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At zone 25 you start getting an upgrade called Gymystic. You will get one:
- every 5 levels starting with zone 25 ending with zone 55
- every 25 levels starting from zone 75 ending with zone 150
The first upgrade makes each gym you have gives a compounding 5% more block (retroactive) to every gym. Subsequent upgrades gives an additional 1% more block. When fully upgraded (11 in total), each gym you have gives a compounding 15% more block to every gym.
What this means is that increasing the number of Gyms not only adds a certain amount of Block, it also increases the Block added by all the Gyms.
Comparing with Shield, buying which only adds Block, leads to the conclusion that buying Gyms is a much more cost-effective method in the late game. This is why most players avoid purchasing the Shieldblock upgrade after their first few runs.
The formula for getting the amount of Block per gym is y = 4 * b * (( 1.04 + 0.01 * n) ^ ( k )), where b is 1.5 if you have Blockmaster and 1 if you don't, n is Gymystic level, k is number of Gyms and y is Block per Gym. ( this formula only works after you have researched Gymystic ). Multiply this by the number of gyms to get the total amount of block provided by gyms. Then you will have to include the effect of trainers.
Each Gymystic upgrade costs 1B wood and 5M science, making it fairly difficult to purchase the first upgrade (at least until you buy a lot of perks), but trivially easy to afford every other one.