4 mouse traps per patrol
a pile of newspapers

One team sets its 4 traps in a line and places its pile of newspaper behind the traps.
The other patrol sets its traps in a line parallel to and about 15 or 20 feet from the first patrol.
Half the scouts of patrol 1 sit at least 5 feet from their traps - their rears must stay on the ground. The other scouts stand behind their traps - they must stay behind the traps.

On "GO", the standing scouts each grab one sheet of newspaper and wad it into a ball. They each throw their ball at the opposing patrol's traps, trying to spring one.
The sitting scouts in the middle try to deflect the thrown balls. They can toss deflected balls back to their own side, but not towards the opposing side.
The standing scouts can not interfer with thrown balls.
The standing scouts can pick up and reuse papers.
If a trap springs for any reason, it is sprung - no resets.
The last patrol with an unsprung trap wins the round.

---

You may have multiple wars going on in different, separated areas with the winners advancing to a championship round. Or, use a round-robin rotation with each patrol getting 1 point for each win.