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Then there is width.
Each time you attack a province you can only have so many troops enter the battle at a time. It is determined by this:
1 Province attacking 1 province = 80 width. Each additional province that attacks adds another 40 width. As a result, if you are attacking from three sides that would be 80+40+40. If you are attacking from two sides then it would be 80+40 = 120.
This pic shows I have a combat width of 120.
http://images.akamai.steamusercontent.com/ugc/271720179210875273/FB7EFD79EB7BFCE145E2EEA5AD6E57733802C53A/
http://steamcommunity.com/sharedfiles/filedetails/?id=701320044
http://steamcommunity.com/sharedfiles/filedetails/?id=701340536
In the picture you will see I am attacking a single lone province and I have two provinces that are joined to that province. As a result, There are two fronts that can attack it. Each province, essentailly, that is joined to province in question (that you wantt to attack) is considered a front ... or add +20 to each province that is joining in on the fight.
So... imagine this... As Germany, you want to attack France, specifically, you want to take the province of Strasbourg FIRST (and only this one lone city and province). If you ALL your troops in 1 province and they attack... you can only get 80 widths worth of troops. However, Germany has, initially, two difference provinces that are adjacent to Strasbourg. If you have troops on both province attack at the same time then you have a total combat width of 120 (80+40). If you captured the French province to the north of Strasbourg and attack Strasbourg with the troops from the north and the 2 adjacent provinces you now have a combat width of 160 (80+40+40). It finally made sense... when I click on the battles and studied it.
If you have units that are exactly 20 width then you can send 4 units (and only 4 units) to attack. However, if your units are optimized BADLY... let us say you have 21 width then you can only send 3 units at a time. 21*3 = 63 (the 4th would be 84 - as a result, your 4th unit would sit and wait until your three units fail and then attack solo 1 to 4 odds).
Is width the only factor? NOPE.... however, it will help you optimize your troops. If you are attacking from three different sides and you had each unit at 20 width then all you need to send it a maximum of 8 units (8*20 = 160) (the province is being attacked by three sides so thats 80+40+40 = 160 width).
Of course, there are other factors... land doctrine, are u crossing a river (thats bad), are they on a hill or mountain ... night time or day time.. etc.
As for the two buttons, one is stop, the other is go. No idea why they have the same tip, it's confusing, I agree.
I did a naval invasion of Hokkaido with the Soviets, and then realized they built no ports on that island, all my divisions were suffering from low organization and unit strength, resulting in a halted advance. Do not invade Japan from the north, horrible idea.
Change to supply area map mode (lower right corner, top icon), mouse over the region that your troops are in that has low supplies, which is most likely red or orange considering how low your supplies are to reduce organization to 25%. The game should show you how much supply that region supports, and how much your troops are using up, and the path supplies are taking from your capital to the front lines. Do you see any red numbers that is NOT your capital? That would be the bottleneck, and you should construct infrastructure there ASAP.
Then make sure infrastructure along the supply path is sufficiently high from the capital to the frontlines, and/or move some troops away from the low supply area temporarily to let them recover.
If you don't own the lands you are fighting on, you won't be able to build infrastructure or ports on them, so you'll need to rely on the infratructure and ports already in place, or hope your allies build more, nothing much you can do about that besides moving some troops away from the low supply regions.
If you run out of supply and equipment... think of it like this. If you had 10,000 troops in a division - but you only had enough rifles and pistols to give HALF of them pistols and rifles. Your 10,000 troops is really 5,000 in attack strength. Then your supply problem is so bad that you can not give them any support equipment or 1 out of 100 equipment... you are fighting at less than 50%... so yeah... dont make all your units too large and too uber. You want keep a large force of them as no frills units (basic infantry with rifles and nothing special) then a few additional good units with a few frills... then a few super uber units that get the best equipment and supplies that the enemy will think are the boogey men when they are encountered.
Germany's SS division was one of those. Well equiped units that always got the best equipment first... and .... well we remember them for some reason.
Overseal battles are weird. Supplies are shipped via the port. The larger the port the more supplies than can enter it. Likewise, once you are on the ground the larger the infra structure (short cut T and infrastructure cost is 3k) the more supplies and troops it can support.
if its not your land and you are lend leasing... you do with what you have... since you arfe not in control. Plus holes, push, and look for enemy weakess and hope you make a difference.