Build a bowling ramp and use it to knock over plastic bottle pins.
3-meter lengths of 4 inch pipe - One for each patrol/team
A collection of Staves - Enough to build a decent structure for each pipe
Sisel/Rope/String - For the staves
Tennis Balls - The more the merrier!
2 litre plastic drinks bottles (raid those recycle bins!). Be careful about milk bottles, they need a clean to not stink! - 10 per patrol (roughly)
Large Score board
Ideally a huge hall!
Place a bucket of tennis balls in the centre and stand up a set of bottles for bowling for each patrol opposite the room from where they will build the ramp. Where they are to build, have the staves and rope ready.
20 minutes: Each patrol must build a bowling ramp, discussing features such as the gradient of the pipe and whether to raise the bottom off the ground.
If you have lots of space and leaders: (You will need one leader to watch each set of bottles plus another to record the scores)
20 minutes (or more!): Have the scouts collect a ball, run to their pipe and bowl it. You need not take turns or have rounds if you have enough leaders - when most of the bottles have been knocked over count them, stand them up and add to the patrol's score - a simple tally of how many bottles are knocked over. If you think you can enforce it, only allow a limited number of balls being carried. When I ran this it was hard to enforce because there's little justification for it.
If you're short on space:
20-30 Minutes; Organise a bowling competition for each patrol. Draw up the scores table and see which patrol gets the most points after X rounds.
To make it more of a challenge:
Don't use the pipe, have them construct a chute from staves.