Numberlink Puzzles

I first encountered Numberlink puzzles in the (numberless) Flow Free app and appreciated them because they were a little more topological than the typical logic puzzle.


Setup

The premise of these puzzles is simple: An \(n \times n\) has \(2k\) of its squares colored with \(k\) colors, two per color. Your job is to connect matching squares by creating a path of each color through the grid. Paths cannot overlap.


Example

Below is an example of a puzzle and its solution:

Some puzzles insist that every square in the grid must be used, a path cannot border itself, and other conditions. Hopefully the puzzles I've made already have those properties, but you can design tougher puzzles that can be solved under these conditions with "false" solutions that don't satisfy them.

While I'm providing some puzzles as sheets that you can do with a pen or by placing tokens, some of the topology might be brought to the fore by using string or a geoboard with rubber bands. Here are some manipulatives that seem like they might play nicely together:

The idea with this set of manipulatives is that the pegs are the grid squares, the rings are used to mark the colored squares, and the rubber bands are to loop around the pegs to form the paths. (For boards of different sizes, you will want to wall in a smaller board with extra rubber bands.)


Challenges

Here are some puzzles that keep the number of colors/numbers small (3-6) and the boards small (5x5 to 7x7).

Those size constraints set a low ceiling, but there are places to get larger puzzles with more colors. Here are a few:

Flow Free has ads unless you pay to unlock it, but the ads weren't too cumbersome to make it unusable in education settings when I've worked with it in the past.


Notes

The main thing I like about this activity, as noted, are the topological ideas that can be used to find solutions. The main ideas are homotopy and Jordan curves.

  • Homotopy: Without sufficient barriers between them, one curve can be morphed continuously into another with the same endpoints.

    While not equivalent as solutions, since solutions will have to pass through particular squares, you can see that the three curves below can be morphed into each other, so we say they're homotopic.

    On the other hand, below are some examples of pairs of curves that are not homotopic because there are blocks obstructing our ability to morph one into the other:

  • Jordan curves: Simple, closed plane curves divide the plane into two regions, and any continuous curve connecting a point of one region to a point of the other must intersect their boundary curve.

    The main way that we make use of Jordan curves is by viewing our paths as part of a loop that includes the path, some blocks, and the border of the puzzle.

    In the pairs of nonhomotopic curves above that connect the \(3\)'s, one divides the board into two regions, each containing a \(6\), while the other divides the board into a region containing two \(6\)'s and a region containing no \(6\)'s. The first can't be right, because the \(6\)'s cannot be connected. Since these are essentially the only two homotopy types of curves, we know the curve must end up looking roughly like the second option.

    Similarly, of our two curves connecting \(4\)'s, one of them cuts off our ability to connect \(1\)'s and \(5\)'s while the other does not.

    Because of the discreteness of the grid, we can also use "near misses" of one curve passing by another in neighboring squares to create borders that wouldn't exist in a continuous setting.


Extra Challenges

  • Students will likely get a kick out of designing their own puzzles. It is tricky to make puzzles with a unique solution that uses every square, but this is always a great discussion point. Some of the puzzles they design may be impossible, and proving that they are is great. Our homotopy and Jordan curve ideas can greatly simplify the analysis.

  • I haven't tried it myself, but this game can be played on triangular and hexagonal grids. Triangular grid cells have only three sides per cell, lowering the number of ways a path could enter and leave a cell. Hexagonal grid cells have six sides per cell, however, which increases the number of ways to enter and leave. There is a hexagonal version of Flow Free available.


This post was sponsored by the Julia Robinson Mathematics Festival

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