Nice Neighbors

a toplogical webgame by Chris Staecker

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Tutorial Graph set: small graphs (easy) 8-vertex graphs 9-vertex graphs Thanks for playing! research mode! enter graph6 for the graph you want:

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How to play

Play the tutorial (button in the upper-right corner)- it only takes about 30 seconds.

Or watch this video (2:30):

Or read these words:

Click and drag to move the dark blue nodes from one spot to another.

You win when one of the background spots is vacant and all lines are blue.

Lines turn red when they don't follow along the lighter connections in the background. They turn blue again if you re-connect them along the light blue lines.

About this game

This game was designed to model a question in digital topology which was unsolved at the time (2014). Each level represents an abstract "digital image", and beating the level is equivalent to showing that the image is reducible by a homotopy equivalence to a smaller image. When the game launched, the original solutions for each level were stored in order to catalog which images were reducible and which were not. Here are some technical mathematical links:

Homotopy equivalence of finite digital images (at arxiv.org), a technical paper by Jason Haarmann, Meg Murphy, Casey Peters, and Chris Staecker which defines and begins to catalog irreducible digital images. This paper only catalogs images up to 7 points. This was published in Journal of Mathematical Imaging and Vision 53 , Issue 3, p 288-302, 2015.

(at arxiv.org), a technical paper by Jason Haarmann, Meg Murphy, Casey Peters, and Chris Staecker which defines and begins to catalog irreducible digital images. This paper only catalogs images up to 7 points. This was published in , Issue 3, p 288-302, 2015. Results of computer enumerations to check the game's results on 8 and 9 points, as well as some related questions.

OEIS sequence A248571, which this game computed for n =8 and n =9.

=8 and =9. Nice Neighbors: A brief adventure in mathematical gamification, a less technical article I published in: Math Horizons 23 no. 4, p 5-7, 2016.

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