How many tiles does it give the discount?
How is the army discount calculated together with other hospital and joint strike?

Hospitals save an increased amount of armies within 4 territories from the hospital in each direction. And a "normal" amount from the 5th territory.
if you unlock your statistics (level 1) with your APs, you can then press on a grey territory (a territory you can see but not yet conquer) and it will show you exactly how much armies will be saved by your hospitals

Don't confuse Hospitals and Field Hospitals cause they both are a thing and I believe Potassium is referring to one of the active artifacts. I haven't replied yet because I have never used them, so I don't know for sure. But I believe that Field Hospitals only affect the immediate neighbors of the territory the Field Hospital is created on. And based on how Hospitals and Joint Strike are calculated, I would assume that you have to use the formula that is to your most disadvantage and saves you the least armies. So, first take the armies that both your Hospitals and your Field Hospitals save you and then calculate how much of the remaining armies are saved by Joint Strike.

Based on the artifact description (Deploy a field hospital that saves XX% from conquering neighboring territories) it only works on neighboring territories.

EDIT (its in XX% cause i don't know the value since they changed lats update)

Edited 4/3/2021 14:23:29

Right, Field Hospital uses a percentage not a fixed number. Then the algorithm PROBABLY is something like: First your Hospitals save you some armies, then Field Hospitals save you some percentage of the remaining, then Joint Strike saves you some of what was left after step 2.

yup, I got it wrong. My bad

It seems the discount is only available to the immediate neighborhood, that is only one tile away from the hospital.

And the army cost to take a territory is (A-B)(1-C%)(1-D%), where A is the original army needed, B is the discount from normal, and C and D are the percentage discount provided by joint strike and field hospital respectively, I guess.

Edited 4/4/2021 04:44:06