Sports Illustrated / Avalon Hill / 10-39 Dice

The family have a couple of tabletop games from the 70s that use funky, six-sided, wooden dice. (Superstar Baseball has a selection of all-time MLB greats, while Bowl Bound has college football teams from the 60’s & 70’s.) There are three dice: one black and two white. There are no pips on the dice—instead numerals are printed on the sides. It’s a bit like Strat-o-Matic. The black-die value is multiplied by ten, and the two white die are added to the total. So, for example, a black 2 and white 3 & 4 represent a value of 27. The faces of the dice are marked as follows:

 BLACK: 1, 2, 2, 3, 3, 3
WHITE1: 0, 1, 2, 3, 4, 5
WHITE2: 0, 0, 1, 2, 3, 4


The values range from 10 to 39, with no gaps. The probabilities are as follows:

VALUE  FREQUENCY  HISTOGRAM
-----  ---------
10      2  0.93%  |**
11      3  1.39%  |***
12      4  1.85%  |****
13      5  2.31%  |*****
14      6  2.78%  |******
15      6  2.78%  |******
16      4  1.85%  |****
17      3  1.39%  |***
18      2  0.93%  |**
19      1  0.46%  |*
20      4  1.85%  |****
21      6  2.78%  |******
22      8  3.70%  |********
23     10  4.63%  |**********
24     12  5.56%  |************
25     12  5.56%  |************
26      8  3.70%  |********
27      6  2.78%  |******
28      4  1.85%  |****
29      2  0.93%  |**
30      6  2.78%  |******
31      9  4.17%  |*********
32     12  5.56%  |************
33     15  6.94%  |***************
34     18  8.33%  |******************
35     18  8.33%  |******************
36     12  5.56%  |************
37      9  4.17%  |*********
38      6  2.78%  |******
39      3  1.39%  |***

The distribution of the 10s, 20s, & 30s is clear from the black-die faces. There are twice as many 20s as 10s, and three times as many 30s as 10s. The white dice create a range of 0-9, so that every value from 10 to 39 is represented.

The white dice form an interesting distribution with the 4 & 5 totals being the most frequent, and the 9 total being the least frequent. This gives quite a variety of probabilities, which is undoubtedly the point.

If we order all the values by frequency, we see the following distribution:

RANK  VALUE  FREQUENCY  HISTOGRAM
----  -----  ---------
 1.      35  18  8.33%  |******************
 1.      34  18  8.33%  |******************
 3.      33  15  6.94%  |***************
 4.      36  12  5.56%  |************
 4.      32  12  5.56%  |************
 4.      25  12  5.56%  |************
 4.      24  12  5.56%  |************
 8.      23  10  4.63%  |**********
 9.      37   9  4.17%  |*********
 9.      31   9  4.17%  |*********
11.      26   8  3.70%  |********
11.      22   8  3.70%  |********
13.      27   6  2.78%  |******
13.      30   6  2.78%  |******
13.      38   6  2.78%  |******
13.      21   6  2.78%  |******
13.      15   6  2.78%  |******
13.      14   6  2.78%  |******
19.      13   5  2.31%  |*****
20.      20   4  1.85%  |****
20.      16   4  1.85%  |****
20.      12   4  1.85%  |****
20.      28   4  1.85%  |****
24.      17   3  1.39%  |***
24.      39   3  1.39%  |***
24.      11   3  1.39%  |***
27.      18   2  0.93%  |**
27.      29   2  0.93%  |**
27.      10   2  0.93%  |**
30.      19   1  0.46%  |*

Interesting. Sort of exponentially decreasing.

The white dice turn out to be quite similar to a pair of normal dice. Here’s the thought process. Normal dice produce eleven different values starting with 2 and continuing through 12 with no gaps. The dice faces look like this:

1, 2, 3, 4, 5, 6
1, 2, 3, 4, 5, 6

It would be nice to have ten different values starting with 0 and continuing through 9 with no gaps. That would give us a a base-ten digit that’s easy for humans to deal with. Well, it’s easy to start with 0—just subtract one from each die face:

0, 1, 2, 3, 4, 5
0, 1, 2, 3, 4, 5

OK, that gives us values from 0 to 10, but we want 0 to 9. We can get there by adding one more 0 face and losing the 5 face from one of the dice:

0, 1, 2, 3, 4, 5
0, 0, 1, 2, 3, 4

Now we’ve got values from 0 to 9 with no gaps. Mission accomplished. This is exactly the configuration of the two white SI/AH dice.

Of course, one can get values from 0 to 9 with a single ten-sided die. (I never saw such a thing in the 70s.) But a ten-sided die produces a uniform distribution, that is, each of the ten values is equally likely. A game designer may prefer a non-uniform distribution, because it allows one to use the less frequent values for the less frequent plays. For example, I think most hit-by-pitch results in Strat are for red-dice totals of 2 or 12. So the white SI/AH dice provide an easy-to-calculate decimal-digit value that does not have uniform probability.

The black SI/AH die is not as exciting. In fact, it could be white if the [1 2 2 3 3 3] were replaced with [10 20 20 30 30 30], in which case once could simply add all three dice.

If you wanted 100 non-uniform values instead of 30, then you could use four dice: the two white dice plus an identical pair colored black. The sum of the black dice would provide the first decimal digit (tens place), and the sum of the two white dice would provide the second decimal digit (units place). That would yield values from 0 to 99 with no gaps.

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