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

[Carl Mungan <mungan@usna.edu>]

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

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.

On Sun, Nov 13, 2016 at 7:40 PM, John Denker <jsd@av8n.com> wrote:

> Here's an amusing little puzzle:
>     I have an unlimited supply of 5¢ and 17¢ stamps.  What is the
>     largest value that I cannot make up using only a combination
>     of these two denominations of stamp?
>
> Q1: What is the numerical answer?              (Hint:  63¢.)
> Q2: Are you sure?  Can you /prove/ your number is correct?
> Q3: See below.
>
> Puzzle rephrased from James Sylvester (late 1800s):
>   http://www-groups.dcs.st-and.ac.uk/history/Biographies/Sylvester.html
>
>
>
> I mention this because previously I wrote:
>
> > >  math proficiency is a very strong predictor of
> > > which students will do well in the physics course.
>
> Note that when I say "math", I mean something special.  I like to
> tell people that:
>  -- arithmetic is about numbers, whereas
>  -- math is about patterns.
>
> Math includes arithmetic, but that's only 0.001% of what math is.
>
> This is tricky, because in grade school, people are taught that
> the word "math" is /synonymous/ with arithmetic.  This is a
> profound misconception, which people need to unlearn, stat.
>
> The "stamps" puzzle illustrates the distinction.  To solve the
> puzzle you need to do a little bit of arithmetic, but only a
> little, and arithmetic is not the interesting part.  You need
> to verify an infinite number of facts.  The direct arithmetical
> approach doesn't work, because it would require an infinite
> amount of arithmetic.  Instead you need to see the /pattern/.
>
> Giving away the answer to Q1 doesn't make the puzzle appreciably
> less interesting.
>
> Note that being faster with arithmetic does not make you a
> better mathematician.  There are plenty of mathematicians who
> have above-average arithmetic skills ... but also plenty who
> have average or below-average arithmetic skills.  For example:
>   "I was unfortunately weak in the multiplication table."
>    http://www-groups.dcs.st-and.ac.uk/history/Biographies/
> Kovalevskaya.html
> Also:
>    http://www-groups.dcs.st-and.ac.uk/history/Biographies/Green.html
>
> The math=arithmetic=speed misconception causes broad and deep
> damage.  Some schools drill, drill, drill kindergärtners and
> then test them on how fast they can do arithmetic.  The kids
> who don't "get it" break down crying.  The whole setup is insane.
> It is at least five jumps removed from sanity:
>  -- It shouldn't focus on arithmetic per se;
>  -- it shouldn't rely on rote to the exclusion of reasoning;
>  -- it shouldn't focus on speed;
>  -- none of this should be inflicted on kindergärtners; and
>  -- no matter what you're doing, it shouldn't make students cry.
> The most important goal, especially in the early grades, is to
> install a love of learning.
>
> Amusing story:  Consider the results of the tripos examination
> at Cambridge in 1837.  The aforementioned James Sylvester came
> in second, George Green came in fourth, and Duncan Gregory fifth.
> J.S. and D.G went on to become eminently successful mathematicians.
> G.G. was in the /middle/ of a distinguished career, having already
> published a tremendously important book, long before he set foot
> in Cambridge.  To this day you can still buy fat books devoted
> primarily to Green functions.
>
> In contrast, the guy who came in first never made any comparably
> important contributions.  According to MacTutor, this was not an
> uncommon result at the time ... because back then the tripos was
> more a test of /speed/ than of insight.
>    http://www-groups.dcs.st-and.ac.uk/history/Biographies/Sylvester.html
>    http://www-groups.dcs.st-and.ac.uk/history/Biographies/Green.html
>
> My point is:  If you're giving a test where *GEORGE GREEN* comes
> in fourth, you're doing something wrong.
>
> On 11/13/2016 11:11 AM, bernard cleyet wrote:
>
> > My score: Quantitative 28 and Literacy 98.
>
> Yeah.  There are still a lot of woefully screwed-up tests out there.
>
> ============
>
> Last but not least:
>
> Q3: How might the "stamps" puzzle sometimes provide an indication
> of real mathematical aptitude, or at least mathematical inclination,
> far beyond what Q2 calls for?
>
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-- 
Carl E. Mungan, Professor of Physics  410-293-6680 (O) -3729 (F)
Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-1363
mailto:mungan@usna.edu     http://usna.edu/Users/physics/mungan/