(a) Hannah gives clues about her six-digit secret number:
a. Clue 1: It is the same number if you read it from right to left.
b. Clue 2: The number is a multiple of 9.
c. Clue 3: Cross off the first and last digits. The only prime factor of the remaining four-digit number is 11.
What is Hannah's six-digit number?
(b) The only way that 10 can be written as the sum of 4 different counting numbers is 1 + 2 + 3 + 4. In how many different ways can 15 be written as the sum of 4 different counting numbers? Enumerate those.
(c) You are given three jugs A, B, and C of capacities 8, 5, and 3 gallons, respectively. A is filled, while B and C are empty. Empty the liquid in A into two equal (or almost equal) quantities. [Hint: Let a, b, and c be the amounts of liquid in A, B, and C, respectively. We have $a + b + c = 8$ at all times. Since at least one of the vessels is always empty or full, at least one of the following equations must always be satisfied: $a = 0$, $a = 8$; $b = 0$, $b = 5$; $c = 0$, $c = 3$]. How many states (situations) in this process with these constraints there are possible states (situations) in this process? If the number is x, represent this problem by means of a directed graph of x nodes and associated edges. Each vertex stands for a state and each edge for a permissible change of states between its two end vertices. Now when you look at this graph, is it possible to go from state (8, 0, 0) to state (4, 4, 0)? If so, enumerate the sequential steps.
