Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

[Phys-l] perfect-square puzzler



I quote from the car guys:
http://www.cartalk.com/content/puzzler/transcripts/200810/

[Each student] is supposed to write a computer program to handle very
large numbers that could not be handled on a typical hand-held
calculator. The teacher told the students to use that program to
determine if a certain very large number is a perfect square. (What
is a perfect square? A perfect square is a whole number or an integer
that is arrived at by squaring another whole number. For example, 900
is a perfect square of 30; 196 is a perfect square of 14. 625 is the
perfect square of 25. So there are no fractions, no decimals, no
nothing. Just whole numbers allowed.)

Each student is assigned a particular number. This kid's number is
334,912,740,121,562. And the teacher wants to know if this is a
perfect square.

His father says, "That's a big number!"

And then out of the inky shadows, who appears but Crusty! And he
says, "Oh, your teacher gave you an easy number."

"She did?" said the kid.

"Oh yeah. I can give you the answer right now."

The question is, what did Crusty know?


I thought that was a nifty puzzler. It's got nothing to do with
physics _per se_ ... but it teaches the kids to /think/, which is
the whole point of school in general ... and the first weeks of
the physics course are remedial mathematics anyway.

In particular, this puzzler provides the opportunity to emphasize
the difference between basic arithmetic and mathematics. Taking
the square root of some particular number is just basic arithmetic,
whereas Crusty's method involves _mathematics_, i.e. rather general
statements about _all_ numbers.


As a pedagogical wrinkle: It could be argued that Tom & Ray
misstated the problem, in that 334912740121562 is /not/ too
large to be crunched by the calculators lying around chez moi.
My calculators use IEEE double-precision floating point, which
can exactly represent (and square-rootify) any integer up to
9007199254740991 i.e. 2^53 - 1 which covers "almost" all the
17-digit numbers. Ditto for spreadsheets. The wrinkle is that
-- narrowly speaking -- this does not detract much from the main
problem at hand; remember the point was not to find the square
root of 334912740121562 but rather to explain Crusty's method.
Crusty isn't the sort of guy to use a spreadsheet.

More broadly speaking, I don't approve of this. Recall my earlier
screed about the perils of using fake data.
https://carnot.physics.buffalo.edu/archives/2008/4_2008/msg00075.html
Fake data that is good enough in one context could be taken out
of context later, leading to all sorts of errors. I don't want
students to take away the idea that 334912740121562 is not
representable.

Constructive suggestion: You could use 339889724436185982 instead.
That's an 18-digit number, definitely not representable in IEEE
double precision floating point.