The Oath combat system
Oath is a great board game, which is in some sense a game about history, and in others a game about empires. If you’re not already familiar with the game, I’d highly recommend that you give it a try.
In this article, I will discuss some of the odds in the oath combat system. I will be relatively light on mathematical details: for a full derivation of the formulas, you can see my writeup.
In the combat heuristics section, I will discuss some good rules for estimating your odds in-game.Thank you to users @steveowen and @Samuel Vriezen for comments on BGG which motivated this section. In particular, the unit loss estimate section gives a quick procedure you can follow to work out how many units you would expect to have left, after combat.
If you instead want to see some pre-computed unit loss tables, here are the links:
If you use Mathematica, I’ve also implemented all the distributions in this Mathematica notebook.
With that out of the way, let’s see some more detail!
Combat heuristics
The main insight from the odds are the zero-loss curves. Looking at the unit difference tables, we can see the various defence dice / attack dice pairings which result in the attacker not having to sacrifice any units 50% of the time:
- 1 vs. 2
- 2 vs. 3
- 3 vs. 4
- 4 vs. 6
- 5 vs. 7
- 6 vs. 9
- 7 vs. 12
- 8 vs. 15
- etc.
Fitting a quadratic to this curve and rounding slightly, we can estimate that, when there are n defending dice, you should have 2+(n2-n)/4 attacking units in order to beat the defending dice roll 50% of the time. Now if you have a different number of attacking units, since there are 3 attacking outcomes worth 0.5, and 3 attacking outcomes worth either 1 or 2 (with unit sacrifice), for every extra attacking dice you have, you expect to get an extra 0.75 attacking power in your roll.
Unit loss estimate
Combining these observations gives the following procedure to estimate the number of attacking units you will have remaining after combat:
- Determine the number of defending dice, n, and compute the attack balance point k=2+(n2-n)/4 (for example, if there are 8 defending dice, the attack balance point is 16)
- Determine your attacking dice count, a, and compute the power adjustment 1+(k-a)·3/4 (for example, if you have 12 attacking dice, the power adjustment is 4)
- Add the defending unit count and add the power adjustment (for example, if there are 8 defending units, the final result is 12)
Now, 50% of the time, you should expect to lose at most 12 units. Let’s compare this to the table computing the unit loss with 8 defenders: this table states that for 8 defending dice, 8 defenders, and 13 attacking dice, you would lose 11 units 50% of the time. This estimate isn’t perfect, but it is relatively easy to compute in-game!
If you’re in a hurry and don’t want to do too much mental math, simply doing (1) will give you a good idea for the number of units you’d want to not have to sacrifice any units 50% of the time. If you want, you can also just memorize the zero-loss curve table to get a more precise estimate.
If you instead want the estimate with ≈80% certainty (instead of 50%), you can perform the same estimate by computing the attack balance point with 5 - 3n/4 + n2/2, rounded down. And for ≈90% certainty, use 5 - 0.9 n + 0.6 n2, rounded down. This number gets very large, very fast!
Also, if you have a card which means you do not sacrifice units when you roll skulls, simply adjust the 3/4 factor to 11/12.
Precise combat odds
Combat mechanics
Combat is an important part of Oath, and the combat system is relatively straightforward, while also being feature-rich with somewhat deceptive odds. Cole Wehrle (the designer of Oath) has written about the development process of the combat system.
During combat, there are two sets of dice: attacking, and defending. The sides of the dice are shown in red and blue respectively in the image below.
The defending distribution is given by taking multiplying the number of shields by 2 for every ×2 dice roll.
Computing the distributions
The full details for the probability computations can be found in this writeup.
The general idea as as follows. A convenient strategy to compute probabilities involving binomial expansions is to use generating functions. While this particular application does not require any sophisticated theory, the approach is conceptually nice.
For example, let’s compute the attack odds by tracking the number of skulls and swords rolled. Consider the polynomial (3x½+2x+x2y)n, where the coefficients are chosen as such since there are 3 dice with a hollow sword, 2 dice with a solid sword, and 1 dice with 2 solid swords (contributing the x2) and 1 skull (contributing the y). We count the hollow swords as ½, which we can round later.
Then, extracting the coefficient of the term xzyk gives the number of ways to roll exactly z attack points (without rounding) with k skulls. Why does this work? When we compute the power of the polynomial raised to the nth power, the resulting polynomial (without collecting terms) contains all possible choices of n dice rolls,
Finally, we need to do some rounding: conveniently, it is alright to round the total, rather than the number of hollow sword rolls, since they are equivalent.
For the defence odds, suppose we are rolling n dice where n=2m·l with l odd. Then the only way to roll exactly n is to have k dice show ×2 (for some k between 0 and m), and then roll 2m-k·l using only shields with the remaining dice. The number of ways to roll k ×2 dice is n choose k, and the number of ways to roll 2m-k·l using only shields is the coefficient of x2m-k·l in the polynomial (2+2x+x2)n-k.
Summary of odds
In each entry, the rows correspond to the number of attacking dice, and the columns correspond to the number of defending dice. The three numbers are in each entry are the 50th, 90th, and 99th percentiles, respectively.
In order to use the unit difference table, choose the row with the number of attacking dice and the column with the number of defending dice. Then, for example, the second entry is is the number of units you will need to sacrifice 90% of the time in order to (strictly) overcome the defence roll power.
Other factors, such as the number of skulls rolled, as well as the defensive unit power, will influence the outcome as well. In order to estimate how many units you will lose, you need to
- Look up the corresponding value in the table for your attack and defence dice counts, as well as your desired percentile.
- Add a number equal to ⅙ of your dice roll (this is the average loss from rolling skulls).
- Add the defensive unit power.
The resulting number is approximately a lower bound on the number of units you will have left over after the combat. Note this does not account for the case where you roll well beyond the defensive strength, but still need to sacrifice units because of skulls. To get an idea of what these numbers look like for certain defensive unit counts, I’ve included 3 additional tables at the end of this article.
Certain cards, such as Rangers and Kindred Warriors, also allow the attacker to roll without sacrificing units from double sword dice rolls.
Since there are many possible combinations, I recommend using the combatSummary function in the Mathematica notebook, which will report the cumulative distribution of the number of attacking units that survive.Eventually, I may write an online Javascript tool to do this computation.
Unit difference percentiles
This is the distribution of the defence roll minus the attack roll. Negative numbers favour the attacker, and positive numbers favour the defender.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1, 2, 3 | 2, 4, 5 | 3, 5, 9 | 4, 8, 13 | 6, 12, 21 | 7, 17, 33 | 9, 24, 48 | 12, 32, 65 | 15, 41, 97 | 19, 56, 129 | 24, 71, 192 | 29, 89, 256 |
| 2 | 0, 1, 2 | 1, 3, 4 | 2, 4, 8 | 3, 8, 12 | 5, 11, 20 | 6, 16, 32 | 8, 23, 47 | 12, 31, 64 | 15, 40, 96 | 18, 55, 128 | 23, 70, 191 | 28, 88, 255 |
| 3 | -1, 1, 2 | 0, 2, 4 | 1, 4, 7 | 2, 7, 12 | 4, 10, 19 | 6, 15, 31 | 8, 22, 46 | 11, 30, 63 | 14, 39, 95 | 17, 54, 127 | 22, 69, 190 | 27, 87, 254 |
| 4 | -2, 0, 1 | -1, 1, 3 | 0, 3, 6 | 1, 6, 11 | 3, 9, 19 | 5, 14, 30 | 7, 21, 45 | 10, 29, 62 | 13, 38, 94 | 16, 53, 126 | 21, 68, 189 | 26, 87, 253 |
| 5 | -3, -1, 0 | -2, 0, 2 | -1, 2, 5 | 1, 5, 10 | 2, 8, 18 | 4, 13, 29 | 6, 20, 45 | 9, 28, 61 | 12, 37, 93 | 15, 52, 125 | 20, 67, 188 | 25, 86, 252 |
| 6 | -4, -2, 0 | -3, 0, 1 | -2, 1, 4 | 0, 4, 9 | 1, 7, 17 | 3, 12, 28 | 5, 19, 44 | 8, 27, 61 | 11, 36, 92 | 15, 51, 124 | 19, 65, 187 | 25, 85, 251 |
| 7 | -4, -2, -1 | -4, -1, 1 | -3, 0, 4 | -1, 3, 8 | 0, 7, 16 | 2, 11, 27 | 4, 18, 43 | 7, 26, 60 | 10, 35, 91 | 14, 50, 123 | 18, 64, 186 | 24, 84, 250 |
| 8 | -5, -3, -2 | -4, -2, 0 | -3, 0, 3 | -2, 2, 8 | -1, 6, 15 | 1, 10, 26 | 3, 17, 42 | 6, 25, 59 | 9, 34, 90 | 13, 49, 123 | 17, 63, 185 | 23, 84, 249 |
| 9 | -6, -4, -3 | -5, -3, -1 | -4, -1, 2 | -3, 1, 7 | -2, 5, 14 | 0, 9, 25 | 2, 16, 41 | 5, 24, 58 | 8, 33, 89 | 12, 48, 122 | 16, 62, 184 | 22, 83, 248 |
| 10 | -7, -5, -3 | -6, -4, -2 | -5, -2, 1 | -4, 0, 6 | -2, 4, 13 | -1, 8, 24 | 1, 15, 40 | 4, 23, 57 | 7, 33, 88 | 11, 47, 121 | 15, 61, 183 | 21, 82, 247 |
| 11 | -8, -6, -4 | -7, -5, -3 | -6, -3, 0 | -5, 0, 5 | -3, 3, 12 | -2, 8, 23 | 1, 14, 39 | 3, 22, 56 | 6, 32, 87 | 10, 46, 120 | 14, 60, 182 | 20, 81, 246 |
| 12 | -9, -7, -5 | -8, -5, -3 | -7, -4, -1 | -6, -1, 4 | -4, 2, 11 | -2, 7, 22 | 0, 13, 38 | 2, 21, 55 | 5, 31, 86 | 9, 45, 119 | 14, 59, 181 | 19, 80, 245 |
| 13 | -10, -7, -6 | -9, -6, -4 | -8, -5, -1 | -7, -2, 3 | -5, 1, 11 | -3, 6, 22 | -1, 12, 37 | 1, 20, 55 | 5, 30, 86 | 8, 44, 118 | 13, 58, 180 | 18, 79, 244 |
| 14 | -11, -8, -6 | -10, -7, -5 | -9, -5, -2 | -8, -3, 2 | -6, 0, 10 | -4, 5, 21 | -2, 11, 36 | 1, 19, 54 | 4, 29, 85 | 7, 43, 117 | 12, 57, 179 | 17, 78, 243 |
| 15 | -12, -9, -7 | -11, -8, -6 | -10, -6, -3 | -8, -4, 2 | -7, -1, 9 | -5, 4, 20 | -3, 10, 35 | 0, 18, 53 | 3, 28, 84 | 6, 42, 116 | 11, 56, 178 | 16, 78, 242 |
| 16 | -13, -10, -8 | -12, -9, -6 | -11, -7, -4 | -9, -5, 1 | -8, -1, 8 | -6, 3, 19 | -4, 9, 34 | -1, 17, 52 | 2, 27, 83 | 5, 41, 115 | 10, 55, 177 | 15, 77, 241 |
| 17 | -14, -11, -9 | -13, -10, -7 | -12, -8, -5 | -10, -6, 0 | -9, -2, 7 | -7, 2, 18 | -5, 8, 33 | -2, 16, 51 | 1, 26, 82 | 5, 40, 115 | 9, 54, 176 | 15, 76, 240 |
| 18 | -14, -12, -9 | -14, -10, -8 | -13, -9, -6 | -11, -7, -1 | -10, -3, 6 | -8, 1, 17 | -6, 8, 32 | -3, 15, 50 | 0, 25, 81 | 4, 39, 114 | 8, 53, 175 | 14, 75, 239 |
| 19 | -15, -12, -10 | -15, -11, -9 | -13, -10, -6 | -12, -7, -2 | -11, -4, 5 | -9, 0, 16 | -7, 7, 31 | -4, 15, 49 | -1, 24, 80 | 3, 38, 113 | 7, 53, 174 | 13, 74, 238 |
| 20 | -16, -13, -11 | -15, -12, -10 | -14, -11, -7 | -13, -8, -3 | -11, -5, 4 | -10, 0, 15 | -8, 6, 30 | -5, 14, 48 | -2, 24, 79 | 2, 37, 112 | 6, 52, 173 | 12, 73, 237 |
| 21 | -17, -14, -12 | -16, -13, -10 | -15, -11, -8 | -14, -9, -4 | -12, -6, 3 | -11, -1, 14 | -8, 5, 30 | -6, 13, 47 | -3, 23, 78 | 1, 36, 111 | 5, 51, 172 | 11, 72, 236 |
| 22 | -18, -15, -13 | -17, -14, -11 | -16, -12, -9 | -15, -10, -4 | -13, -7, 2 | -11, -2, 13 | -9, 4, 29 | -7, 12, 47 | -4, 22, 77 | 0, 35, 110 | 4, 50, 171 | 10, 71, 235 |
| 23 | -19, -16, -13 | -18, -15, -12 | -17, -13, -10 | -16, -11, -5 | -14, -8, 2 | -12, -3, 12 | -10, 3, 28 | -8, 11, 46 | -5, 21, 76 | -1, 34, 109 | 4, 49, 170 | 9, 70, 234 |
| 24 | -20, -17, -14 | -19, -16, -13 | -18, -14, -11 | -17, -12, -6 | -15, -9, 1 | -13, -4, 12 | -11, 2, 27 | -9, 10, 45 | -5, 20, 75 | -2, 33, 108 | 3, 48, 169 | 8, 70, 234 |
| 25 | -21, -18, -15 | -20, -16, -14 | -19, -15, -11 | -18, -13, -7 | -16, -9, 0 | -14, -5, 11 | -12, 1, 26 | -9, 9, 44 | -6, 19, 75 | -3, 32, 107 | 2, 47, 168 | 7, 69, 233 |
Unit loss with 3 defenders
This is a table of the attacking unit loss given that there are 3 defenders.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 4, 5, 6 | 5, 7, 8 | 6, 9, 12 | 7, 11, 16 | 9, 15, 24 | 11, 20, 36 | 13, 27, 51 | 15, 35, 68 | 19, 44, 100 | 23, 59, 132 | 27, 75, 195 | 32, 92, 259 |
| 2 | 3, 5, 5 | 4, 6, 7 | 5, 8, 11 | 7, 11, 15 | 8, 15, 23 | 10, 19, 35 | 12, 26, 51 | 15, 34, 67 | 18, 43, 99 | 22, 58, 131 | 26, 74, 194 | 31, 91, 258 |
| 3 | 3, 4, 5 | 4, 5, 7 | 5, 7, 10 | 6, 10, 15 | 7, 14, 22 | 9, 18, 34 | 11, 26, 50 | 14, 34, 66 | 18, 42, 98 | 21, 58, 130 | 26, 73, 194 | 30, 90, 258 |
| 4 | 2, 3, 4 | 3, 5, 6 | 4, 6, 9 | 5, 9, 14 | 7, 13, 22 | 9, 17, 33 | 10, 25, 49 | 13, 33, 66 | 17, 41, 97 | 20, 57, 129 | 25, 72, 193 | 30, 90, 257 |
| 5 | 1, 3, 3 | 2, 4, 5 | 3, 6, 9 | 4, 8, 13 | 6, 12, 21 | 8, 17, 33 | 10, 24, 48 | 12, 32, 65 | 16, 41, 97 | 19, 56, 129 | 24, 72, 192 | 29, 89, 256 |
| 6 | 1, 2, 3 | 2, 3, 4 | 3, 5, 8 | 4, 8, 13 | 5, 11, 20 | 7, 16, 32 | 9, 23, 48 | 12, 31, 64 | 15, 40, 96 | 19, 55, 128 | 23, 71, 191 | 28, 88, 255 |
| 7 | 1, 2, 4 | 2, 3, 4 | 2, 4, 7 | 3, 7, 12 | 4, 11, 20 | 6, 15, 31 | 8, 22, 47 | 11, 30, 63 | 14, 39, 95 | 18, 54, 127 | 23, 70, 190 | 27, 88, 255 |
| 8 | 1, 3, 4 | 1, 3, 4 | 2, 4, 7 | 3, 6, 11 | 4, 10, 19 | 5, 14, 30 | 7, 22, 46 | 10, 30, 63 | 14, 38, 94 | 17, 54, 127 | 22, 69, 190 | 27, 87, 254 |
| 9 | 1, 3, 4 | 1, 3, 4 | 2, 4, 6 | 2, 5, 10 | 3, 9, 18 | 5, 14, 30 | 7, 21, 45 | 9, 29, 62 | 13, 38, 94 | 16, 53, 126 | 21, 68, 189 | 26, 86, 253 |
| 10 | 2, 3, 5 | 2, 3, 5 | 2, 4, 5 | 2, 5, 10 | 3, 8, 17 | 4, 13, 29 | 6, 20, 45 | 9, 28, 61 | 12, 37, 93 | 16, 52, 125 | 20, 68, 188 | 25, 86, 252 |
| 11 | 2, 3, 5 | 2, 3, 5 | 2, 4, 5 | 2, 4, 9 | 3, 8, 17 | 4, 12, 28 | 5, 19, 44 | 8, 27, 60 | 11, 36, 92 | 15, 51, 124 | 19, 67, 187 | 25, 85, 251 |
| 12 | 2, 4, 5 | 2, 4, 5 | 2, 4, 5 | 2, 4, 8 | 3, 7, 16 | 3, 11, 27 | 5, 18, 43 | 7, 27, 60 | 11, 35, 91 | 14, 51, 124 | 19, 66, 186 | 24, 84, 251 |
| 13 | 2, 4, 6 | 2, 4, 6 | 2, 4, 6 | 2, 4, 8 | 3, 6, 15 | 3, 11, 27 | 4, 18, 42 | 7, 26, 59 | 10, 35, 91 | 13, 50, 123 | 18, 65, 186 | 23, 83, 250 |
| 14 | 2, 4, 6 | 2, 4, 6 | 2, 4, 6 | 2, 4, 7 | 3, 6, 14 | 3, 10, 26 | 4, 17, 42 | 6, 25, 58 | 9, 34, 90 | 13, 49, 122 | 17, 64, 185 | 22, 83, 249 |
| 15 | 2, 4, 6 | 2, 4, 6 | 2, 4, 6 | 2, 5, 7 | 3, 6, 14 | 3, 9, 25 | 4, 16, 41 | 5, 24, 58 | 8, 33, 89 | 12, 48, 121 | 16, 63, 184 | 22, 82, 248 |
| 16 | 3, 5, 7 | 3, 5, 7 | 3, 5, 7 | 3, 5, 7 | 3, 5, 13 | 3, 8, 24 | 4, 15, 40 | 5, 23, 57 | 8, 32, 88 | 11, 47, 121 | 16, 63, 183 | 21, 81, 248 |
| 17 | 3, 5, 7 | 3, 5, 7 | 3, 5, 7 | 3, 5, 7 | 3, 6, 12 | 3, 8, 24 | 4, 15, 39 | 5, 23, 56 | 7, 32, 88 | 10, 47, 120 | 15, 62, 183 | 20, 81, 247 |
| 18 | 3, 5, 7 | 3, 5, 7 | 3, 5, 7 | 3, 5, 7 | 3, 6, 12 | 3, 7, 23 | 4, 14, 39 | 5, 22, 55 | 6, 31, 87 | 10, 46, 119 | 14, 61, 182 | 19, 80, 246 |
| 19 | 3, 5, 7 | 3, 5, 7 | 3, 5, 7 | 3, 5, 7 | 3, 6, 11 | 3, 7, 22 | 4, 13, 38 | 5, 21, 55 | 6, 30, 86 | 9, 45, 118 | 13, 60, 181 | 19, 79, 245 |
| 20 | 3, 6, 8 | 3, 6, 8 | 3, 6, 8 | 3, 6, 8 | 3, 6, 10 | 4, 7, 21 | 4, 12, 37 | 5, 20, 54 | 6, 29, 85 | 8, 44, 118 | 13, 59, 180 | 18, 78, 244 |
| 21 | 3, 6, 8 | 3, 6, 8 | 3, 6, 8 | 3, 6, 8 | 3, 6, 10 | 4, 7, 21 | 4, 12, 36 | 5, 20, 53 | 6, 29, 85 | 8, 44, 117 | 12, 59, 179 | 17, 78, 244 |
| 22 | 4, 6, 8 | 4, 6, 8 | 4, 6, 8 | 4, 6, 8 | 4, 6, 10 | 4, 7, 20 | 4, 11, 35 | 5, 19, 52 | 5, 28, 84 | 7, 43, 116 | 11, 58, 179 | 16, 77, 243 |
| 23 | 4, 6, 8 | 4, 6, 8 | 4, 6, 8 | 4, 6, 8 | 4, 6, 10 | 4, 7, 19 | 4, 10, 35 | 5, 18, 52 | 5, 27, 83 | 7, 42, 115 | 10, 57, 178 | 16, 76, 242 |
| 24 | 4, 6, 9 | 4, 6, 9 | 4, 6, 9 | 4, 6, 9 | 4, 7, 9 | 4, 7, 18 | 4, 9, 34 | 5, 17, 51 | 6, 27, 82 | 7, 41, 115 | 10, 56, 177 | 15, 76, 241 |
| 25 | 4, 7, 9 | 4, 7, 9 | 4, 7, 9 | 4, 7, 9 | 4, 7, 9 | 4, 7, 18 | 4, 9, 33 | 5, 17, 50 | 6, 26, 82 | 7, 41, 114 | 9, 56, 176 | 14, 75, 241 |
Unit loss with 5 defenders
This is a table of the attacking unit loss given that there are 5 defenders.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 6, 7, 8 | 7, 9, 10 | 8, 11, 14 | 9, 13, 18 | 11, 17, 26 | 13, 22, 38 | 15, 29, 53 | 17, 37, 70 | 21, 46, 102 | 25, 61, 134 | 29, 77, 197 | 34, 94, 261 |
| 2 | 5, 7, 7 | 6, 8, 9 | 7, 10, 13 | 9, 13, 17 | 10, 17, 25 | 12, 21, 37 | 14, 28, 53 | 17, 36, 69 | 20, 45, 101 | 24, 60, 133 | 28, 76, 196 | 33, 93, 260 |
| 3 | 5, 6, 7 | 6, 7, 9 | 7, 9, 12 | 8, 12, 17 | 9, 16, 24 | 11, 20, 36 | 13, 28, 52 | 16, 36, 68 | 20, 44, 100 | 23, 60, 132 | 28, 75, 196 | 32, 92, 260 |
| 4 | 4, 5, 6 | 5, 7, 8 | 6, 8, 11 | 7, 11, 16 | 9, 15, 24 | 11, 19, 35 | 12, 27, 51 | 15, 35, 68 | 19, 43, 99 | 22, 59, 131 | 27, 74, 195 | 32, 92, 259 |
| 5 | 3, 4, 5 | 4, 6, 7 | 5, 8, 11 | 6, 10, 15 | 8, 14, 23 | 10, 19, 35 | 12, 26, 50 | 14, 34, 67 | 18, 43, 99 | 21, 58, 131 | 26, 74, 194 | 31, 91, 258 |
| 6 | 2, 4, 5 | 3, 5, 6 | 4, 7, 10 | 6, 10, 15 | 7, 13, 22 | 9, 18, 34 | 11, 25, 50 | 14, 33, 66 | 17, 42, 98 | 21, 57, 130 | 25, 73, 193 | 30, 90, 257 |
| 7 | 2, 3, 4 | 3, 4, 6 | 4, 6, 9 | 5, 9, 14 | 6, 13, 22 | 8, 17, 33 | 10, 24, 49 | 13, 32, 65 | 16, 41, 97 | 20, 56, 129 | 25, 72, 192 | 29, 90, 257 |
| 8 | 2, 3, 4 | 2, 4, 5 | 3, 5, 9 | 4, 8, 13 | 6, 12, 21 | 7, 16, 32 | 9, 24, 48 | 12, 32, 65 | 16, 40, 96 | 19, 56, 129 | 24, 71, 192 | 29, 89, 256 |
| 9 | 2, 3, 4 | 2, 4, 5 | 3, 5, 8 | 4, 7, 12 | 5, 11, 20 | 7, 16, 32 | 9, 23, 47 | 11, 31, 64 | 15, 40, 96 | 18, 55, 128 | 23, 70, 191 | 28, 88, 255 |
| 10 | 2, 3, 5 | 2, 3, 5 | 2, 4, 7 | 3, 7, 12 | 4, 10, 19 | 6, 15, 31 | 8, 22, 47 | 11, 30, 63 | 14, 39, 95 | 18, 54, 127 | 22, 70, 190 | 27, 88, 254 |
| 11 | 2, 3, 5 | 2, 3, 5 | 2, 4, 6 | 3, 6, 11 | 4, 10, 19 | 5, 14, 30 | 7, 21, 46 | 10, 29, 62 | 13, 38, 94 | 17, 53, 126 | 21, 69, 189 | 27, 87, 253 |
| 12 | 2, 4, 5 | 2, 4, 5 | 2, 4, 6 | 3, 5, 10 | 3, 9, 18 | 5, 13, 29 | 6, 20, 45 | 9, 29, 62 | 13, 37, 93 | 16, 53, 126 | 21, 68, 188 | 26, 86, 253 |
| 13 | 2, 4, 6 | 2, 4, 6 | 2, 4, 6 | 3, 5, 10 | 3, 8, 17 | 4, 13, 29 | 6, 20, 44 | 8, 28, 61 | 12, 37, 93 | 15, 52, 125 | 20, 67, 188 | 25, 85, 252 |
| 14 | 2, 4, 6 | 2, 4, 6 | 2, 4, 6 | 3, 5, 9 | 3, 7, 16 | 4, 12, 28 | 5, 19, 44 | 8, 27, 60 | 11, 36, 92 | 15, 51, 124 | 19, 66, 187 | 24, 85, 251 |
| 15 | 2, 4, 6 | 2, 4, 6 | 2, 4, 6 | 3, 5, 8 | 3, 7, 16 | 4, 11, 27 | 5, 18, 43 | 7, 26, 60 | 10, 35, 91 | 14, 50, 123 | 18, 65, 186 | 24, 84, 250 |
| 16 | 3, 5, 7 | 3, 5, 7 | 3, 5, 7 | 3, 5, 8 | 3, 6, 15 | 4, 10, 26 | 5, 17, 42 | 6, 25, 59 | 9, 34, 90 | 13, 49, 123 | 18, 65, 185 | 23, 83, 250 |
| 17 | 3, 5, 7 | 3, 5, 7 | 3, 5, 7 | 3, 5, 8 | 3, 6, 14 | 4, 10, 26 | 4, 17, 41 | 6, 25, 58 | 9, 34, 90 | 12, 49, 122 | 17, 64, 185 | 22, 83, 249 |
| 18 | 3, 5, 7 | 3, 5, 7 | 3, 5, 7 | 3, 5, 8 | 3, 6, 14 | 4, 9, 25 | 4, 16, 41 | 6, 24, 57 | 8, 33, 89 | 12, 48, 121 | 16, 63, 184 | 21, 82, 248 |
| 19 | 3, 5, 7 | 3, 5, 7 | 3, 5, 7 | 3, 5, 8 | 3, 6, 13 | 4, 8, 24 | 4, 15, 40 | 5, 23, 57 | 7, 32, 88 | 11, 47, 120 | 15, 62, 183 | 21, 81, 247 |
| 20 | 3, 6, 8 | 3, 6, 8 | 3, 6, 8 | 3, 6, 8 | 3, 6, 12 | 4, 8, 23 | 4, 14, 39 | 5, 22, 56 | 7, 31, 87 | 10, 46, 120 | 15, 61, 182 | 20, 80, 246 |
| 21 | 3, 6, 8 | 3, 6, 8 | 3, 6, 8 | 3, 6, 8 | 4, 6, 11 | 4, 7, 23 | 4, 14, 38 | 5, 22, 55 | 6, 31, 87 | 9, 46, 119 | 14, 61, 181 | 19, 80, 246 |
| 22 | 4, 6, 8 | 4, 6, 8 | 4, 6, 8 | 4, 6, 8 | 4, 6, 11 | 4, 7, 22 | 4, 13, 37 | 5, 21, 54 | 6, 30, 86 | 9, 45, 118 | 13, 60, 181 | 18, 79, 245 |
| 23 | 4, 6, 8 | 4, 6, 8 | 4, 6, 8 | 4, 6, 8 | 4, 6, 10 | 4, 7, 21 | 4, 12, 37 | 5, 20, 54 | 6, 29, 85 | 8, 44, 117 | 12, 59, 180 | 18, 78, 244 |
| 24 | 4, 6, 9 | 4, 6, 9 | 4, 6, 9 | 4, 6, 9 | 4, 7, 10 | 4, 7, 20 | 5, 11, 36 | 5, 19, 53 | 6, 29, 84 | 8, 43, 117 | 12, 58, 179 | 17, 78, 243 |
| 25 | 4, 7, 9 | 4, 7, 9 | 4, 7, 9 | 4, 7, 9 | 4, 7, 10 | 4, 7, 20 | 5, 11, 35 | 5, 19, 52 | 6, 28, 84 | 7, 43, 116 | 11, 58, 178 | 16, 77, 243 |
Unit loss with 8 defenders
This is a table of the attacking unit loss given that there are 8 defenders.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 9, 10, 11 | 10, 12, 13 | 11, 14, 17 | 12, 16, 21 | 14, 20, 29 | 16, 25, 41 | 18, 32, 56 | 20, 40, 73 | 24, 49, 105 | 28, 64, 137 | 32, 80, 200 | 37, 97, 264 |
| 2 | 8, 10, 10 | 9, 11, 12 | 10, 13, 16 | 12, 16, 20 | 13, 20, 28 | 15, 24, 40 | 17, 31, 56 | 20, 39, 72 | 23, 48, 104 | 27, 63, 136 | 31, 79, 199 | 36, 96, 263 |
| 3 | 8, 9, 10 | 9, 10, 12 | 10, 12, 15 | 11, 15, 20 | 12, 19, 27 | 14, 23, 39 | 16, 31, 55 | 19, 39, 71 | 23, 47, 103 | 26, 63, 135 | 31, 78, 199 | 35, 95, 263 |
| 4 | 7, 8, 9 | 8, 10, 11 | 9, 11, 14 | 10, 14, 19 | 12, 18, 27 | 14, 22, 38 | 15, 30, 54 | 18, 38, 71 | 22, 46, 102 | 25, 62, 134 | 30, 77, 198 | 35, 95, 262 |
| 5 | 6, 7, 8 | 7, 9, 10 | 8, 11, 14 | 9, 13, 18 | 11, 17, 26 | 13, 22, 38 | 15, 29, 53 | 17, 37, 70 | 21, 46, 102 | 24, 61, 134 | 29, 77, 197 | 34, 94, 261 |
| 6 | 5, 7, 8 | 6, 8, 9 | 7, 10, 13 | 9, 13, 18 | 10, 16, 25 | 12, 21, 37 | 14, 28, 53 | 17, 36, 69 | 20, 45, 101 | 24, 60, 133 | 28, 76, 196 | 33, 93, 260 |
| 7 | 5, 6, 7 | 6, 7, 9 | 7, 9, 12 | 8, 12, 17 | 9, 16, 25 | 11, 20, 36 | 13, 27, 52 | 16, 35, 68 | 19, 44, 100 | 23, 59, 132 | 28, 75, 195 | 32, 93, 260 |
| 8 | 4, 5, 6 | 5, 7, 8 | 6, 8, 12 | 7, 11, 16 | 9, 15, 24 | 10, 19, 35 | 12, 27, 51 | 15, 35, 68 | 19, 43, 99 | 22, 59, 132 | 27, 74, 195 | 32, 92, 259 |
| 9 | 3, 5, 6 | 4, 6, 7 | 5, 8, 11 | 6, 10, 15 | 8, 14, 23 | 10, 19, 35 | 12, 26, 50 | 14, 34, 67 | 18, 43, 99 | 21, 58, 131 | 26, 73, 194 | 31, 91, 258 |
| 10 | 3, 4, 5 | 3, 5, 7 | 4, 7, 10 | 6, 10, 15 | 7, 13, 22 | 9, 18, 34 | 11, 25, 50 | 14, 33, 66 | 17, 42, 98 | 21, 57, 130 | 25, 73, 193 | 30, 91, 257 |
| 11 | 2, 4, 5 | 3, 5, 6 | 4, 6, 9 | 5, 9, 14 | 6, 13, 22 | 8, 17, 33 | 10, 24, 49 | 13, 32, 65 | 16, 41, 97 | 20, 56, 129 | 24, 72, 192 | 30, 90, 256 |
| 12 | 2, 4, 5 | 3, 4, 6 | 3, 6, 9 | 4, 8, 13 | 6, 12, 21 | 7, 16, 32 | 9, 23, 48 | 12, 32, 65 | 16, 40, 96 | 19, 56, 129 | 24, 71, 191 | 29, 89, 256 |
| 13 | 2, 4, 6 | 3, 4, 6 | 3, 5, 8 | 4, 7, 13 | 5, 11, 20 | 7, 16, 32 | 9, 23, 47 | 11, 31, 64 | 15, 40, 96 | 18, 55, 128 | 23, 70, 191 | 28, 88, 255 |
| 14 | 2, 4, 6 | 2, 4, 6 | 3, 5, 7 | 4, 7, 12 | 4, 10, 19 | 6, 15, 31 | 8, 22, 47 | 11, 30, 63 | 14, 39, 95 | 18, 54, 127 | 22, 69, 190 | 27, 88, 254 |
| 15 | 2, 4, 6 | 2, 4, 6 | 3, 5, 7 | 3, 6, 11 | 4, 10, 19 | 5, 14, 30 | 7, 21, 46 | 10, 29, 63 | 13, 38, 94 | 17, 53, 126 | 21, 68, 189 | 27, 87, 253 |
| 16 | 3, 5, 7 | 3, 5, 7 | 3, 5, 7 | 3, 6, 10 | 4, 9, 18 | 5, 13, 29 | 7, 20, 45 | 9, 28, 62 | 12, 37, 93 | 16, 52, 126 | 21, 68, 188 | 26, 86, 253 |
| 17 | 3, 5, 7 | 3, 5, 7 | 3, 5, 7 | 3, 6, 10 | 4, 8, 17 | 5, 13, 29 | 6, 20, 44 | 9, 28, 61 | 12, 37, 93 | 15, 52, 125 | 20, 67, 188 | 25, 86, 252 |
| 18 | 3, 5, 7 | 3, 5, 7 | 3, 5, 7 | 3, 6, 9 | 4, 8, 17 | 4, 12, 28 | 6, 19, 44 | 8, 27, 60 | 11, 36, 92 | 15, 51, 124 | 19, 66, 187 | 24, 85, 251 |
| 19 | 3, 5, 7 | 3, 5, 7 | 3, 5, 7 | 3, 6, 9 | 4, 7, 16 | 4, 11, 27 | 5, 18, 43 | 7, 26, 60 | 10, 35, 91 | 14, 50, 123 | 18, 65, 186 | 24, 84, 250 |
| 20 | 3, 6, 8 | 3, 6, 8 | 3, 6, 8 | 3, 6, 9 | 4, 7, 15 | 4, 10, 26 | 5, 17, 42 | 7, 25, 59 | 9, 34, 90 | 13, 49, 123 | 18, 64, 185 | 23, 83, 249 |
| 21 | 3, 6, 8 | 3, 6, 8 | 3, 6, 8 | 3, 6, 9 | 4, 7, 14 | 4, 10, 26 | 5, 17, 41 | 6, 25, 58 | 9, 34, 90 | 12, 49, 122 | 17, 64, 184 | 22, 83, 249 |
| 22 | 4, 6, 8 | 4, 6, 8 | 4, 6, 8 | 4, 6, 9 | 4, 7, 14 | 4, 9, 25 | 5, 16, 40 | 6, 24, 57 | 8, 33, 89 | 12, 48, 121 | 16, 63, 184 | 21, 82, 248 |
| 23 | 4, 6, 8 | 4, 6, 8 | 4, 6, 8 | 4, 6, 9 | 4, 7, 13 | 4, 9, 24 | 5, 15, 40 | 6, 23, 57 | 8, 32, 88 | 11, 47, 120 | 15, 62, 183 | 21, 81, 247 |
| 24 | 4, 6, 9 | 4, 6, 9 | 4, 6, 9 | 4, 6, 9 | 4, 7, 12 | 4, 8, 23 | 5, 14, 39 | 6, 22, 56 | 7, 32, 87 | 10, 46, 120 | 15, 61, 182 | 20, 81, 246 |
| 25 | 4, 7, 9 | 4, 7, 9 | 4, 7, 9 | 4, 7, 9 | 4, 7, 12 | 4, 8, 23 | 5, 13, 38 | 6, 22, 55 | 7, 31, 87 | 9, 46, 119 | 14, 61, 181 | 19, 80, 246 |