Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 653 — #661
15.11. References 653
(c) Map the i.p.
to the triangle
fhu—vi ; hv—wig
fhu—vi ; hv—wi ; hu—wig:
Notice that multicolored i.p.’s map to multicolored triangles. Explain why this
mapping is 2-to-1 on these multicolored objects.
(d) Show that at most six multicolored i.p.’s can have the same center. Conclude
that there are at most 36 possible multicolored i.p.’s.
Hint: A vertex incident to r red edges and b blue edges is the center of r b different
multicolored i.p.’s.
(e) If two people are not friends, they are called strangers. If every pair of people
in a group are friends, or if every pair are strangers, the group is called uniform.
Explain why parts (a), (c), and (d) imply that
Every set of six people includes two uniform three-person groups.
Exam Problems
Problem 15.32.
There is a robot that steps between integer positions in 3-dimensional space. Each
step of the robot increments one coordinate and leaves the other two unchanged.
(a) How many paths can the robot follow going from the origin .0; 0; 0/ to .3; 4; 5/?
(b) How many paths can the robot follow going from the origin .i; j; k/ to .m; n; p/?
Problems for Section 15.7
Practice Problems
Problem 15.33.
Indicate how many 5-card hands there are of each of the following kinds.
(a) A Sequence is a hand consisting of five consecutive cards of any suit, such as
5~ 6~ 7 8 9|:
Note that an ace may either be high (as in 10-J-Q-K-A), or low (as in A-2-3-4-5),
but can’t go “around the corner” (that is, Q-K-A-2-3 is not a sequence).
How many different Sequence hands are possible?