1. Matt Baker The Dollar Game on a Graph (with 9 surprises!)
Georgia Institute of Technology
Based on joint work with
Serguei Norine
Princeton University
Gathering for Gardner 9
Atlanta, Georgia, March 25th, 2010

2. The game
The dollar game is a game of solitaire played on a graph.

3. The game The dollar game is a game of solitaire played on a graph.
To play, begin by selecting a graph G (the game board) and an initial configuration of dollars on G.
(This means we assign an integer to each vertex of G.)

4. The rules There are two kinds of legal moves in the game:
A lending move consists of selecting a vertex and
having it give $1 to each of its neighbors.

5. The rules There are two kinds of legal moves in the game:
A lending move consists of selecting a vertex and
having it give $1 to each of its neighbors.
A borrowing move consists of selecting a vertex and
having it borrow $1 from each of its neighbors.

6. Example of a lending move5 2 $ $ $ -1 7 6 1 -4 1 4 -4 4 -5

7. The goal The goal of the dollar game is to get all the
vertices of G out of debt by a sequence of
legal moves.

8. The goal The goal of the dollar game is to get all the
vertices of G out of debt by a sequence of
legal moves.

9. Some observations Since the total amount N of dollars in the game
never changes, if N < 0 then the game is not
winnable.

10. Some observations Since the total amount N of dollars in the game
never changes, if N < 0 then the game is not
winnable.
If N ≥ 0 then the game may or may not be 1 -1 2 -1 -1
Not winnable
Winnable

11. 9 surprises

12. Surprise #1 If the game is winnable, then it can always be won
using only borrowing moves.

13. Surprise #1 If the game is winnable, then it can always be won
using only borrowing moves.
More precisely, any winnable game can be won
using the following borrowing binge strategy:
any time there are vertices in debt, pick one of them
and do a borrowing move. Repeat until everyone is
out of debt!

14. Surprise #2 The total number of moves required to win the
game when playing the borrowing binge strategy
is independent of which borrowing moves you do
in which order!

15. Surprise #2 The total number of moves required to win the
game when playing the borrowing binge strategy
is independent of which borrowing moves you do
in which order!
Note, however, that it is usually possible to win in
fewer moves by employing lending moves in
combination with borrowing moves.

16. Surprise #3 Definition: The Euler number of a graph is
g = # edges - # vertices + 1.
Example:
g = = 6

17. Surprise #3 Let N be the total number of dollars present in the
game (at any time). If N ≥ g, then the game is
always winnable.
(This result is best possible: one cannot replace g by
any smaller integer.)

18. Surprise #3 If N ≥ g, then the game is always winnable.
(Here N=1 and g=1, so the game must be winnable.)

19. Surprise #4 Theorem: The number of inequivalent configurations of any
given degree is equal to the number of spanning trees in the
graph. 1 1 1

20. Surprise #5 Theorem: There are distinct but equivalent debt-free
configurations of degree k on G if and only if G can be
disconnected by removing at most k edges. 2 1 1

21. Surprise #5 Theorem: There are distinct but equivalent debt-free
configurations of degree k on G if and only if G can be
disconnected by removing at most k edges. 2 1 1
Thus the dollar game encodes the edge connectivity
of G.

22. Surprise #7
There is no surprise #6.

23. Surprise #8 Consider the following two-player variant of the
dollar game:
Riemann takes away some number m of dollars
from the initial configuration D, trying to prevent his
opponent Roch from winning the resulting game of
solitaire. 1

24. Surprise #8 Consider the following two-player variant of the
dollar game:
Riemann takes away some number m of dollars
from the initial configuration D, trying to prevent his
opponent Roch from winning the resulting game of
solitaire.
Define h0(D) to be the smallest integer m for which
Riemann can prevent Roch from winning.
(So h0(D) ≥ 0 for all D, and h0(D) ≥ 1 if and only if
D is winnable.) 1

25. Surprise #8 The canonical configuration K on a graph G assigns
to each vertex v of the graph
(# of neighbors of v) - 2
dollars. 1 1 1 -1 1 1 2 1 1 1 1

26. Surprise #8 For any configuration D on any graph G,
Riemann-Roch Theorem for Graphs:
For any configuration D on any graph G,
h0(D) - h0(K-D) = degree(D) g

27. Surprise #8 For any configuration D on any graph G,
Riemann-Roch Theorem for Graphs:
For any configuration D on any graph G,
h0(D) - h0(K-D) = degree(D) g
(This is an analog of the famous Riemann-Roch
Theorem in algebraic geometry.)

28. Surprise #8 For any configuration D on any graph G,
Riemann-Roch Theorem for Graphs:
For any configuration D on any graph G,
h0(D) - h0(K-D) = degree(D) g
(This is an analog of the famous Riemann-Roch
Theorem in algebraic geometry.)
In particular, we recover the fact that if
degree(D) ≥ g then the game is winnable
(i.e. h0(D) ≥ 1).

29. Surprise #8 For any configuration D on any graph G,
Riemann-Roch Theorem for Graphs:
For any configuration D on any graph G,
h0(D) - h0(K-D) = degree(D) g
It also follows that if degree(D) = g-1,
then D is winnable if and only if K-D
is winnable. (Duality principle)

30. Surprise #9 Brill-Noether Conjecture for Graphs:
Every graph has a configuration D of degree at most
[(g+3)/2] such that h0(D) ≥ 2.
(It’s known that [(g+3)/2] cannot be replaced by
anything smaller.)

31. Thank you! Reference: M. Baker and S. Norine, “Riemann-Roch
and Abel-Jacobi theory on a finite graph”, Advances
In Mathematics 215 (2007),
Adam Tart’s Java simulation of the dollar game: