practice problems pending from Q2.

Q1. There is a 5x10 chocolate bar. P1 and P2 are 2 players cutting it through the grid lines.
In a single move you can take either a complete row or a complete column.
Whoever picks up a 1x1 piece first, wins.

S1.
Since one of the dimensions is even there is a guaranteed winner here: Player 1.
If both dimensions were odd there won't be any guaranteed winner.

So player 1 will split it into 2 equal halves of 5x5.
And from there will simply mirror whatever player 2 does on the symmetric half.
But as soon as player 2 reduces 1 dimension of a grid to 1, player 1 picks up that piece and wins. 

Q2.

S2.
1. For the 2x1 domino case Player 2 will win as explained in Q4 here.
2. For the 3x1 Tromino case, again player 2 will win using the same reflection strategy as done for 2x1 domino.

In both 1. and 2. the 2x1, 3x1 pieces cannot overlap their own reflection across the center.
But in 3. it's possible.
Remember that reflection across center formula here is 11-r,11-c

3. 

Q3.
"The nos. 1 through 20 on a board in a row. Two players are putting "+" and "-" in between two no. If the result is even player 1 would win, otherwise player 2. Who would win?"

S3.
Starting parity is even, i.e. sum of all numbers from 1 to 20 is even.
Now if you flip a plus to minus, the sum changes by an even number only.
So the final result will always be even. P1 would win.

Q4.

There are two piles of 7 stones each. At each turn, a player may take as many stones as he chooses, but only from one of the piles. The loser is the player who cannot move. Who can win this game and how?

S4.
Player 2 will always win by mirroring.
And if piles are of different size, the player 1 will first make them of same sizes and then win by mirroring.