Small Engines (Lawn Mowers, etc.): Sears lawn tractor starting issues, battery cables, negative cable

Question
I installed a new battery, a new solenoid, and a new starter on my Sears lawn tractor (model 917.273392) with an 18hp briggs & stratten engine.  I get a succession of fast clicks.  I tried a bypass procdure by jumping the starter and the solenoid directly from the battery.  Neither caused the engine to turn.  If I hand crank the engine to where it turns easily and try to start it, the starter turns the engine only to the first time it tries catch.  
What now????

Answer
Is the battery fully charged.  I know you said it is new, but sometimes batteries sit on seller's shelves a while and loose some charge.

If you remove the spark plugs will the starter turn the engine?

TIP:  Make sure the battery connections are clean.  Dis-connect the battery cables and clean the battery terminal posts and the battery cable ends with a wirebrush.  Make sure both ends of the ground (negative) cable are clean and making good connection.  I have fixed many "clicking" starting issues just by cleaning the cables and charging the battery.

Let me know what happens.
Eric