Major Q9
Heroscape is a tabletop war game released in 2004 and discontinued in 2010. Wikipedia, fan site. Attack dice show a skull 50% of the time. Defense dice show a shield 33% of the time. Each excess skull rolled counts as one wound. Disclaimer: I'm not particularly aware of any existing theory/math for this game. Topics like this may have been covered in the past. This is just something I got interested in doing for myself, in my own style.
Queglix Gun Special Attack Range 6. Attack 1, 2, or 3.
Major Q9 starts each turn with 9 attack dice. Choose any figure within range and attack by rolling 1, 2, or 3 attack dice. Major Q9 may keep making special attacks with 1, 2, or 3 attack dice until he has rolled all 9 attack dice. Major Q9 may target the same or different figures with each attack.
Major Q9's special attack has a decent degree of flexibility, allowing you to split your dice into 4 different reasonable schemes. But which one's best, and when? Should you ever just use his basic attack? The full card of the unit in question.
First, let's look at the average damage dealt by his attack options, versus variable enemy defense.
Assuming no weird special abilities are in play, things are quite simple: use the Queglix Gun, 3x3. The only exception is if you have height advantage, and the enemy has 8 or more defense.
However, this is assuming all your attacks are against an infinite-health target. That's fine enough versus heroes, but squads should be considered separately - they all have 1 life, so anything more than 1 wound is always wasted.
Unsurprisingly, 1x9 is now best at low defense values. At 2 defense, 1x9/2x4+1x1/2x3+3x1 are all effectively equal. (Their average squads killed at that point are respectively ~2.00/2.00/1.96.) At 3 defense and beyond, 3x3 is best. Even with height advantage, normal attacks are never worth it against squads.
Extreme height and other uncommon buffs are not accounted for. You can probably imagine about where the line would fall if you add another die to his regular attacks.