Mathematics Problem 117
**************************

What is the largest number of regions  r(n)  that a plane is divided into
by  n  straight lines in the plane?
Give  r(n) as a function of  n  and explain why your answer is correct.

**********************
Advanced Problem 117A 
********************* 

What is the largest number of regions  r(n, d)  that
d-dimensional Euclidean space is divided into by  n  hyperplanes?   
Give r(n, d)  as a function of  n  and  d,  
and find formulas for b(n, d), the number of  regions that are bounded, and  
u(n, d), the number of regions that are unbounded.
Of course,  r(n, d) = b(n, d) + u(n, d).  
Explain why these numbers are correct.


Each midshipman submitting a correct solution to Problem 117 by noon on
Sunday 1 April 2001 will win a cookie, or two cookies for the regular
problem and the advanced problem in the case  d = 3, or three cookies for
the advanced problem in its general form.  Submit solutions to Prof. Wardlaw
at mathprob@usna.edu (please no attachments!) or via his mailbox in
Chauvenet 301.

Correct solutions to Mathematics Problem 116 were submitted by Midshipmen
Aron T. Foster, Aaron Geary, and Andrew J. Privette.  Midn. Geary's solution
is posted on the board.  Professor Mark Kidwell submitted a correct solution
to Advanced Problem 116A.