19. Here is a trick to “prove” that you can calculate faster than a person with a calculator. Turn your back and ask a friend to write down any two positive integers, then add them to get a third, then add the second and third to get a fourth, and so on, adding each time the last two integers until there are 10 numbers. Have your friend show you the list, whereupon you write down right away the total of all 10, while your friend begins to add them up on the calculator (to prove that you’re right). The secret: The total is always 11 times the seventh number, and multiplying by 11 is fairly easy to do in your head—just add each pair of neighboring digits, carrying if necessary. Suppose that your friend writes down and as the first two numbers. Show that indeed the total of all 10 numbers is 11 times the seventh number. (Adapted from Martin Gardner, Mathematical Circus, Knopf, New York, 1979.)