Lights out variation proofs? I would like some help solving these questions regarding a specific variation of the lights out game where all lights are initially off.
The game can be played here (by double clicking edit then click play for all lights out at start):
http://www.ueda.info.waseda.ac.jp/~n-kato/lightsout/
and these are the questions:

I know that proofs exist for part a) but I do not fully understand them; if someone could explain one in simple terms, it would be great. 
 A: In response to question (a),
The total number of possible configurations is $2^{25}$ (each light ON or OFF)
The total number of sequences of buttons pressed is also $2^{25}$ (each button pressed or not pressed).
Hence, exactly one of the following must be true:
(i) All possible configurations can be achieved by pressing buttons starting from the configuration of all squares OFF.
(ii) There is at least one configuration which can be achieved through two distinct sequences of button presses.
If we prove (ii), then (i) is false.
As an example, consider the configuration
$$\left( \begin{array}[ccccc] \\ 0&1&0&1&0 \\ 0&1&1&1&1 \\ 1&1&0&1&1 \\ 1&1&1&1&1 \\ 1&1&1&1&1 \end{array}\right)$$ This configuration can be achieved beginning from all squares off with either of the following sequences $$ \left( \begin{array}[ccccc] \\ 0&P&0&P&0 \\ P&0&0&0&P \\ P&P&P&P&0 \\ 0&0&P&P&P \\ 0&P&0&P&P \end{array}\right) \,\,\,\,\,,\,\,\,\,\, \left( \begin{array}[ccccc] \\ 0&0&P&0&0 \\ 0&0&P&0&0 \\ 0&0&P&0&P \\ P&0&0&P&0 \\ 0&0&P&0&P \end{array}\right)$$
Hence (ii) is true and (i) must be false.
In response to (b),
the method to achieve the first configuration (with only the middle light OFF) is 
$$ \left( \begin{array}[ccccc] \\ P&0&P&0&P \\ 0&P&0&P&0 \\ P&0&0&0&P \\ 0&P&0&P&0 \\ P&0&P&0&P \end{array}\right) $$
