In this page I will study the Game System of BOW with some very basic mathematical knowledge. I will also explain why, in my opinion, this game system is well balance and will justify the choice of some rules. Of course you definitely don't need to read this page to play BOW but you can read it by curiosity or if you plan to add your own rules or units to the BOW universe.

1. Dice

A normal dice is a uniform cube where each face has the same probability to appear when you throw it (1 chance on 6, i.e. probability = 1/6). But in BOW, the dices are replaced by a 2x2 brick. This brick is not a uniform cube but a parallelepiped which height is only 60% of its length or width. Honestly I don't know how to compute the probability of each face for such a volume because the physics and the dynamic of the throw will clearly influence the result. Simply use the ratio between the sizes of the different faces will not lead to a correct estimation.

But it remains the empirical way, which means throwing a lot of brick 2x2 and observing the result. You can do it yourself, it is not very difficult: take 20 bricks 2x2 in your hand, just drop them on the table, and note on a paper the number of bricks on their edge. Repeat this operation 5 times, and finally you will have launch 100 bricks 2x2. Sum the 5 results to know the probability; normally you should get a number around 26. That's what I did 10 times to get in final 1000 bricks thrown.

Total: 21292237272225232828
Average Probability: 26.2

This result is also in concordance with the result found by someone else (27%) as mentioned in this lugnet thread. The random system in BOW is based on a throw of several independent bricks. Since the probabilities for each brick are independent from each other, we can sum the probabilities to evaluate the chances of the global result. For example if you launch 4 bricks, the global probability to have at least one brick sideway is 26 + 26 + 26 + 26 = 104% that means you are quite sure to succeed your throw. In other words, the average number of bricks that will stop sideway is 1 when you launch 4 bricks. Of course that doesn't mean you always will get 1 brick sideway when you launch 4 bricks, because we are talking about probabilities here, so sometimes you will get 0, sometimes you will get more than 1, but in average that's the result you can expect! In the following table, I did this very simple calculation for the different number of bricks you can throw.

Number of
Bricks thrown
Probability of success
(at least 1 Brick sideway)
Average Number of
Bricks sideway
1 26%0 (0.26)
226 + 26 =52%0 (0.52)
326 + 26 + 26 =78%0 (0.78)
44 x 26 =104%1 (1.04)
55 x 26 =130%1 (1.30)
66 x 26 =156%1 (1.56)
77 x 26 =182%1 (1.82)
88 x 26 =208%2 (2.08)

As you can see a BR of 1 or 2 as poor chances to succeed, whereas a BR of 3, 4 or 5 bricks has quite good chances to succeed, and for a BR of 6 or more the chances of success are very strong. That's why in BOW the abilities of 1 or 2 are reserved to the basics troopers or the weak unit. An ability of 3, 4 or 5 can often be found on an average unit such as a minor hero, a commander, or a well equipped trooper. And the abilities of 6 or more are reserved to the heroes.

But also, don't forget that in BOW, all the random throws are done in opposition. That means, if two enemy heroes are fighting against each other, they can fight with very high BR but the probability of win stays equivalent for both of them. In the same way, the chances that a basic trooper kills your hero with a lucky draw are very low. Indeed, since the throws are made in opposition, to know the probability of win, you should look at the difference of bricks thrown before referring to the previous table.

Your probability
of win

2. Round and Turn

In BOW, during your Turn, you don't play all your army but only one minifig. I know that this is unusual for a wargame, but this point is quite important for the balance of the game. In a classical wargame where the players plays all their units during their Turn, the first player that start the game has a huge advantage because he can destroy quite a big part of the army of his opponent before taking any damage. After the first Turn, it is like the second player start with a smaller army and of course this smaller army will inflict smaller damages. To give you an idea of the unfairness of this system, imagine that the two players have exactly the same army, and at the beginning of the game, the two armies are facing each other in a desert battlefield and in range of the weapons. The first player will of course make fire all his minifigs. Since the two armies have the same power the average number of kill will be 50%, because each minifig of the attacker that fires on one minifig of the defender has 50% of chance to win (since attacker and defender have the same power). During the second Turn, the second player will do the same thing but with an army reduced of 50%. Look in the following table how the two armies are reduced Turn after Turn (the bold indicates whose Turn it is):

TurnMinifig count of
first player
DamagesMinifig count of
second player
0100not started yet100
1100== 100 fires, 50 kills ==>50
275<== 50 fires, 25 kills ==50
375== 75 fires, 37 kills ==>13
469<== 13 fires, 6 kills ==13
569== 69 fires, 34 kills ==>0

So the result of this frontal confrontation is that the first player wins with 69% of his army still alive. Of course this situation is quite exaggerated, in reality the second army uses the obstacles of the battlefield as a cover, and the first army probably needs mainly to move during the first Turn. But still, the advantage of the first army is disproportionate.

The board game designer Richard Borg is famous to have created a new system known as "Command and Colors System" for his popular casual wargames like "BattleLore" or "Memoire 44". In this system, the players don't play their full army during their Turn, but only a subset of it (typically from one to four units). This system, just like in BOW, avoid the unbalanced advantage of the first player and reduce the duration of each Turn which helps keeping the players active. I'm convinced that this is the main reason of the success of this game system.

When I designed BOW at one moment, I tested a system where the players were playing all their army during their Turn, but the dead minifigs were not removed from the battlefield and were allowed to fight until the end of the Round. Then the dead minifigs of all the players were removed altogether at the end of the Round when all the players played their army. But this system required a lot of marking bricks (who is played, who is dead, etc...) and leads to a messy battlefield. Also I had the feeling that the immersive experience was quite bad, not to mention that you get bored/frustrated during the long Turn of your opponent.

Playing only one minifig during your Turn is my response to all this issues, even if I know that this solution requires some marking bricks to remember who is played and who is not. Also at one moment I thought of removing the Round concept and leave the players play the unit the want every Turn, but in that system it is obvious that the players will play their hero (or their most powerful unit) every Turn and never play their troopers, and of course we don't want to go in such an extremity.

3. Cost Point

Like in every classical wargame, BOW provide a Cost Point mechanism. This mechanism is very useful for the players to create balanced armies very quickly. The Cost Point mechanism depends strongly on the game system, that's why it is different in every wargame. Here are the CP rules as a reminder:

If the minifig has a close fight weapon:
CP = (Action Budget + Move + Dodge/Armour + Attack/Damage + Treat/Repair) x Health Point

If the minifig has a range weapon:
CP = (Action Budget + Move + Dodge/Armour + Attack/Damage + Treat/Repair) x Health Point x Damage

Summing the values of all the abilities of the character is in my opinion a good start to evaluate the power of a unit. Because the values of the abilities represent how many bricks the unit can launch, there is an obvious link with its power. Now the reason why I multiply the ability sum by the HP can also be explained quite easily. Imagine that you have two different units with the same ability values, except that one has 2 HP instead of 1. Since this unit has 2 HP we will need to damage it two times to destroy it completely (and a wounded/damaged unit doesn't suffer any penalty, it can still fight with a full power). So it is like I need to kill 2 identical units with 1 HP.



Of course this is not exactly true, because with one powerful attack you can inflict 2 damage points and kill the unit with 2 HP in one Turn, whereas you need to perform 2 different attacks to kill the 2 units with 1 HP. So in theory this give a little advantage to the 2 separate units, but on the other hand a unit with only 1 HP can not be treated/repaired, since he died immediately after it was damaged.

Now for the range weapon multiplication, this is more something empirical. I noticed during the test phase that the range weapon units were too powerful compared to the close combat units. Because the range weapon units can inflict damage on close combat units without suffering any back-damage from them. In BOW, units with a range weapon can use them to fire at point-blank range without any penalty. Indeed this true for a gun, but also for a bow (unless your opponent literally grab you). I think in some other wargame they introduce some penalty rules for the range units when they are in contact to balance their advantage on the close-combat units. My opinion is that this kind of specific rules is complicated to remember during the game, so I preferred put this balance factor in the CP itself.

Finally, if a range unit and a close combat unit are in contact, there is no advantage for both of them. In this situation the type of weapon doesn't matter. So logically the CP would be the same if the range units were always fight at short range. The real advantage of the range units depends in fact of their distance of fight. The farer they can fire, the more powerful they are, because a victim close-combat unit will need a lot of Turn to reach the attacking range unit and finally fight back. In BOW the distance of fire of the range weapon is linked to the damage of the weapon (range of the weapon = damage x 15 studs). That's why finally I decided to multiply the CP by the damage of the weapon if the weapon is a range one.

4. Conclusion

With the quite secure probabilities of the Brick Roll System that avoid too big lucky or unlucky draw, and with the Round and Turn System that suppress the advantage of the first player, I tend to believe that the game system of BOW is well balanced. Actually I found during my tests that it is so balanced that a basic confrontation on a desert battlefield will lead to a tie with both armies completely destroyed (or almost, since of course there's one last man standing that defines the winning army).

With such balanced game system, the interest of BOW can be found in the tactic of each players, in the battlefield layout and in the scenario that are not necessary symmetrical. Of course, even if the game system is balanced, you can create an unbalanced scenario, or play with two unbalanced armies! :-)

I hope you didn't find this page too technical, and that didn't give you a feeling of complexity. Personally I think it is crucial to study the mathematical probability of your game system if you want to be sure to produce a fun and balanced gameplay. Usually the final player doesn't have any idea of the analysis that was done on the game system, but I wanted to give you some info on how I designed this game system. Now forget it, and have fun with BOW!