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We can do the same as the top of the page, but add a new symbol, X, which we place in the top right corner of the puzzle in some position, e.g. at (1,1,1) of our 7×7×7 cube. 
Now the solved puzzle has two different configurations in which X can go to:
The top right corner of the puzzle is in one of the corners of a smaller cube
or
The top right corner is in the middle of an edge of a smaller cube.
The latter option is not necessarily clear, so here is a more convincing example. 
Here is a more detailed step by step description of the solution. The example is intentionally chosen such that it would fit into a small window.
First we place the X in the upper left corner. 
Next we look for a piece with edges on axes 7,8,9,10, i.e. in two of the corners of a smaller cube, but as close as possible to the top right corner of the puzzle. 
By rotating the smallest cube we can do this. 
Finally, we have the configuration shown on the right.
More examples and further explanations
Diagram of Rubik's Cube
The way of thinking in terms of this notation is to use the example of Rubik's Cube, where one face has the symbol X.
In this case the symbol is placed in a corner of the cube and in the solved state it is placed in a corner of a smaller cube.
In general this notation can be used for any puzzle where the symbol goes to a corner of a smaller cube. 
The symbol can be used in any position of the puzzle, as long as it will be in a corner of a smaller cube when solved.
However, it is sometimes useful to have the symbol at the middle of an edge of a smaller cube. Then the symbol is at the middle of a "corner" of a smaller cube. 
It is the same as drawing a "corner of a corner" instead of X.
Another notation
If we want to be formal about it, we can call X the middle of the edge of the smaller cube. 
Then the picture becomes this. 
A:
I think this explanation sounds convincing to me, but I can't find the official explanation.
Whew, who did the counting, I just looked and saw 3125841983
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