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Here you see my Rubik's $10\times10\times10$

I don't know how to solve this using an algorithm, also the same thing happens to my $6\times6\times6$.

What algorithm should I use?

(The whole cube is solved except 2 edges)

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You have both OLL and PLL parity that can occur on all even layered cubes (4x4, 6x6, 8x8, etc.)

Use the following algorithm to fix the OLL parity:

enter image description here

r2 B2 U2 l U2 r' U2 r U2 F2 r F2 l' B2 r2

And the following algorithm to fix the PLL parity:

enter image description here

L2 D (Ff)2 (Ll)2 F2 l2 F2 (Ll)2 (Ff)2 D' L2
performed as: L2 D x [(Uu)']2 [(Ll)']2 (U')2 ([(Ll)']2 L2) (U')2 [(Ll)']2 [(Uu)']2 x' D' L2

Source

PS: A small letter means just the inner layer on a 4x4x4. In your case of the 10x10x10, you have the following layers:

enter image description here


As to why this occurs? Parity on twisty puzzles is unrelated to the mathematical term parity. On twisty puzzles the term is used to indicate a problem that shouldn't be possible within the 'law of cubes'.

This 'law' state that it is never possible to have a single swap (like your PLL parity), regardless of the puzzle. Why does parity occur then, if it isn't possible? Because there are sometimes hidden pieces, equivalent pieces or orientless pieces.
In case of the even layered Cubes there are hidden centers (imagine your 4x4 as a 5x5 without centers). Even though you don't see it on the outside, one of the inner edge centers is flipped, causing the parity on the 4x4.

Here is some more information about parity on twisty puzzle in general:

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  • 2
    $\begingroup$ Just a suggestion to execute it faster, you could do whole slices instead of slice move and also solve OLL at the same time, a double parity algorithm might be useful when you know there is too. $\endgroup$ – Ariana Aug 22 '16 at 15:54
  • $\begingroup$ @ArianaGrande +1 It's indeed possible to use a single algorithm for both the OLL and PLL case at the same time. I'm personally no speedcuber though, and since I need to learn both the OLL and PLL parity algorithms separate anyway when I have those cases, I prefer to use two algorithms instead of one slightly bigger algorithm. For 4x4/6x6 speedcubing it's indeed better to use a single algorithm for any OLL/PLL or combined OLL&PLL case. $\endgroup$ – Kevin Cruijssen Aug 22 '16 at 17:04

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