*very\* effective against heavy dice, and they can blind you to everything but the high-valued Poisons if you’re all Shadow.

– Against Trip dice, abandon hope. You \*must\* take them every chance you get, but against a savvy player your only chance is right at the beginning. Against Hope, Peace and Wastenott are guaranteed to be doomed, unless they roll two ones and an additional number below 4 right from the start.

Trip dice prey on the weakness of Shadow dice. Let us imagine that Hope and Wastenott are battling. Wastenott has been doing pretty well and has managed to wipe out Hope down to her 1-sided trip die without losing any dice himself. None of his dice are showing a 1. Wastenott is now utterly, completely hosed. Wastenott passes. Hope trips a random die. If that die comes up less than or equal , Hope takes it. Otherwise, it rolled something over 1. Wastenott passes. Hope continues. Each attack sequence has a probability of capturing, and Wastenott is completely locked down. He will eventually be wiped out. This will occur essentially whenever an all-Shadow character must pass against a character with Trip Dice. (We first encountered this effect when taking Faith vs. Wastenott, and all Wastenott’s values were above 2.)

I did some basic probability calculations on the various shadow dice for chances of being Blind, Active or Hosed against dice of various other sizes. I didn’t really know what I’d be doing with these when I wrote the program, but some interesting results emerged. With an equal die distribution amongst the six major die sizes, the 8-sided die is active the most, with the 10-sided being a close second. If the d16 is thrown into the mix, the 10-sided becomes marginally more active. Likewise, the 4-sided die is blind the least, and the 20-sided die is never hosed. When given a Shadow Swing die, my Shadowy friends and I tend to pick in the 8-12 range. When playing Peace or Wastenott, I tend to like 10-sided shadow dice, as they can target the heavier dice often without being too juicy a target themselves. I would suppose that a stalemate-oriented player would prefer 20-sided Shadows, but they’d be real liabilities unless you’re very good at Change Killing.

Here are the probability tables my program computed:

Averages without d16:

Die | Blind | Active | Hosed

—–+——-+——–+——-

4 | 0.194 | 0.323 | 0.483

6 | 0.316 | 0.376 | 0.308

8 | 0.417 | 0.394 | 0.189

10 | 0.500 | 0.389 | 0.111

12 | 0.567 | 0.367 | 0.067

20 | 0.725 | 0.275 | 0.000

Averages with d16:

Die | Blind | Active | Hosed

—–+——-+——–+——-

4 | 0.179 | 0.299 | 0.521

6 | 0.293 | 0.353 | 0.354

8 | 0.389 | 0.378 | 0.233

10 | 0.469 | 0.382 | 0.149

12 | 0.535 | 0.372 | 0.093

16 | 0.634 | 0.337 | 0.029

20 | 0.704 | 0.296 | 0.000

Shadow d4

Die | Blind | Active | Hosed

—–+——-+——–+——-

4 | 0.375 | 0.625 | 0.000

6 | 0.250 | 0.417 | 0.333

8 | 0.188 | 0.312 | 0.500

10 | 0.150 | 0.250 | 0.600

12 | 0.125 | 0.208 | 0.667

16 | 0.094 | 0.156 | 0.750

20 | 0.075 | 0.125 | 0.800

—–+——-+——–+——-

Avg | 0.179 | 0.299 | 0.521

Shadow d6

Die | Blind | Active | Hosed

—–+——-+——–+——-

4 | 0.583 | 0.417 | 0.000

6 | 0.417 | 0.583 | 0.000

8 | 0.312 | 0.438 | 0.250

10 | 0.250 | 0.350 | 0.400

12 | 0.208 | 0.292 | 0.500

16 | 0.156 | 0.219 | 0.625

20 | 0.125 | 0.175 | 0.700

—–+——-+——–+——-

Avg | 0.293 | 0.353 | 0.354

Shadow d8

Die | Blind | Active | Hosed

—–+——-+——–+——-

4 | 0.688 | 0.312 | 0.000

6 | 0.562 | 0.438 | 0.000

8 | 0.438 | 0.562 | 0.000

10 | 0.350 | 0.450 | 0.200

12 | 0.292 | 0.375 | 0.333

16 | 0.219 | 0.281 | 0.500

20 | 0.175 | 0.225 | 0.600

—–+——-+——–+——-

Avg | 0.389 | 0.378 | 0.233

Shadow d10

Die | Blind | Active | Hosed

—–+——-+——–+——-

4 | 0.750 | 0.250 | 0.000

6 | 0.650 | 0.350 | 0.000

8 | 0.550 | 0.450 | 0.000

10 | 0.450 | 0.550 | 0.000

12 | 0.375 | 0.458 | 0.167

16 | 0.281 | 0.344 | 0.375

20 | 0.225 | 0.275 | 0.500

—–+——-+——–+——-

Avg | 0.469 | 0.382 | 0.149

Shadow d12

Die | Blind | Active | Hosed

—–+——-+——–+——-

4 | 0.792 | 0.208 | 0.000

6 | 0.708 | 0.292 | 0.000

8 | 0.625 | 0.375 | 0.000

10 | 0.542 | 0.458 | 0.000

12 | 0.458 | 0.542 | 0.000

16 | 0.344 | 0.406 | 0.250

20 | 0.275 | 0.325 | 0.400

—–+——-+——–+——-

Avg | 0.535 | 0.372 | 0.093

Shadow d16

Die | Blind | Active | Hosed

—–+——-+——–+——-

4 | 0.844 | 0.156 | 0.000

6 | 0.781 | 0.219 | 0.000

8 | 0.719 | 0.281 | 0.000

10 | 0.656 | 0.344 | 0.000

12 | 0.594 | 0.406 | 0.000

16 | 0.469 | 0.531 | 0.000

20 | 0.375 | 0.425 | 0.200

—–+——-+——–+——-

Avg | 0.634 | 0.337 | 0.029

Shadow d20

Die | Blind | Active | Hosed

—–+——-+——–+——-

4 | 0.875 | 0.125 | 0.000

6 | 0.825 | 0.175 | 0.000

8 | 0.775 | 0.225 | 0.000

10 | 0.725 | 0.275 | 0.000

12 | 0.675 | 0.325 | 0.000

16 | 0.575 | 0.425 | 0.000

20 | 0.475 | 0.525 | 0.000

—–+——-+——–+——-

Avg | 0.704 | 0.296 | 0.000

## Odd Dice Math (Ryan Keane)

On the Cheapass e-group, people are always asking how to hunt down non-standard dice, so I thought I'd write up a quick list for the Beat People Up site on how to fulfill your odd dice needs by using blank dice. Credit should go to Glen Barnett and Ryan McGuire, among others, who also came up with many of these ideas.

For most of these, there are several ways to do it. I've chosen the method that uses the least dice, uses only blank d6's when possible, and/or I just prefer. Blank d20's might be easier to find than blank d10's, so you can just use a blank d20 whenever I call for a blank d10. Since some of the larger dice build upon the smaller ones, for brevity I just refer to smaller non-standard dice when necessary.

Cheap blank d6's are available for $0.15 each from http://members.aol.com/pigonmath/page3.html

Chessex has blank d20's for $0.40 each at http://www.chessex.com/Special\_Sets\_Blank.htm

Here's the list:

* d1: anything
* d2: d6 (1,1,1,2,2,2)
* d3: d6 (1,1,2,2,3,3)
* d4: available
* d5: d10 (1,1,2,2,3,3,4,4,5,5)
* d6: available
* d7: \*
* d8: available
* d9: d3 + d6 (0,0,3,3,6,6)
* d10: available
* d11:
* d12: available
* d13: \*\*\*
* d14: \*
* d15: d5 + d6 (0,0,5,5,10,10)
* d16: d8 + d6 (0,0,0,8,8,8)
* d17: \*\*
* d18: d6 + d6 (0,0,6,6,12,12)
* d19:
* d20: available
* d21: \*
* d24: d8 + d6 (0,0,8,8,16,16)
* d25: d5 + d10 (0,0,5,5,10,10,15,15,20,20)
* d26: \*\*\*
* d27: d9 + d6 (0,0,9,9,18,18)
* d28: \*
* d29:
* d30: semi-available [d10 + d6 (0,0,10,10,20,20)]

\* There are d7's around if you can find them. They are said to roll fair, with the length-width ratio having been optimized for randomness, but I haven't seen any tests (check them out at http://www.multimania.com/arjan/pip07.htm). You might be able to get these from http://www.brsnasis.com/vendor/catGMS.htm, but I think they're out of stock. Their source, who is retired, is:

* Lou Zocchi
* Gamescience
* 7604 Newton Dr, N. Biloxi MS 39532
* phone 228 392 4177

However, it is possible to make home-made fair rolling 7-sided dice spinners by cutting a heptagon out of board and placing a rod through the center (you can do this for any # of sides - the internal angles of an n-sided polygon are 180(n-2)/n) (take a look at a 3-sided spinner http://beta.content.communities.msn.co.uk/DiceManiacsClubakaTheRandomFandom/ alexandersdice.msnw?action=ShowPhoto&PhotoID=105).

If you do have a d7, you can make d14's [d7 + d6 (0,0,0,7,7,7)], d21's [d7 + d6 (0,0,7,7,14,14)], and d28's [(d7 + d20 (0,0,0,0,0,7,7,7,7,7,14,14,14,14,14,21,21,21,21,21)].

Also, I've seen a few d14's around (http://beta.content.communities.msn.co.uk/DiceManiacsClubakaTheRandomFandom/ kevincooksuncommonsideddice.msnw?action=ShowPhoto&PhotoID=2281;

http://members.aol.com/dicetalk/polymor2.htm;

http://beta.content.communities.msn.co.uk/DiceManiacsClubakaTheRandomFandom/ nicksdice.msnw?action=ShowPhoto&PhotoID=111) -

the first one may roll fair,but none of these have been mass-produced as far as I know.

\*\* d34's are semi-available, but not as blanks. Just subtract 17 whenever you roll >17. You can get them for $3.00 each at http://www.chessex.com/Special\_d34.htm

\*\*\* Gamescience made blank d26's

(http://beta.content.communities.msn.co.uk/DiceManiacsClubakaTheRandomFandom/ kevincooksblankdice.msnw?action=ShowPhoto&PhotoID=1705

http://beta.content.communities.msn.co.uk/DiceManiacsClubakaTheRandomFandom/ nicksdice.msnw?action=ShowPhoto&PhotoID=164),

which would also provide you with d13's, but I don't think they roll fair.

Also, check out http://members.aol.com/dicetalk/polyh6.htm. With large dipyramids and trapezohedra, fair rolling dice of all even sizes (and thus all odd sizes with proper numbering) are possible - maybe some company will get around to making them.

There - with a bunch of blank d6's and some d20's, you should be able to fulfill all your swing die needs. In the 1-20 range (where most BM swing's fall within), there's only one gap greater than 1 die (d12 to d15). So go and beat people up now that you can optimize your swing die choice.

1. ↑ https://web.archive.org/web/20180725002525/http://beatpeopleup.cheapass.com/strategy/