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+---
+title: "Exploring Toggle Games on Graphs"
+date: 2023-12-17T23:27:37-05:00
+summary: "Study of *Toggle*, an impartial game"
+abstract: >
+    In the commercial one-player game Lights Out™ a grid of lights is randomly generated with some lights on and some lights off.
+    The player can press a light to flip its on/off state as well as the state of its neighbors.
+    Toggle seeks to transform Lights Out™ into a variety of impartial two-player games.
+    Two players take turns toggling the on/off state of lights in an attempt to leave the other player with no available legal moves.
+    We analyze Toggle on various finite simple graphs and use impartial game theory to determine which player has a winning strategy given an initial Toggle configuration.
+    Finally, we prove that determining the winning player given an arbitrary Toggle configuration is PSpace-complete.
+categories:
+- "Moravian REU 2023"
+tags:
+- "Combinatorics"
+- "Game Theory"
+- "Computational Complexity"
+---
+
+Peer Researchers
+: [Nathan Hurtig](https://nathanhurtig.com/)
+: *Djenaba Djeob*
+
+Mentors
+: Dr. Eugene Fiorini
+: Dr. Andrew Woldar
+: Dr. Patrick Cesarz
+
+{{< presentations >}}
+## Presentations
+
+* Poster
+    * [SACNAS NDiSTEM 2023](/research/posters/toggle_ndistem23.pdf)
+{{< /presentations >}}
+
+
+{{< publications >}}
+## Publications
+
+* [A363934](https://oeis.org/A363934) Table read by ascending antidiagonals. T(n,k) is the Sprague-Grundy value for the Heat Toggle game played on an n X k grid where each vertex has initial weight 1. 
+* [A364489](https://oeis.org/A364489) Values of n for which the Sprague-Grundy value of Heat-Charge Toggle on an (n+2)-vertex path with initial weights -1,1^n,-1 is evil for odd n or odious for even n.
+* [A364503](https://oeis.org/A364503) Sprague-Grundy values for Heat-Charge Toggle on paths from A364489 where paths with an even number of vertices are odious, or paths with an odd number of vertices are evil.
+{{< /publications >}}