New Toads and Frogs Results

1. New Toads and Frogs Results By Jeff Erickson Presented by Nate Swanson

2. Overview • Notation and Game Rules • Basic Simplification Techniques • Ways of Calculating Knot Values

3. Notation and Game Rules • One-dimensional board • Left = Toads Right = Frogs • Toads move to the right, Frogs move to the left • A toad may either push to an empty square, or jump a single frog and land on an empty square

4. Notation and Game Rules

5. Basic Simplification Techniques • Dead Pieces: • Any piece in a contiguous sequence starting with 2 toads (or the left edge of the board), and ending with 2 frogs (or the right edge of the board) • Any other piece is alive • We may remove any dead pieces

6. Basic Simplification Techniques

7. Death Leap Principle • Isolated- • None of its neighboring squares is empty • Any position in which the only legal moves are jumps into isolated spaces has value zero

8. Death Leap Principle • Proof – suppose it’s Left’s turn: • If she has no move, she loses • Otherwise, she must jump into an isolated space • Right responds by pushing the jumped frog • This leaves the board in the same situation

9. Death Leap Principle Any board that has none of the following positions has value zero:

10. Terminal Toads TheoremandFinished Frogs Formula Proof: Show 2nd wins on

11. Terminal Toads TheoremandFinished Frogs Formula • Mirror strategy: • X is responded in (-X) • Last toad in 1st compartment is marked with * • Any move in the third component is answered by moving the marked T, and visa versa • Enough to show Left loses going 1st; 2 special cases for Right • Similar argument for Fin. Frogs Form.

12. Terminal Toads TheoremandFinished Frogs Formula

13. Ways to Calculate Knot Values • Knot – when all toads and frogs form a contiguous sequence • Need only to consider positions that start with a single toad and end with a single frog • Lemma 1 (all superscripts positive)

14. Ways to Calculate Knot Values • Lemma 2 Proof: By case analysis of Lemma 1 and TTT

15. Lemma 2 Case Analysis

16. Ways to Calculate Knot Values • Lemma 3 Proof: By case analysis of Lemma 1 and TTT (every position 3 moves away is an integer).

17. Ways to Calculate Knot Values • Lemma 4 Proof: Show 2nd wins on Base Case: b=2, Lemma 3 Similar argument for reverse game

18. Ways to Calculate Knot Values • Lemma 5 • If neither player can move from the position Then:

19. Lemma 5 • Proof: induct on a • Left moving 1st • Left must jump; Right responds by pushing jumped frog • By TTT, this equals (b-1) • By induction, this game equals 0

20. Lemma 5 • Right moving 1st : counting argument • Left’s toads will move at least b times, for a total of ab moves • Right’s frog will move at most a moves, which is if Right never jumps, leaving a(b-1) + a= ab • Therefore, Right will lose

21. Ways to Calculate Knot Values • Lemma 6 • If neither player can move from the position Then T F

22. Ways to Calculate Knot Values • Lemma 7 Proof: It suffices to prove that, We then induct on c (like before), and symmetrically do the same for the other side.

23. Lemma 7 • Both players mark their respective single piece, and makes sure that that piece never jumps (best strategy) • Left gets cd + b + d + 1 in the 1st component and ab + a + c in the 2nd. • Right gets ab + a + c + 1 in the 1st component, cd – d +b + 1 in the 2nd, and d – 1 in the 3rd • Base Case: Lemma 1

24. Conclusion • Lemmas cover each case for knotted games • Each knotted game has an integer value • Each knotted game’s value can be computed directly without evaluating any of the followers