This does it, I am going to create a website giving detailed instruction on doing hand calculations.
Looking over the archives there are a number questions on performing sprinkler calcs.
Any designer worth his salt should be able to calculate a system using a cheap (under $20) scientific calculator, yellow legal pad, a pencil and table of pipe size/ID'S.
I did my first hydraulic calculations in 1976 well before the advent of computers. Trees were simple, loops were not hard to do and grids were not defined so starting out we didn't do those.
But you should be able ot do simple ladder grids by hand.
We also didn't have the 1.2 and 1.4 * sq rt. of design ereas either. What we did was calculate the entire branch line. If you had a branch line with 9 heads one side and 8 on ther other you calculated all 17 sprinklers down through the riser nipple.
I have always been thankful for the eight years I did hand calculations it taught me how to balance a system and how, just by having one head at a higher elevation then another, a system could end up being really screwed up.