Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 183 — #191
6.4. The Stable Marriage Problem 183
right-hand and the bottom into your left. Then you shuffle the two halves together
so that the cards are perfectly interlaced; that is, the shuffled deck consists of one
card from the left, one from the right, one from the left, one from the right, etc. The
top card in the shuffled deck comes from the right-hand in an outshuffle and from
the left-hand in an inshuffle.
(a) Model this problem as a state machine.
(b) Use the Invariant Principle to prove that you cannot make the entire first half
of the deck black through a sequence of inshuffles and outshuffles.
Note: Discovering a suitable invariant can be difficult! This is the part of a
correctness proof that generally requires some insight, and there is no simple recipe
for finding invariants. A standard initial approach is to identify a bunch of reachable
states and then look for a pattern—some feature that they all share.
Problem 6.6.
Prove that the fast exponentiation state machine of Section 6.3.1 will halt after
dlog 2 ne C 1 (6.3)
transitions starting from any state where the value of z is n 2 Z C .
Hint: Strong induction.
Problem 6.7.
Nim is a two-person game that starts with some piles of stones. A player’s move
consists of removing one or more stones from a single pile. Players alternate moves,
and the loser is the one who is left with no stones to remove.
It turns out there is a winning strategy for one of the players that is easy to carry
out but is not so obvious.
To explain the winning strategy, we need to think of a number in two ways: as
a nonnegative integer and as the bit string equal to the binary representation of the
number—possibly with leading zeroes.
For example, the XOR of numbers r; s; ::: is defined in terms of their binary representations:
combine the corresponding bits of the binary representations of r; s; :::
using XOR, and then interpret the resulting bit-string as a number. For example,
2 XOR 7 XOR 9 D 12