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 Post subject: Soving the Deepkey - Easy color schemePosted: Tue Apr 03, 2012 2:24 pm

Joined: Tue Apr 27, 2010 10:38 am
Background on the Deepkey

The first puzzle I was able to produce in virtual form (I did in fact get the laser cut pieces for a physical Deepkey I mentioned, but there is still too much play so binds and pops make it basically unplayable) after this discussion of deeper-than-deepcut puzzles is the Deepkey:

It has 4-fold symmetry like the Rashkey, which is an exactly deep cut puzzle like a 2x2x2, but the Deepkey has the two circles overlapped even further, so center pieces reappear but are no longer fixed. The Deepkey has most of the pieces from the Rashkey, and two new ones:

The green triangles in the above picture are equivalent to the Rashkey's triangle pieces and the yellow squares are equivalent to the Rashkey's as well. The blue point pieces are new to the Deepkey, as are the red centers. The thin diamond pieces from the Rashkey have been eliminated by choosing the cut depth for the Deepkey to reduce them to zero area where two points and two squares come together.

The first color scheme I tried with the Deepkey was a two color sun-moon style:

I believe it should solve similarly to the Rashkey 2, but I have yet to actually do so. The biggest difference is of course the point pieces, so I added an easier color scheme similar to the Rashkey 1. It only uses centers and points, and only a few of each:

Soving the Deepkey - Easy color scheme

Note: I'm using the standard notation of L indicates a clockwise quarter turn of the left circle, R a clockwise quarter turn of the right circle, a move followed by ' indicates a counterclockwise turn and a move followed by ² indicates a 180° turn.

My solution has two main phases, building the lenses and building the crosses. Each phase has two parts, one for each circle. The first phase, building the lenses, can be done intuitively as there is a great deal of freedom due to the many identical parts. The goal of the first phase is to get all the pieces that make up the right cross into the lenses of the left circle and all the pieces that make up the left cross into the lenses of the right circle:

The lenses of the left circle are in red above and the lenses of the right circle are in blue. The pairs light blue and light red points combined with the lenses in between each pair make up the left and right crosses respectively. Starting from a scrambled puzzle:

The first step is to chose one circle or the other, and bring the pieces that will make up that circle's cross into the other circle's lenses as one solid lens and one hollow lens adjacent to each other. Choosing the left cross to start with, after the first part of phase one the puzzle should look something like this:

The second part of phase one is slightly more restricted, as neither of the completed lenses should ever be moved into their cross. This means that and points that need to be rotated 180° will have to be placed in the cross, rotated 90°, moved out of the cross, the circle with completed lenses reset, the point placed back in the cross, and finally rotated another 90°. The lenses of the second circle should also end up adjacent to each other then phase one is complete:

The second phase starts by placing one of the hollow lenses in its cross, and rotating that circle so the hollow lens is horizontal. Then the solid lens of that circle will be inserted into the hollow lens, completing that cross. The sequence I use for this is LR²L'R²L, its inverse, or the right circle version of one of those. The main effect of this sequence is to cycle the lenses of the circle turned first; using LR²L'R²L as an example, the lenses of the left circle will cycle as so:

Note that each lens is reversed when it moves into the cross, and reversed again when it moves out. This is not important in the easy color scheme, as all pieces of a lens are the same color, but is relevant to more complex color schemes. The other effect of this sequence is to swap the left and right points of both crosses, and 5-cycle several other points:

Since the lenses are 4-cycled and the points are 5-cycled, using this sequence 5 times will advance the lenses one position and leave the remaining points unchanged, as long as the left and right points of both crosses are the same color. Thus, to complete the first step of phase two from the last position shown, R'L' would setup the hollow lens and (R'L²RL²R')x5 would insert the solid lens, completing the left cross:

The last step of phase two is to complete the other cross, which requires more careful setup than the first step, to prevent breaking up the first cross. To start the last step the solid lens of the incomplete cross should be at the far left or far right of the puzzle. Then the completed cross must be broken up such that putting it back together also places the hollow lens into its cross and rotates it into a horizontal position. From the last position shown, RL'R will break up the left cross, then LR'LR' will rebuild it and place the hollow lens:

Finally, the last solid lens can be inserted into the hollow lens, in this case by (LR²L'R²L)x5.

Final Thoughts

This (much) easier color scheme for the Deepkey achieved my main goal of getting more familiar with how the new pieces move compared to the Rashkey, and may be the first puzzle I've come up with a solution for from scratch. This is also my first writeup of a solution, so if anyone has any feedback please let me know. Both versions of the Deepkey, as well as quite a few other circle puzzles, can be played at circlepuzzles.staticloud.com. I'm currently working on a rewrite that will allow new puzzles to be added more easily, and add some visual polish such as animated moves. If you give it a try please let me know if you experience any issues, or can think of anything I should add to the next version.

Eddie Lee

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