Six-Sided Gods Part 1 (Jan 98)
Wally's defence of his dissatisfaction with six-sided die is valid as far as it goes. Unfortunately very small gradations of result are possible, and 1/6 or 16.66666% is not an absolute limit. For what follows, however, I cannot take credit. It is actually a restatement of an article by Bob Cory that appeared in Jack Scruby's Table Top Talk in January of 1965. It was a masterful article, and I would reprint it for you, but unfortunately I lent some of my copies of Table Top Talk to a child who was doing a report on Wargames, and the malignant little maggot cut up many of them for pictures so he could glue it on poster-board for a school project. He cut out one on the page before that mutilated most of the article, and most of the mathematical table that came with it. I have, however, been able to reconstruct the author's method. My apologies to Mr. Cory... wherever you are.
Basically the method resolves around ascending combinations of numbers to achieve equivalent percentile results. Since the result of a total of 2 on two dice can be achieved with only 1 possibility in 36, this is a 1/36 or .0278% I assume this is a small enough gradation of result even for you, Wally.
| THE CORY TABLE OF DICE RESULTS | ||||
|---|---|---|---|---|
| Odds | Dec. Equiv | % Equiv | Fract Equiv | Total of Two dice to make |
| 1/36 | .0278 | 2.7% | 1/36 | 2 |
| 2/36 | .0556 | 5.5% | 1/18 | 3 |
| 3/36 | .0833 | 8.3% | 1/12 | 4 |
| 4/36 | .1111 | 11.11% | 1/9 | 5 |
| 5/36 | .1389 | 13.89% | 5/36 | 6 |
| 6/36 | .1667 | 16.67% | 1/6 | 7 |
| 7/36 | .1944 | 19.44% | 7/36 | 17 or 2 |
| 8/36 | .2222 | 22.22% | 2/9 | 7 or 3 |
| 9/36 | .2500 | 25.00% | 1/4 | 7 or 4 |
| 10/36 | .2778 | 27.78% | 5/18 | 7 or 5 |
| 11/36 | .3056 | 30.56% | 11/36 | 7 or 6 |
| 12/36 | .3333 | 33.33% | 1/3 | 7,6 or 2 |
| 13/36 | .3611 | 36.11% | 13/36 | 7,6 or 3 |
| 14/36 | .3889 | 38.89% | 7/18 | 7,6 or 4 |
| 15/36 | .4167 | 41.67% | 5/12 | 7,6 or 5 |
| 16/36 | .4444 | 44.44% | 4/9 | 7,6,5 or2 |
| 17/36 | .4722 | 47.22% | 17/36 | 7,6,5 or3 |
| 18/36 | .5000 | 50% | 18/36 | 7,6,5,or4 |
| 19/36 | .5278 | 52.78% | 19/36 | 7,6,5,4 or 2 |
| 20/36 | .5556 | 55.56% | 5/9 | 7,6,5,4 or 3 |
| 21/36 | .5833 | 58.33% | 7/12 | 7,6,5,4,3 or 2 |
| 22/36 | .6111 | 61.11% | 11/18 | 7,6,5,4,3,2 or 12 |
| 23/36 | .6389 | 63.89% | 23/36 | 7,6,5,5,3,2 or 11 |
| 24/36 | .6667 | 66.67% | 2/3 | 7,6,5,4,3,2 or 10 |
| 25/36 | .6944 | 69.44% | 25/36 | 7,6,5,4,3,2 or 9 |
| 26/36 | .7222 | 72.22% | 13/18 | 7,6,5,4,3,2 or 8 |
| 27/36 | .7500 | 75.00% | 3/4 | 8,7,6,5,4,3,2 or12 |
| 28/36 | .7778 | 77.78% | 7/9 | 8,7,6,5,4,3,2 or11 |
| 29/36 | .8056 | 80.56% | 29/36 | 8,7,6,5,4,3,2 or10 |
| 30/36 | .8333 | 83.33% | 5/6 | 8,7,6,5,4,3,2 or9 |
| 31/36 | .8611 | 86.11% | 31/36 | 9,8,7,6,5,4,3,2 or12 |
| 32/36 | .8889 | 88.89% | 8/9 | 9,8,7,6,5,4,3,2 or11 |
| 33/36 | .9167 | 91.67% | 11/12 | 9,8,7,6,5,4,3,2 or10 |
| 34/36 | .9444 | 94.44% | 17/18 | 10,9,8,7,6,5,4,3,2or 12 |
| 35/36 | .9722 | 97.22% | 11/12 | 10,9,8,7,6,5,4,3,2or 11 |
| 36/36 | 1.000 | 100% | 1/1 | all |
When I f irst saw this article I was a wee lad, in fact a Malignant little Maggot myself, so I did not appreciate the beauty of it. It may seem cumbersome at first, but a little thought and you can see that Mr. Cory was quite clever. It basically relies upon the critical group of base integers of 2 through 7, and that, probability wise, 2 is equivalent to 12 and so on ... or ....
There is the same chance of getting a
- 2 as a 12
3 as an 11
4 as a 10
5 as a 9
6 as a 8
Seven of course has the most number of chances, that is 6 out of 36. By adding repeated series of numbers to achieve pyramiding possible outcomes he has constructed a table that has a constant incremental value in steps of 2.78%! This yields in effect, 36 possible results, which should be more than enough to sate the tastes of even Wally Simon. At the same time, this is readily translatable both to percentage results and to fractional results for ease of reference.
I freely admit however that APPARENT difficulty is the nonlinearality of the results. That is, we are accustomed to the numerical result equaling or bearing a close relationship to the proportional result, that is, We expect that if we have 47.22% chance of hitting something then any result we make on our random number generator (dice, card, throwstick, random-number-generator, computer) will be inclusive in the range of I to 47.22. Mr. Cory's table of 3,5,6,7 is not! 1,2, and 4 are not there. 1, of course, cannot ever be produced with the roll of 2 die, and 2 and 4 are disallowed.
Yet this is a problem ONLY if you frame your rules in a certain way. If you must use an arithmetically calculated percentage, that is, something the person must factor in his head to yield some number, than it is an extra step to look up on the Cory table to get the results needed. You might as well roll percentile dice. However, considering the amount of tables gamers look up already I don't see how one more would hurt. On the other hand, making up pre-determined situational charts completely eliminates this objection. The Cory chart is used only as a reference prior to the game to craft the chart, and in no way enters the play of the game. Consider the following situation... a table where troops of A are assaulting those of B.
| 2 | A surrenders | 2.7% |
| 3 | A is horribly schlonged | 5.5% |
| 4,7 | A is schlonged | 25% |
| 5,6 | B is schlonged | 25% |
| 8 | B is lightly schlonged and falls back 1 move | 13.89% |
| 9,10 | B is horribly schlonged but holds the ground | 19.44% |
| 11 | A and B are schlonged | 5.5% |
| 12 | B surrenders | 2.7% |
You see, one of the objections one might make to the Cory Chart is that it allows for only a unary result, that is a simple pass/fail. Yet this is not true. It, in fact, because of the very non linearity of its numbers, allows for a multiplicity of results! There are, in fact, all those numbers not used for success. For example in the 44.44% result a 7,6,5 or 2 will give success, which means that 3, 4, 8, 9, 10, 11, or 12 will not. This means that you can have it any way you want. You can use it as a simple unary result (pass - fail) or you can have separate results tied to specific rolls of the die for either failure or success or both. That is, while 7,6,5, or 2 can be success, and success might be the same every time, the specific failure rolls, 3,4,8,9,10,11,and 12 might each give different types of failure, or on the other hand, the result of failure might be common, and the results of success differentiated. Of course, there is always the possibility that both failure and success might have differentiated results, in fact, right down to 36 different ones on both types of the spectrum for one side, the other or both. It all depends on how you wish to figure the charts and how you wish to arrange the probabilities.
Here is the heart of the issue... it isn't the dice, or the number of faces on them, it's the rule designers. Most designers are simply sloppy designers and they design sloppy rules. That is, they take the easy way out and end their thinking at the approximation of a percentage a certain event will happen in war. But they have NOT given any thought to an easy or effective way to calculate or arrive at that decision in a game. How many rules do you know where it says "roll to see if you can fire ... roll to see if you can see the target ... roll to see if you hit the target ... roll to see if you penetrate the target... roll to see where you penetrate on the target .... roll to see... " So far that is six rolls and six decisions. For the sake of argument assume each roll is a 50/50% chance, that is an accumulated percentage of 1.5% to actually do something. At those rates you should not be firing at all, you're just wasting ammunition. Now... on the other hand, if a single roll, by delineating all the possibilities of what could happen, from missing (and let's face it, if you can't see the target or can't fire you've missed it anyway) to blowing it sky high, then you've accomplished something.
Yet the utility of The Cory Chart is not yet exhausted! Let us pyramid the values!
| DIE TOTAL | # OF DICE COMBINATIONS TO MAKE TOTAL | Decimal Eq. Comb/36 | TOTAL OF NUMBER PLUS ALL LOWER NUMBERS |
| 2 | 1 | .0278 | .0278 |
| 3 | 2 | .0556 | .0834 |
| 4 | 3 | .0833 | .1667 |
| 5 | 4 | .1111 | .2778 |
| 6 | 5 | .1388 | .4166 |
| 7 | 6 | .1667 | .5833 |
| 8 | 5 | .1388 | .7501 |
| 9 | 4 | .1111 | .8332 |
| 10 | 3 | .0833 | .9165 |
| 11 | 2 | .0556 | .9721 |
| 12 | 1 | .0278 | 1.000 |
I have not expanded these simple calculations to percentages, those are readily observable. What is interesting is the difference in percentages when viewed linearly. There is a 58% chance of success if you roll a 7 or less, while there is a 91% chance of success if you roll a 10 or less. The point is that once again, by grouping the more likely result at the 7 and other results further out from it you have another means of distribution of results. At the same time, it is obvious that many of the number combinations in The Cory Chart could be changed, for example, not only is 8 interchangeable with 6, but the decimal equivalent of 7 (.1667) is identical to the sum of the decimal equivalents of 5 (.1111) and 2 (.0556). This means that you do not necessarily have to start your numbers range of success with 7, but could substitute other numbers. At the same time, this 7 could be broken down to two results of 5 and 2. That is, if you wanted to have a general success rate of 16.67% but wanted two possibilities of success then you could use a 2 or 5. Granted they would be two quite different chances of specific success, but nevertheless, it is in no way set in stone.
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© Copyright 1998 Wally Simon
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