[math-fun] Pizza hut / Conway problems, with alleged solutions (or partially)
For problem B (at least in the list=set version I did) we can indeed prove uniqueness for the solution "19": As mentioned before, 20 is not a solution; now add "n" to every sum=20 and multiply every product by n, to see that 20+n also is not a solution. This rules out every integer >19, except that there are a few cases where n was already used in the proof 20 was not a solution, hence adding n is forbidden since "set," not "multiset," of numbers summing to 20. But, all those cases, as it happens, were already ruled out anyhow by the computation I originally did. QED. Somebody might also want to work out the list=multiset version of Pizzahut problem B... the answer(s) if any, must be either 0,1,2,... or 19. I'd also be interested to know what Dan Asimov concluded about problem C, his post was rather mysterious. My own investigation of C was likely flawed and/or incomplete in some manner; I think I had some valid thoughts, but they probably were more "a good start" than "the full answer."
http://blog.pizzahut.com/flavor-news/national-pi-day-math-problems-solved/ now posts answers to A & B. It says the answer to B is 12, not (as I'd said) 19, because "list" indeed meant "multiset" not "set". Problem C is still listed as "unsolved" and no solution is posted. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
On 3/22/16, Warren D Smith <warren.wds@gmail.com> wrote:
http://blog.pizzahut.com/flavor-news/national-pi-day-math-problems-solved/
now posts answers to A & B. It says the answer to B is 12, not (as I'd said) 19, because "list" indeed meant "multiset" not "set".
Problem C is still listed as "unsolved" and no solution is posted.
"Jeff" objected to Conway's solution "12" of problem B. I had assumed that by "numbers" Conway meant "nonnegative integers." But Jeff assumed they meant "integers" and points out 12=3+3+2+2+2=18-2-2-1-1. Both have product=72. As opposed to Conway's 12=6+2+2+2=4+4+3+1 with product 48. Therefore, Jeff argues 12 is not a valid solution. But by that reasoning you could always add +2-2 to all sums, yielding same sums but product multiplied by -4. Therefore Jeff would prove there is no solution! Which would sort of allow us to deduce that Conway had not intended to allow negative numbers. (Part of the puzzles is figuring out what Conway meant.) My misinterpretation was more elegant since for it there actually is a unique solution... Anyway, funsters might want to work on C. Far as I can tell, I came closer to solving C than anybody else, but what I did evidently still is inadequate.
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Warren D Smith