Howe Wins Brooklyn Hights After Long Island

Suppose that the fog failed to completely cover George Washington's retreat. He ends up leaving 3,230 men behind all artillery, but took plenty of cannonballs for the canons he left behind (hey, they were leaving in the night with no sleep).

The British descend on them before most of them even can shoot back. One commander raises the white flag while a subordinate in confusion shoots a British colonel. The disorganized troops are stabbed by furious British until the officer gets up (he only got bruised falling off his horse and stabbed in the leg in the melee confusion) and gives the remaining Patriots a chance to surrender.

All survivors are captured. 750 intact, thousands with bayonet wounds, and a large number of Patriots are dead.

But Washington still has 10K (or 7K or 6K depending on your history book) troops left.

How will Howe react? He beat over 3K troops with almost no casualties (a disorganized foe vs an organized one can often end up with lopsided kill ratios, and casualty ratios which count surrenders instead of only kills can get even more lopsided)

Assuming Burgoyne still opens up the Saratoga campaign, can the Patriots beat them?
 
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