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 Theorem
and
Finished Frogs Formula
Proof: Show 2nd wins on
Terminal Toads Theorem
and
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 Theorem
and
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 moves
away 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
TF
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.
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