TITLE: Seeking a combinatorial proof of the identity$1+3+\cdots+(2n-1)=n^2$
QUESTION [9 upvotes]: I would appreciate if somebody could help me with the following problem
Q: Seeking a combinatorial proof
$$1+3+\cdots+(2n-1)=n^2$$

REPLY [5 votes]: $$\matrix{a&b&c&d&e&\dots\cr b&b&c&d&e&\dots\cr c&c&c&d&e&\dots\cr d&d&d&d&e&\dots\cr e&e&e&e&e&\dots\cr\vdots&\vdots&\vdots&\vdots&\vdots&\dots\cr}$$