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todo.md

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TODO:

  • Next meeting: 7/5 @8PM
  • Make test_games() command-line based (more interface-y too)
  • Get lots of data
  • Fix min/max move shenanigans (done)
  • Come up w/own questions
    • Write up coherent doc w/questions, conjectures, answers
      • Try to prove conjectures
  • Find various patterns ("done")

    • 1, x, x + 1 (proven)

      • Odd: 0, 2, ... x - 1 mod 2x + 1
      • Even: 0, 2, ... x - 2 mod 2x

    • 1, x, x + 2

      • Odd: 0, 2, ... 2x mod 2x + 2
      • Even: 0, 2, ... x - 2, x + 1, x + 3 ... 2x - 1 mod 2x + 2

    • 1, x, x + 3

      • Odd: 0, 2, ... x + 1 mod 2x + 3
      • Even: ???? mod x + 4??

    • 1, x, x + 4

      • Odd: 0, 2, ... 2x + 2 mod 2x + 4
      • Even: 0, 2, ... x - 2, x + 1, x + 3 ... 2x - 1 mod 2x + 4 - (except x = 2)

    • 1, x, x + a?

      • a is Odd:
        • Odd: n ≡ {0, 2, ... x + (a − 2)} (mod 2x + a)
        • Even: ???
      • a is Even:
        • Odd: 0, 2, ... 2x + (a - 2) mod 2x + a
        • Even: 0, 2, ... x - 2, x + 1, x + 3 ... 2x - 1 mod 2x + a (mostly)

    • 1, x, x + 1, x + 2 (empirically)

      • Odd: 0, 2, ... x - 1 mod 2x + 2

      • Even: 0, 2, ... x - 2 mod 2x + 1

        If x is odd, then Player 2 wins the Base 3 game of NIM with n objects iff: n ≡ {0, 2, ... x − 2} (mod 2x + 1) If x is even, then Player 2 wins the Base 3 game of NIM with n objects iff: n ≡ {0, 2, ... x − 2} (mod 2x)

        If x is odd, then Player 2 wins the Base 4 game of NIM with n objects iff: n ≡ {0, 2, ... x − 1} (mod 2x + 2) If x is even, then Player 2 wins the Base 4 game of NIM with n objects iff: n ≡ {0, 2, ... x − 2} (mod 2x + 1)

        If x is odd, then Player 2 wins the Base 5 game of NIM with n objects iff: n ≡ {0, 2, ... x − 1} (mod 2x + 3) If x is even, then Player 2 wins the Base 5 game of NIM with n objects iff: n ≡ {0, 2, ... x − 2} (mod 2x + 2)

        If x is odd, then Player 2 wins the Base a game of NIM with n objects iff: n ≡ {0, 2, ... x − 1} (mod 2x + i - 2) If x is even, then Player 2 wins the Base a game of NIM with n objects iff: n ≡ {0, 2, ... x - 2} (mod 2x + i - 3) (based only on the above empirically verified rules)

    • More TODOS:

      • Potentially try to prove (by induction) Base 4
      • Alt games:
        • n pieces on board: move 1, 2, ... floor(sqrt(n))
        • Nim with cash?