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With masses of open land, modern technology and a few colonial claims, Texas has the potential to grow, or would do if it didn't have a large powerful neighbor to the north and a greedy neighbor to the south each with different ideas of where Texas should be heading.
Her small modern Army is capable of fighting a limited war against either, but if war continues too long then defeat is certain.
Texan resources are somewhat lacking, but the budget can be managed without undue strain.
Starting the scenario at war with Mexico, the biggest challenge you will face is just to stay complete, if Texas can manage to pull the USA into an alliance then perhaps she can put a break on Mexican expansionism, but their may be a quid pro quo.
Played well budding Texans ought to finish in or around the top 15.
Becoming Independent (Tribulation's Strategy)
The key to becoming independent begins with holding your position across the Rio Grande. Send all of your units except for the smaller cavalry unit to San Antonio, and merge them all into one army under the command of General Houston. This should hold the Mexicans for quite some time.
Then, send the smaller cavalry unit and capture Denver. Sell Denver to the United States for their two Canadian colonies. Then, sell these Canadian colonies to Britain for claims in Washington and Idaho. Take the remaining Mexican colony in Colorado, then send the smaller cavalry unit to merge with your remaining soldiers who are holding the Rio Grande under Houston.
Wait for the next diplomatic action, then trade the Washington and Oregon colonies to the United States for the Colorado and Oklahoma colonies. Claim them and (if you want) grant them statehood. This will give you roughly 80 prestige, but it does not give you peace with the Mexicans. You must now wait for your next diplomatic action. Keep reinforcing your infantry division with the artillery brigade after every battle, and eventually your line will be broken by superior Mexican numbers (unless you want to sacrifice laborers and turn them into soldiers, but the point is moot.)
As soon as you get your new diplomatic action, request a defensive alliance with America. They should accept, as you have over 180 relations bonus. As soon as this happens, begin to sue for peace with the Mexicans on terms of no tribute. They may accept immediately, but they may decline. Continue to spam them with no tribute peace terms until they accept (they should eventually) and you can now begin to expand your Texan Republic without fear of American expansion into your southern cores, as well as a size triple of what you started with and a huge immigration bonus.