It ended the Confederate threat around Washington, D.C.
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He was the Union commander at the Battle of Gettysburg.
Grant's decisive victory at the Third Battle of Chattanooga opened to door to the invasion of Georgia and to the deep Union penetration into the core of the South. Last but not least, the vital railroad linking the Confederate western states east from Mississippi and those of the eastern front had been interrupted for the rest of the war.
With the Battle of Vicksburg, the South lost 40,000 solders which would be hard to replace. It opened up the Mississippi river to shipping by the North. That action caused the Northern Rail Roads to lower their rates. It also enabled the North to attack the South from more directions. The Battle of Gettysburg was more of a morale booster than of great military importance. True, it was both a strategic and tactical victory. Still, it did nothing to conquer the South. Yet, it was a fantastic morale booster. It showed that the great Lee was human. That he could be defeated. It showed that a small cavalry unit under Custer could defeat a larger unit under the great Jeb Stuart. With the right general, the South could be defeated.
The pigeonhole principle is merely the following observation - "If we are to place N balls into M boxes where N > M, at least one box will contain at least two balls." A generalized version of the pigeonhole principle says that if we place at least nk + 1 balls into n boxes, then at least one box will contain at least k+1 balls. I say "at least" a lot because these numbers are arbitrary and lower bounds.
He barely did. In fact, his army was ready to revolt against him. Times were hard and tough. Luckily Washington was an effective leader and general. Not the exact historical way of putting it, but it's accurate enough to be correct on a test. Sorry, I haven't read up on Valley Forge in awhile. I can tell you no battle (or at least any of significance) happened here.