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This question is inspired by the Chameleon 8x8 tour puzzle.

A knight is a chess piece that moves by jumping to a square $\sqrt 5$ units from its location. (The more conventional way to put it is that it can move two steps horizontally and one step vertically, or two steps vertically and one step horizontally.)

A camel is a "fairy chess piece" not present in the regular rules of chess; it moves by jumping to a square $\sqrt{10}$ units from its location. (The more conventional way to put it is that it can move three steps horizontally and one step vertically, or three steps vertically and one step horizontally.)

In this puzzle, the piece you have is a "camel-eon". The camel-eon starts out as a knight, but it will transform into a camel after making its first move, then transform back into a knight after making its second move, and so on: transforming after every move. For example, here is a diagram of several moves of a camel-eon starting from the bottom left corner of a chessboard. Camel moves are marked in red, knight moves are marked in blue.

enter image description here

Can you find a closed camel-eon tour of the $8 \times 8$ chessboard? That is, can you find a sequence of moves (starting from anywhere you like - it doesn't matter) that visit every square exactly once, and return to the starting point?

To be clear, both pieces move by jumping: they go from the starting square of the jump directly to the ending square, without passing through any of the squares in between.

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  • $\begingroup$ Nice puzzle! We can probably make puzzles for all piece combinations. $\endgroup$
    – Dmitry Kamenetsky
    Dec 3, 2021 at 7:16
  • $\begingroup$ This is the only one with a pun, though :) $\endgroup$
    – Misha Lavrov
    Dec 3, 2021 at 15:12

1 Answer 1

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After much effort, I have managed to tame this beast. Here it is, with the red/blue color scheme:

enter image description here

And here is more colorful version which should make following the path a bit easier:

enter image description here

Of course, there may be billions of closed camel-eon tours, though most of them won't have such symmetry.

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  • $\begingroup$ Well done! I agree that the symmetry is nice. $\endgroup$
    – Misha Lavrov
    Dec 5, 2021 at 3:36
  • $\begingroup$ Nice work! Did you solve it using some kind of DFS taking into account symmetry? $\endgroup$
    – Dmitry Kamenetsky
    Dec 13, 2021 at 13:27
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    $\begingroup$ @DmitryKamenetsky No, this one was done manually in Paint, though I used Geogebra to draw the points and lines for the final image. I did do a bit of DFS coding afterward: millions of symmetric paths and no way to judge them for aesthetic value. $\endgroup$
    – Daniel Mathias
    Dec 14, 2021 at 0:19

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