1. Timur thought of a three-digit number and told it to Anton, who then wrote down three numbers that result from replacing one of the digits of the original number with 1 (for example, if the number 215 was thought of, Anton would write down 115, 215, 211). The sum of all the numbers written down by Anton turned out to be 1243. What number could Timur have thought of? Provide all possible options, and prove that there are no others.
Solution: Notice that from the number abc, we get the sum $1 b c + a 1 c + a b 1 = 111 + 2^{\star} a b c$. From the equation $1243 = 111 + 2 * a b c$, it follows that $a b c = 566$.