What is the least number of armies needed to take down a Size 10 Wasteland?
Assume standard offense/defense/luck settings from ladder and that number of turns is not an issue, its just a territory in the background that you would just like to have knocked down so you could get the bonus, but don't have a follow-on destination for the reserve armies nor a pressing need to present them on the other side.
I never did the math for multi-turn attacks, but my instinct says even 9 might be enough in theory:
On your first attack you kill a maximum of 60% * 9 * 25% + 9 * 75% = 8,1 "=" 9 defenders (but that has an incredibly low chance of happening, so let's say you kill 8 (that has approximately a 3% chance). You'll lose at least 70% * 10 * 25% + 0 * 75% = 1,75 = 1 attackers (but let's say it's 2). That means in the second turn you have a 7 vs 2 attack (> 99% success rate).
So finding the theoretical minimum is pointless: it has a ridiculously low chance of actually working out that way and adding even a single extra army will drastically increase your success rate.
What strategy employs the least number of armies?
One that is utterly unlikely to work. So instead, I'll answer a related question: what strategy would I recommend.
Remember that every defender not killed will get another 70% kill rate against your attackers next turn, so try to kill as many as you can. Also, attacking with X armies and failing will not get you the bonus. Attacking with X + bonus-value will typically increase your chance of success (on this turn) hugely, giving you another bonus-value armies to deploy not only the very next turn, but every turn after that. So unless you really can't spare the additional armies (in which case you have no business breaking the blockade in the first place), your "return on investment" is one turn, so incredibly short.
Lesson: don't try to work it out so you prevent armies "getting stuck" far away from the front line; having (for instance) two armies stuck finishing a bonus worth five armies is completely worth it!
My thoughts, based on previous trial & error is to keep attacking the neutral with about 125% of the armies in the neutral.
For example: Turn 1, attack Size 10 Wasteland with 12-13 armies, which will leave about 2-3 neutrals behind and about 5-6 armies to take the neutral on the next turn.
Less than 12, and you end up not doing enough damage and killing all of your attacking men. More than 13, too many excess leading to overkill on next turn.
I'd personally err on the side of a slight overkill, just to get it over with and have one less thing to worry about.
-Open to your thoughts/opinions.
If you weren't, there wouldn't be much point asking the question. ;)
Follow on questions to spur additional debate:
-What else should you consider strategically in taking down wastelands to make it worth the army investment?
The value of the bonus you're completing. If it's very low, it might be better to invest the armies in completing other (bigger) bonuses elsewhere (even if they are on the front line).
One major factor would also be nested bonuses; if the wasteland doesn't merely "break" a small bonus, but that bonus is also the last incomplete one in a superbonus otherwise entirely in your control, breaking a wasteland suddenly becomes a much better idea. (This is probably not relevant to you, since you seem to be mostly concerned with Season III, but I wanted to mention it anyway, for completeness' sake.)
-Does having 3 Neutrals in all other territories with Size 10 Wastelands alter the discussion on if its worth while taking down wastelands? (Season III)
Yes, since every other (neutral) bonus on the board will take a much bigger investment in armies to complete, essentially reducing the "price" of breaking the wasteland in comparison.
-What if you "Forget" where the wastelands were and already killed 5/6 of the bonus before finding the wastland? Ditch it or swallow your pill?
Ehm wait, I thought we were talking about the situation where the wasteland is the only territory missing from completing a bonus...?? If, after the wasteland, you still need to take some other territories as well, I'd say that in most practical situations your armies would be better spend elsewhere. Also, there's no reason to be conservative about it: any armies "left over" can be spent taking the remaining neutrals in the bonus.
-Will the above thumbrule apply to large blockades of 100+?
Instinctively I'd say you should increase the 125% multiplier a bit. Remember, x defenders kill 70% * x attackers on average, attacking with 125% armies means you're expected to kill 60% * 125% * x = 75% * x, barely more than you're losing.