New Toads and Frogs Results

By Jeff Erickson

Presented by Nate Swanson

Overview

• Notation and Game Rules

• Basic Simplification Techniques

• Ways of Calculating Knot Values

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

Notation and Game Rules

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

Basic Simplification Techniques

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

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

Death Leap Principle

Any board that has none of the following positions has value zero:

Terminal Toads Theoremand

Finished Frogs Formula

Proof: Show 2nd wins on

Terminal Toads Theoremand

Finished 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.

Terminal Toads Theoremand

Finished Frogs Formula

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)

Ways to Calculate Knot Values

• Lemma 2

Proof:By case analysis of Lemma 1 and TTT

Lemma 2 Case Analysis

Ways to Calculate Knot Values

• Lemma 3

Proof:By case analysis of Lemma 1 and TTT (every position 3 movesaway is an integer).

Ways to Calculate Knot Values

• Lemma 4

Proof: Show 2nd wins on

Base Case: b=2, Lemma 3

Similar argument for reverse game

Ways to Calculate Knot Values

• Lemma 5

• If neither player can move from the position

Then:

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

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

Ways to Calculate Knot Values

• Lemma 6– If neither player can move from the position

Then

T F

Ways to Calculate Knot Values

• Lemma 7

Proof: It suffices to prove that,

We then induct on c (like before), and symmetrically do thesame for the other side.

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

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