[math-fun] math fun from politics
Texas University Shuts Down Bake Sale By Associated Press September 24, 2003, 11:17 PM EDT DALLAS -- Southern Methodist University shut down a bake sale Wednesday in which cookies were offered for sale at different prices, depending on the buyer's race or gender. The sale was organized by the Young Conservatives of Texas, who said it was intended as a protest of affirmative action. A sign said white males had to pay $1 for a cookie. The price was 75 cents for white women, 50 cents for Hispanics and 25 cents for blacks. . . . The group sold three cookies during its protest, raising $1.50. What are the possible distributions of buyers? Will this increase interest in politics among math students and increase interest in math among politically minded students.
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 You obviously can't have more than one of either 1 or .75. Once you get 1, you can't have either .75 or .5. Once you get .75, you have to get another .75 in 2 pieces, and the only way to do that is with .5+.25. If you start with .5, you have to make up 1 with only 2 pieces left, but you've already eliminated the case with .75, so you have only one way with cookies <= .5. I think that there is an elegant way to do this with generating functions, but I can't recall it right now. At 09:45 AM 9/25/03 -0700, John McCarthy wrote:
Texas University Shuts Down Bake Sale
By Associated Press
September 24, 2003, 11:17 PM EDT
DALLAS -- Southern Methodist University shut down a bake sale Wednesday in which cookies were offered for sale at different prices, depending on the buyer's race or gender.
The sale was organized by the Young Conservatives of Texas, who said it was intended as a protest of affirmative action.
A sign said white males had to pay $1 for a cookie. The price was 75 cents for white women, 50 cents for Hispanics and 25 cents for blacks. . . . The group sold three cookies during its protest, raising $1.50.
What are the possible distributions of buyers?
Will this increase interest in politics among math students and increase interest in math among politically minded students.
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
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
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The true part of my last letter yesterday was
[It's been a long day!]
The feeling that I'd expressed too strongly my criticisms of Jud McCranie's proposed attack on the kissing number problem somehow convinced me that they were wrong, and led me to apologise. In fact, they were quite valid, except that I think I mistakenly described the number of cases for each sphere (after the first few) as "a 3-dimensional continuum", when it's really only 2-dimensional. Jud's method is indeed fallacious. In 3 dimensions, it would probably get the correct kissing number of 12, and so maybe leave the fallacy undetected, but almost certainly the fallacy would be made manifest in the 4-dimensional case by the fact that it would "prove" the kissing number to be < 24. John Conway
participants (5)
-
Henry Baker -
John Conway -
John McCarthy -
R. William Gosper -
Thane Plambeck