Let n = 2r-1, and consider a binary word of block length n. How many binary words are there that differ from the given word in not more than one place? Given 2k distinct binary words of block length n with the property that any one of the words with one bit changed differs from every other word in at least two bit positions, how big can k be? Prove that no binary single-error-correcting code can have fewer check symbols than the Hamming code.