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Re: [Phys-L] placement / evaluation tests ... or not



On 11/13/2016 07:15 PM, Carl Mungan wrote:
Anyways, 5 cents makes it easier to understand the pattern. Assuming the
same kind of pattern holds true for other integers, I hypothesize that
given denominations of a and b cents, where a and b are relatively prime,
the largest total you cannot make is ab-a-b. Obviously that's not divisible
by either a or b so you cannot make it. To complete the proof, I need to
show I can make all integers from ab-a-b+1 up to ab-a-b+min(a,b). I'll work
on it in my spare time, unless someone else first replies and gives me an
example to show my hypothesis is bunk and so I shouldn't bother.

Not bunk.

ab-a-b. Obviously that's not divisible
by either a or b so you cannot make it.

That part of the proof needs work.
22 is not divisible by 5 or by 17, but you can make it.

Spoiler: magic word is "congruence":
https://www.google.com/search?q=theory+of+congruences
which leads to things like:
http://www.math.cornell.edu/~csheridan//Math1350Schedule_files/LinearCongruences.pdf

You can't use the simple proof verbatim (because of the minus
sign) but you can swipe the general idea.

And oh, by the way, that's the answer to Q3. If the customers
are not content to answer the particular case (a=5) and (b=17)
but spontaneously want to generalize to all (a) and (b), or at
least relatively prime (a) and (b), then you know they really
understand the distinction between arithmetic and math.

FWIW this is one of the maxims in Pólya's book: Always look
for the generalization. Some people argue that Pólya is too
advanced for everybody except those who have already figured
it all out on their own ... but I strongly suspect there is
a substantial segment of people in between those extremes,
who would benefit from reading Pólya's exposition.

===================

Fabulous problem.

Yeah.

Never heard of it before. (How did you?)

Typical minestrone: A friend of mine was ranting about people not
getting proper credit for their contributions. That led to Stigler's
law of eponymy (see next msg) which led to Kovalevskaya which led to
Sylvester (who stuck up for K.) and then also to Green (at Cambridge
with S.) which gave me two biographical examples and one mathematical
example that were relevant to the message from BC, so I decided to
write it all up.

Classical minestrone doesn't have any fixed recipe; it's made from
whatever bits are lying around.

Another term for it is "just in time inventory". The concept is applied
systematically in modern manufacturing, but in this case it refers to a
rather less systematic extemporaneous conglomeration of factoids.

Sometimes when I'm too burned out to do anything serious I unwind by
reading biographies. There's a ton of good stuff at
http://www-groups.dcs.st-and.ac.uk/history/BiogIndex.html

They claim to collect only mathematicians, but apparently they count
theoretical physicists as honorary mathematicians.