DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

show that the truth of S(N + 1) follows. Because you know S(N − 2)
is true, postage amounting to N − 2 cents can be made and adding one
3-cent stamp to this allows N + 1 cents of postage to be made.
Step 4: Inductive Conclusion.
By the Principle of Mathematical Induction, using only 3- and 5-cent
stamps you can make n cents in postage for every n ≥ 8 .
•51. What postage do you think can be made using only 5-cent and 6-cent
stamps? Do you think that there is a number N such that for any
n ≥ N, you can make n cents worth of postage? Either explain why
such an integer N does not exist or give an inductive process inherent
in the problem from which a proof by induction could be constructed.
(A complete proof by induction is not required for this problem.)
•52. Explain why it is true that for any integer N there always exists n > N
for which it is impossible to make n-cents worth of postage using only
2-cent and 4-cent stamps. Generalize this observation.
→53. Suppose a and b are two positive integers for which there exists an
integer N such that for any n ≥ N, n-cents of postage can always be
made using only a-cent and b-cent stamps. Using the information you
have collected in the preceding problems, conjecture a value of N which
has this property. (N will depend on a and b). Test your conjecture
on some more examples.
54. Divide your group into pairs to play three or four games of Sylver
Coinage (a.k.a Postal Kiosk), 1 in which players take turns naming positive
integers n for which n-cents of postage cannot be made using only
numbers which have already been named. The player who chooses 1
loses. For instance, 3, 4, 5, 2, 1 is a possible game, but 3, 5, 7, 6, 4, 1 is
not (why?). Without referring to human mortality, explain why any
game of Sylver Coinage ends after finitely many steps.
55. Suppose that f is a function which is defined by the inductive process
f(1) = 1 and f(n) = n + f(n − 1). For practice, find f(2), f(3),
and f(4), and then prove that f(n) = n(n + 1)/2. Notice that this
gives another proof of (2.1), because the sum there satisfies the two
conditions defining the function f.
•56. What properties in the definition of function allow you to say that
every function from [m + 1] to [10] is built up from a function from
[m] to [10] in a unique way? Let Sn be the set of all functions from [n]
1 The authors of the book Winning Ways for Your Mathematical Plays attribute this
the game of Sylver Coinage to a 1884 article by the mathematician Joseph J. Sylvester.
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