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 Theorem and Finished Frogs Formula Proof: Show 2 nd wins on

11. Terminal Toads Theorem and Finished Frogs Formula Mirror strategy: –X is responded in (-X) –Last toad in 1 st 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 1 st ; 2 special cases for Right –Similar argument for Fin. Frogs Form.

12. Terminal Toads Theorem and Finished 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 2 nd 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 1 st 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 1 st : 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 TF

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 1 st component and ab + a + c in the 2 nd. Right gets ab + a + c + 1 in the 1 st component, cd – d +b + 1 in the 2 nd, and d – 1 in the 3 rd 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