SOAR Homework One
These homework problems are meant to expand your understanding of what
goes on during class. Any you turn in will be graded and returned to
you. Answers may or may not be posted on the web, depending on demand.
Problems


In tictactoe, the best first move for X is a
corner. Is there a best first move for X in
Ctictactoe?

In tictactoe, the best first movie for O is
the center (in response to the best first move for
O in (a)). Is there a best first response for
O in Ctictactoe?

Tictactoe usually ends in a draw. Is it possible for
Ctictactoe to end in a draw?

Repeat the previous question for Ttictactoe.
 Is there a best first move for X in
Ttictactoe?

Is there a best first response for
O in Ttictactoe?

Is it possible for Ttictactoe to end in a draw?

Recall from class that an nsided regular polygon has all n
interior angles equal to θ_{n} = 180° − [(360°)/n].
 Show that a vertex of type (n_{1},n_{2},…,n_{k}) must
satisfy the equation
(1/2  1/n_{1})
+ (1/2  1/n_{2})
+ …+
(1/2  1/n_{k})
= 1.
or, equivalently,
(n_{1}  2)/n_{1}
+ (n_{2}  2)/n_{2}
+ …
+ (n_{k}  2)/n_{k}
= 2.

There is only one vertex type where two of the ns are at least 12.
What is this vertex type?

Show that at most one of the ns in a vertex can be greater than
12. (You're showing that your answer to the previous part is
the only possible answer.)

What is the largest possible value of k for a vertex type
(n_{1},n_{2},…,n_{k})? That is, what is the largest number of
regular polygons that can meet at a vertex?

Write a short computer program to determine all possible vertex
types.
These problems are also available as a pdf file.
Solutions
Please email Peter
if you are interested in answers or solutions for the web. Thanks.
SOAR Winter 2003 Course Homepage