Symbolism, Imagery, Allegory
Through the Looking-Glass is structured like a chess game: the pieces become characters, Alice herself a pawn, and all her adventures are simply complicated dramatizations of different moves in the game. In fact, most editions of the book include a chart of the chessboard and the moves, which Lewis Carroll himself put together. As he explains in his note, the red and white pieces don't take proper turns, but otherwise the moves in the game are directly related to the moves the characters make in the book.
This explains, for example, why Alice has to cross six brooks: as a pawn, she starts in the second square and has to cross into the next six before getting "promoted" to queen. She moves through the third square rapidly on a train journey because pawns can move two squares on their first move, essentially skipping or flying through a square. The chess metaphor also explains why the Queens move so quickly and erratically, since Queens can move in any direction and in any number of squares in chess, and why the White Knight has trouble riding in a straight line, since chess Knights move in an L-shape.
If traveling on a bizarre and uncertain journey in order to become a queen instead of a pawn sounds a bit like a metaphor for growing up, well, who are we to say no? And that might also explain why the White Knight is so sad to see Alice cross over into the eighth square, where he can't follow. It's more than a little reminiscent of Lewis Carroll's own reluctance to let go of the real-life Alice Liddell when she grows up. Beyond that more specific, biographical connection, the book implies that life is just a game, full of arbitrary rules and not nearly as meaningful as we like to pretend.
Alice's Adventures in Wonderland also makes references to a game underlying the action – in this case, cards. The Queen, King, and courtiers that Alice meets are all in the suit of hearts, while, fittingly, the spades end up as gardeners and the clubs as soldiers. However, unlike Through the Looking-Glass, the first Alice book isn't structured by the moves of a game. There isn't any particular card game being "played," metaphorically or literally, behind the scenes of Alice's experience in Wonderland. Instead, the personified cards mingle with talking animals as part of the childish fantasy context. Playing cards, pet mice, wild rabbits, and housecats are all elements of Alice Liddell's everyday life that Lewis Carroll draws on to populate the fantasy world.
Games crop up in other areas of Wonderland, too – for example, in the strange game of croquet that everyone plays in Chapter 8. In this case, instead of life resembling a game, a game comes to life. All the pieces, from the croquet hoops (soldiers) to the mallets (flamingos) to the balls (hedgehogs) are living creatures. This seems rather cruel to us today, but Carroll means for it to be funny. As the croquet game proceeds, the complexity of the situation is delightfully silly. And, of course, the croquet game is rigged, since the Queen of Hearts arrests and sentences to death anyone who might beat her. Talk about being a sore loser!