Analysis of HeroScape

The total number of possible results of a set of events is equal to the number of outcomes for the first event multiplied by the number of outcomes for the second event, the result of the previous calculation multiplied by the number of outcomes for the third event, and so on for each of the events. The result of the last multiplication is the total number of results. So if X is the number of outcomes of the first event, Y is the number of outcomes of the second event, and Z is the number of outcomes of the third event, the total number of possible results for the set of events (X, Y, Z), is X x Y x Z.


Each die represents a separate event. The attack and defense dice have six sides. Thus, the total number of results is equal to 6 to the power of the number of dice that are being rolled. The following table lists the total number of results for 2 through 13 dice.
6x6 =36
6x6x6 =216
6x6x6x6 =1296
6x6x6x6x6 =7776
6x6x6x6x6x6 =46656
6x6x6x6x6x6x6 =279936
6x6x6x6x6x6x6x6 =1679616
6x6x6x6x6x6x6x6x6 =10077696
6x6x6x6x6x6x6x6x6x6 =60466176
6x6x6x6x6x6x6x6x6x6x6 =362797056
6x6x6x6x6x6x6x6x6x6x6x6 =2176782336
6x6x6x6x6x6x6x6x6x6x6x6x6 =13060694016


In HeroScape only a small fraction of the results on the die represent hits or blocks. If we wanted to calculate the total number of results of a set of dice that consisted solely of blocks we would take the number of blocks on each die to the power of the number of dice that are rolled. The following table lists the total number of such results for 2 through 13 dice.
2x2 =4
2x2x2 =8
2x2x2x2 =16
2x2x2x2x2 =32
2x2x2x2x2x2 =64
2x2x2x2x2x2x2 =128
2x2x2x2x2x2x2x2 =256
2x2x2x2x2x2x2x2x2 =512
2x2x2x2x2x2x2x2x2x2 =1024
2x2x2x2x2x2x2x2x2x2x2 =2048
2x2x2x2x2x2x2x2x2x2x2x2 =4096
2x2x2x2x2x2x2x2x2x2x2x2x2 =8192


Similarly, if we wanted to calculate the total number of results of a set of dice that consisted solely of hits we would take the number of hits on each die to the power of the number of dice that are rolled. The following is a table of such calculations.
3x3 =9
3x3x3 =81
3x3x3x3 =256
3x3x3x3x3 =243
3x3x3x3x3x3 =729
3x3x3x3x3x3x3 =2187
3x3x3x3x3x3x3x3 =6561


Similarly, if we wanted to calculate the total number of results of a set of dice that consisted solely of misses we would take the number of misses on each die to the power of the number of dice that are rolled. The following is a table of such calculations.
4x4 =16
4x4x4 =64
4x4x4x4 =256
4x4x4x4x4 =1024
4x4x4x4x4x4 =4096
4x4x4x4x4x4x4 =16384
4x4x4x4x4x4x4x4 =65536
4x4x4x4x4x4x4x4x4 =262144
4x4x4x4x4x4x4x4x4x4 =1048576
4x4x4x4x4x4x4x4x4x4x4 =4194304
4x4x4x4x4x4x4x4x4x4x4x4 =16777216
4x4x4x4x4x4x4x4x4x4x4x4x4 =67108864


For example, the number of results of two dice consisting of 2 shield blocks is equal to the number of blocks on each die multipled together (or 2 (the number of shields) to the second power (the number of dice)). The result is 4. To calculate the percentage of the total number of possibilities we would divide this result by 36. The result is 11.111%.

If we wanted to calculate the number of 1 shield blocks on two dice we would multiply the number of blocks on one die by the number of misses on the other. We also need to consider all the unique permutations of the dice. The number of permutations is equal to n! (where n is the number of dice that are being rolled) Thus, for two dice there are two permutations. We multiply the initial result by the number of permutations. The sum of the two results divided by the total number of possibilities gives us the percentage of hits. ((Blocks x Hits) x n!)/ Number of Possibilities = Percentage
((2 x 4) x 2)/ 36 = 44.444%

When rolling more dice the consideration of the number of possibilities is somewhat more complex. The number of permutations of a set of numbers where some of the numbers may be equal is equal to n!/m! x o! ... (where n is the number of dice that are being rolled) (m is the number of dice with a particular result and o is the number of dice with a different particular result) For example if we were rolling for dice and wanted to calculate the number of results consisting of two shields, We would multiply (2 x 2 x 4 x 4) x (24/(2 x 2))/1296=
(64 x 6)/1296 = 29.62963%

The following are tables of the results for sets of dice.

2Defense Dice
36Possible Results
16No Shields Results
161 Shield Results
42 Shields Results


2Defense Dice
44.444%No Shields Results
44.444%1 Shield Results
11.111%2 Shields Results


3Defense Dice
216Possible Results
64No Shields Results
961 Shields Results
482 Shields Results
83 Shields Results


3Defense Dice
29%No Shields Results
44%1 Shields Results
22%2 Shields Results
3%3 Shields Results


4Defense Dice
1,296Possible Results
256No Shields Results
5121 Shield Results
3842 Shields Results
1283 Shields Results
164 Shields Results


4Defense Dice
19%No Shields Results
39%1 Shield Results
29%2 Shields Results
9%3 Shields Results
1%4 Shields Results


5Defense Dice
7,776Possible Results
1,024No Shields Results
2,5601 Shield Results
2,5602 Shields Results
1,2803 Shields Results
3204 Shields Results
325 Shields Results


5Defense Dice
13%No Shields Results
32%1 Shield Results
32%2 Shields Results
16%3 Shields Results
4%4 Shields Results
Less than 1%5 Shields Results


6Defense Dice
46,656Possible Results
4,096No Shields Results
12,2881 Shield Results
15,3602 Shields Results
10,2403 Shields Results
3,8404 Shields Results
7685 Shields Results
646 Shields Results


6Defense Dice
8%No Shields Results
26%1 Shield Results
32%2 Shields Results
21%3 Shields Results
8%4 Shields Results
1%5 Shields Results
Less than 1%6 Shields Results


7Defense Dice
279,936Possible Results
16,384No Shields Results
57,3441 Shield Results
86,0162 Shields Results
71,6803 Shields Results
35,8404 Shields Results
10,7525 Shields Results
1,7926 Shields Results
1287 Shields Results


7Defense Dice
5%No Shields Results
20%1 Shield Results
30%2 Shields Results
25%3 Shields Results
12%4 Shields Results
3%5 Shields Results
Less than 1%6 Shields Results
Less than 1%7 Shields Results


8Defense Dice
1,679,616Possible Results
65,536No Shields Results
262,1441 Shield Results
458,7522 Shields Results
458,7523 Shields Results
286,7204 Shields Results
114,6885 Shields Results
28,6726 Shields Results
4,0967 Shields Results
2568 Shields Results


8Defense Dice
3%No Shields Results
15%1 Shield Results
27%2 Shields Results
27%3 Shields Results
17%4 Shields Results
6%5 Shields Results
1%6 Shields Results
Less than 1%7 Shields Results
Less than 1%8 Shields Results


9Defense Dice
10,077,696Possible Results
262,144No Shields Results
1,179,6481 Shield Results
2,359,2962 Shields Results
2,752,5123 Shields Results
2,064,3844 Shields Results
1,032,1925 Shields Results
344,0646 Shields Results
73,7287 Shields Results
9,2168 Shields Results
5129 Shields Results


9Defense Dice
2%No Shields Results
11%1 Shield Results
23%2 Shields Results
27%3 Shields Results
20%4 Shields Results
10%5 Shields Results
3%6 Shields Results
Less than 1%7 Shields Results
Less than 1%8 Shields Results
Less than 1%9 Shields Results




2Attack Dice
36Possible Results
9No Skulls Results
181 Skull Results
92 Skulls Results


2Attack Dice
25%No Skulls Results
50%1 Skull Results
25%2 Skulls Results


3Attack Dice
216Possible Results
27No Skulls Results
811 Skull Results
812 Skulls Results
273 Skulls Results


3Attack Dice
12%No Skulls Results
37%1 Skull Results
37%2 Skulls Results
12%3 Skulls Results


4Attack Dice
1,296Possible Results
81No Skulls Results
3241 Skull Results
4862 Skulls Results
3243 Skulls Results
814 Skulls Results


4Attack Dice
6%No Skulls Results
25%1 Skull Results
37%2 Skulls Results
25%3 Skulls Results
6%4 Skulls Results


5Attack Dice
7,776Possible Results
243No Skulls Results
1,2151 Skull Results
2,4302 Skulls Results
2,4303 Skulls Results
1,2154 Skulls Results
2435 Skulls Results


5Attack Dice
3%No Skulls Results
15%1 Skull Results
31%2 Skulls Results
31%3 Skulls Results
15%4 Skulls Results
3%5 Skulls Results


6Attack Dice
46,656Possible Results
729No Skulls Results
4,3741 Skull Results
10,9352 Skulls Results
14,5803 Skulls Results
10,9354 Skulls Results
4,3745 Skulls Results
7296 Skulls Results


6Attack Dice
1%No Skulls Results
9%1 Skull Results
23%2 Skulls Results
31%3 Skulls Results
23%4 Skulls Results
9%5 Skulls Results
1%6 Skulls Results


7Attack Dice
279,936Possible Results
2187No Skulls Results
15,3091 Skull Results
45,9272 Skulls Results
76,5453 Skulls Results
76,5454 Skulls Results
45,9275 Skulls Results
15,3096 Skulls Results
2,1877 Skulls Results


7Attack Dice
0.78125%No Skulls Results
5.46875%1 Skull Results
16.40625%2 Skulls Results
27.34375%3 Skulls Results
27.34375%4 Skulls Results
16.40625%5 Skulls Results
5.46875%6 Skulls Results
0.78125%7 Skulls Results


8Attack Dice
1,679,616Possible Results
6,561No Skulls Results
52,4881 Skull Results
183,708 2 Skulls Results
367,4163 Skulls Results
459,2704 Skulls Results
367,4165 Skulls Results
183,7086 Skulls Results
52,4887 Skulls Results
6,5618 Skulls Results


8Attack Dice
0.00390625%No Skulls Results
3.125%1 Skull Results
10.9375%2 Skulls Results
21.875%3 Skulls Results
27.34375%4 Skulls Results
21.875%5 Skulls Results
10.9375%6 Skulls Results
3.125%7 Skulls Results
0.00390625%8 Skulls Results


We may learn all kinds of things about the game from these statistics.

All of the defense distributions are skewed to the left, i.e. they favor rolling fewer shields. One can also see how unlikely it is to roll many shields. The attack distributions on the other hand are symmetrical, i.e. you are equally likely to roll all skulls or no skulls. The defense distribution is skewed to the extent that an attack of 3 dice ends up having a higher average than a defense of 4 dice. As the number of dice increase this skewing favors attacking even more.

Given that the game favors attacking, it is important to attack first. The charts also illustrate the importance of rolling an extra die. Given that you can roll an extra die for being on a higher level than your opponent, taking a high position is very important.

We may wish to calculate the joint occurence of two events. In such an instance multiply the probability of event 1 with that of event 2. The result will be much smaller than the probabilities of the separate events.

We may wish to determine whether it is more advantagous to attack first, or to seek higher ground. To do this we need to compare the odds of wounding our opponent in both scenarios. If both characters had 3 attacks and 4 defense and we attack first, the odds of wounding our opponent are 38.93%. (This is calculated by multiplying the percentage of a higher attacking result by that of a lower defending result for all such results, and summing the results) If we seek higher ground and attack second (we now have 4 attacks and 5 defense), the unmodified odds of wounding our opponent would be 53.72%. However, our opponent attacks first. The odds of being wounded are 30.70%. Thus, the odds of wounding are equal to 53.72% of 69.7% (the remaining) which is 37.44284% Thus, we see that in this circumstance attacking first is the better option. In the case of an attack 3, defense 6 character versus a 3/4 character, heading for higher ground is the better option.

The odds of an attack 3 figure wounding an Izumi Samurai in close combat are slightly less than the odds of the Izumi Samurai wounding the character in the process.

The odds of an attack 3 figure wounding a Deathwalker are less than 1 in 10.



The Horde Strategy

A squad of four characters will result in at least 10 attacks, as the following chart illustrates

4 attacks versus 1 attack
3 attacks versus 1 attack
2 attacks versus 1 attack
1 attack versus 1 attack
Total Attacks
10 versus 4

A squad of three characters will result in at least 6 attacks. Here the difference between three and four characters ganging up on a character is made clear, as 10 attack is almost double 6 attacks. Obviously not more than 6 characters can gang up on a single normal sized character in hand to hand combat.

If a character with attack 3 fought a character with defense 4 38.93% of the attacks would wound the opponent.
3 attack vs 4 defense
.37x.19=.0703
.37x.39=.1443
.37x.19=.0703
.12x.29=.0348
.12x.39=.0468
.12x.19=.0228
xxxxxxxx.3893


In 10 such attacks we can expect to have 3.893 successful attacks. Each one of these will give at least 1 wound. There is about a 15% chance that an attack will result in 2 or 3 wounds being dealt. That is more than enough to kill Agent Carr or Sgt. Alexander. In 6 attacks we can expect at least 2.3358 successful attacks.

One of the difficulties in the horde strategy is utilizing all of your figures. There generally are so many that most will not have a chance to attack. It is much easier to get the figures into combat if they have a ranged attack. This allows even more than 6 figures to attack a single opponent at once. Thus, taking ranged figures is important in horde strategy.

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