Since the "a" row is missing "c" and "f" and since "af" cannot equal "f" (or "a" would be equal to "e", when we know them to be distinct), we can conclude that "af" = "c". Left multiplication by "a" then yields "f" = "ac", which we may multiply on the right by "c" to give us "fc" = "a". Multiplying this on the left by "d" gives us "c" = "da", which we can multiply on the right by "a" to obtain "ca" = "d". Similarly, multiplying "af" = "c" on the right by "d" gives us "a" = "cd". Updating the table, we have the following, with the most recent changes in blue: