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The Angel Problem

The Angel Problem. John H. Conway Presented by Aaron Nickel and Carson English. How to Play. The angel and the devil alternate turns on an infinite chess board The angel move to any square in (for example) 1000 kings moves of its position The devil moves by eating away any square

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The Angel Problem

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  1. The Angel Problem John H. Conway Presented by Aaron Nickel and Carson English

  2. How to Play • The angel and the devil alternate turns on an infinite chess board • The angel move to any square in (for example) 1000 kings moves of its position • The devil moves by eating away any square • The angel wins if he can move forever • The devil wins if he strands the angel

  3. Can the angel defeat the devil? • Berlekamp has proven a chess king can be defeated on a 32 x 33 board • That’s about all we know. • Conway is offering cash to solve the problem

  4. Flawed Potential Strategies • Potential Functions • Overly Sensitive • Counter Strategy • Depends on Angel’s

  5. Only Fools Rush In A fool is an angel that is required to increase his y-coordinate. Theorem: A devil can catch a fool

  6. As the angel moves, he is confined to smaller and smaller areas.

  7. Lax Fools • Non-decreasing in one direction • Convert to Plain Fool • Of much higher power • 1,000 becomes 8,000,000,000 • Can still be trapped

  8. A Lax Fool traveling north

  9. Relaxed Fools • Fool has a Limited Decrease • Convert to Plain Fool • Of even higher power • Can trap any size laxity

  10. Funneling the Relaxed Fool

  11. Out-and-Out Fools • Distance from start strictly increases • Divide plane into sectors • Divide moves among sectors • Relaxed Out-and-Out Fool

  12. The Kaleidoscope Strategy

  13. Diversion Theorem: For each point P and distance D, no matter how the angel moves, there will be two times at the latter of which the angel will be at least D units nearer to P than he was at the former.

  14. His Own Worst Enemy • Angel eats millions • Burn every space he could have moved to • Is no worse than before for the angel • Angel returns to area finite times • Better for angel to not return

  15. Before you sell your soul… While the diverting strategy goads the angel into where he might not want to go, in arbitrarily large journeys, the diversion will appear inconsequential.

  16. Conclusion • Still unproved either way • Prizes available • $100 for Angel • $1000 for Devil

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