The aim of the Nine Dots Puzzle is to draw a path connecting 9 dots arranged in a $3 imes 3$ grid using 4 continuous straight lines, never lifting the pen/pencil from the piece of paper. A solution is displayed below (spoiler alert):

In the Wikipedia page linked above, the solution is labeled as “One of many solutions to the puzzle…” However, this page claims it”s the unique solution to the puzzle. In my cursory exploration of the puzzle, all the solutions I have found are either some rotation of the above solution or the same solution with a different starting point.

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For example, the following solution:

is just a $90,^{circ}$ rotation of the first.

Alternatively, if we number the dots like so:

the first solution can be generated using the paths $$9

ightarrow 5

ightarrow 1

ightarrow 2

ightarrow 3

ightarrow 6

ightarrow 8

ightarrow 7

ightarrow 4

ightarrow 1 ag{1}label{a} $$and$$9

ightarrow 5

ightarrow 1

ightarrow 4

ightarrow 7

ightarrow 8

ightarrow 6

ightarrow 3

ightarrow 2

ightarrow 1 ag{2}label{b} $$which are just reflections about $y=-x$ (if the 1-dot is the origin) of each other.

Another path (and its reflection) which generates the first solution is:$$1

ightarrow 2

ightarrow 3

ightarrow 6

ightarrow 8

ightarrow 7

ightarrow 4

ightarrow 1

ightarrow 5

ightarrow 9 ag{3}label{c}$$

My initial thought is that, *if* the first solution is indeed the unique solution, then the Wikipedia article may be counting all the rotations, reflections, and alternate paths as different solutions to the puzzle because I can”t think of another solution.

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**Question:**

Treat solutions which are rotations or reflections of each other or solutions which generate the same “picture” despite their exact path as equivalent solutions. How many solutions are there and how would you prove (other than through brute force) that the puzzle has only $K$ solution(s)?

This question may require a more rigorous definition of “same picture” but I”m unable to think of a better way of describing it.

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