This truly fits nowhere else.

I have been playing in Excel lately. Trying to find out things about my board game, mathematical speaking.

The sheet that I created has the option to change a d6 in any custom dice. And the unit that is attacked by these dice can go up to 20 health.

Then, in the long run, you can see the chance of the unit still living. And a total sum of the unit still being alive. This will result in an end durability. Not to be compared with the average durability, but interesting nonetheless. Since, well, the end durability is the true average durability.

To test the sheet, I tried an unit with 1 health. And then a die roll of 50% to kill it (000111). Most people consider 1,5 as the average. Thus the unit dies immediately or after 1 more round. This is incorrect, math people know that you need to continue adding the chances. The end result is 2. Which is correct. You see a neat 1 - 1,5 - 1,75 - 1,875 etc list. And it ends with 2.

I got confidence that my sheet works properly.

My board game was also put to an test. To see if I could discover something interesting. And I did.

The die is changed to 001234. Average damage of 5/3.

And I checked all the health from 1 to 20.

What I discovered was a neat expected health value (which is 0,6 times the health) plus 0,6. In other words. All units have their health plus 1 added. Then multiplied by 0,6 (or by 3/5).

The unit with 1 health has even a whopping 0,9 added. The 20 health mark received 12,6.

So, I created some curves. The most important one is the efficiency of end durability divided by expected durability. Most of my battles where tested on expected durabilities. But now, I can do them again with the end durability.

Then it hit me. If the addition to the expected value has a neat 3/5 added. Just like the invert of the die damage. Would this hold true for the bonus die? 012345.

Average 2,5 damage or inverted 0,4.

As it turns out, the addition is not 0,4. I did not get 8,4 on the 20 health mark. But 8,53354 instead.

I am completely puzzled. This makes no sense at all to what I have seen so far. But chances are that my list is too short. It only goes to 20 health. Maybe the theory holds true at a higher health. If not, then the *0,6 and +0,6 are a coincidence.

My initial goal with this research is to see if I can discard the usage of bonus dice. Which is to support the little guys, but a lot of hassle so far. But if they are more durable due to a rule change on the damage by dice. Then this is more then welcome.