# 5x5x5 Programming Discussion

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I decided to work on a 5x5x5 Brute Force Solver, setting aside the 4x4x4 project for the time being.

I guess the logical place to start is with the count of nodes as a function of depth, as we have done before. This is what I have so far:

 n Unique Position Counts 1 45​ 2 1620​ 3 ?????​ 4 ?????​

Moves that are discounted:

1. Sequences such as U and U' which just negate one another.
2. Transposing moves such as U u then u U, parallel rotations which create identical positions.
3. At ply 3, moves such as U u E (if E turns the same way as U) that can create the same position as D' d' on the 5x5x5.

Item #3 brings up an interesting point. How many legal 3-ply moves of the same parallel face are there that do not duplicate a position that can be created with fewer turns? And is there an easy way to determine this programmatically?

Intuitively, it seems that if there are 3 different colors on 1 face, the move sequence must represent a "legal" set of moves that the program should allow to be made. The question is, can a 2-move sequence be derived for one if a 3-move sequence created it?

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