Small Engines (Lawn Mowers, etc.): Key Switch, neg terminal, safety switches

Question
I have a 1973 Wheel Horse Raider with K241 is it possible that a bad key switch would not allow it to start? I have replaced the solenoid, tested starter, safety switches are good., battery is fine but when I turn the key nothing. There is voltage to the solenoid then to the starter and everywhere else but no click or crank. I tried jumping over both the solenoid and starter but nothing. Now heres the weird thing. Hope I can explain this clearly. I have a 3 post solenoid. 2 Big 1 little. 1 big post has cable from positive terminal on battery other  big post cable to starter. small post to safety switch on clutch/brake. If I take the cable that goes from solenoid to starter off on the solenoid side. I connect that to the neg terminal on battery so now the cable goes from there to starter. I turn the key now the solenoid clicks but starter dont move.  But if I take a jumper from pos terminal on battery to outside case of star the starter will spin. I cant get the solenoid and starter to go together. It almost seems as something is backwards because I cant traditionally jump the solenoid or starter. I have checked every wire I have seen all will carry current I am just wondering if a key switch migth be the problem. I even have voltage to the coil. lights are disconnected so no short there. Any input night keep me from going crazy. I have seen every problem with this tractor except this. Thanks for your suggestions.

Answer
Is the battery new?  We had a couple of batteries reverse polarity recently.  

How if the ground connections?  Remove the negative cable from the battery and frame.  Clean all mounting surfaces with a wirebrush.

It's possible a battery cable is bad.  Did you check the resistance or voltage drop across the cables?

Let me know.
Eric