There's also problem #1 in Polya & Szego, Problems and Theorems in Analysis: 1. In how many ways can you change one dollar? [The solution is given using generating functions---actually more in problem 2 than problem 1 I guess, now that I look at it] This sent me in search of other Problem #1's. Here's a random selection Knuth Vol I: 1.1.1 [10] Show how the values (a,b,c,d) of four variables can be rearranged to (b,c,d,a) by a sequence of replacements. In other words, the new value of a is to be the original value of b, etc. Try to use the minimum number of replacements. Coxeter, Introduction to Geometry: 1. Using rectangular Cartesian coordinates, show that the reflection in the y-axis (x=0) reverses the sign of x. What happens when we reflect in the line x=y? Marshall Hall Jr, Theory of Groups: Show that from the associative law (ab)c = a(bc) it follows that all methods of bracketing a_1...a_n without altering the order of factors, yield the same product. William Feller, An Introduction to Probability Theory and its Applications: 1. Among the digits 1,2,3,4,5 first one is chosen, and then a second selection is made among the remaining four digits. Assume that all twenty possible results have the same probability. Find the probability that an odd digit will be selected (a) the first time, (b) the second time, (c) both times. Donald J. Newman, A Problem Seminar: Derive the operations +, -, x, and / from - and reciprocal. Gerhard Ringel, Map Color Theorem: Does there exist a graph with one vertex of valence five and all the others of valence four? Courant and Robbins, What is Mathematics? Consider the question of representing integers with the base a. In order to name the integers in this system we need words for the digits 0,1,...a-1 and for the various powers of a: a, a^2, a^3,... How many different number words are needed to name all the numbers from zero to one thousand, for a=2,3,4,5,...15? Which base requires the fewest? Richard K Guy, Unsolved Problems in Number Theory: Are there infinitely many primes of the form a^2+1? (I was doing pretty well on these problems until that one) Clifford and Preston, Algebraic Theory of Semigroups 1(a) If e is an indempotent element of a left cancellative semigroup S, then e is a left identity element of S. (This way of stating exercises, without the imperative, has always seemed to me to have a certain majesty to it. It sure beats the "Shew that..."s) It would be interesting to know the first sentence of the Bible that might be reasonably construed as a mathematics problem. Genesis I,i is certainly a candidate; just put that "Shew that..." in front... Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/ehome.htm ----- Original Message ----- From: "R. William Gosper" <rwg@tc.spnet.com> To: <math-fun@mailman.xmission.com> Sent: Thursday, September 25, 2003 10:14 PM Subject: Re: [math-fun] math fun from politics
HGB wrote
I think that there is an elegant way to do this with generating functions, but I can't recall it right now.
Let the coeff of q^k represent the ways of spending k cents. Let the coeff of p^k represent the ways of making k purchases. " " " " b^k " purchases including k blacks. " " " " h^k " " " k hispanics. " " " " w^k " " " k women. " " " " m^k " " " k white males.
Then the generating function is (c152) 1/(1-m*q^100*p)/(1-p*w*q^75)/(1-p*h*q^50)/(1-p*b*q^25)
1 (d152) -------------------------------------------------------- 25 50 100 75 (1 - b p q ) (1 - h p q ) (1 - m p q ) (1 - p q w)
Through 150 cents and 3 purchases, (c153) taylor(%,q,0,150,p,0,3);
25 2 2 50 (d153)/T/ 1 + (b p + . . .) q + (b p + h p + . . .) q
3 3 2 75 + (b p + b h p + w p + . . .) q
2 3 2 2 100 + (b h p + (b w + h ) p + m p + . . .) q
2 2 3 2 125 + ((b w + b h ) p + (h w + b m) p + . . .) q
2 3 3 2 2 150 + ((b h w + b m + h ) p + (w + h m) p + . . .) q + . . .
(c154) coeff(coeff(%,q,150),p,3)
2 3 (d154)/R/ b h w + b m + h Confirming Henry's
Since the report said exactly 3 cookies were sold, and the prices were 1, .75, .5, and .25, I get only 3 possibilities:
1+.25+.25, .75+.5+.25 and .5+.5+.5 Privately, I received [John's questions inspired me to format them in a modern academic format. I daren't try Rich's moderation by replying with this to the list, but thought you might enjoy it...] The miscreant's transgressions and identity are (temporarily) available at www.ippi.com/rwg/smu.text --rwg
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun