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Vol. 1, No. 1, June 2012 | At Right Angles 65
What is a problem? It refers to a task or situation where
you do not know what to do; you have no way already
worked out to deal with the situation, no ‘formula’; you
have to discover the way afresh, by thinking on the spot.
In this sense, a problem is not the kind of exercise you
meet at the end of a chapter. On completing a chapter on
quadratic equations one may be assigned a list of ten or
twenty quadratic equations to solve. But these are ‘drill
exercises’ — they must not be called ‘problems’. A prob- lem is essentially non-routine. You have to throw yourself
at it in order to solve it.
In the history of mathematics it has happened time and
again that problems posed by mathematicians — to
themselves or to others — would lie unsolved for a long
time. Perhaps the most famous instance of this is that
of Fermat’s Last Theorem (‘FLT’), whose origin lies in
a remark casually inserted by Fermat in the margin of
a mathematics book he was reading; the eventual solu- tion to the problem came after a gap of three and a half
centuries! (See the Review of Fermat’s Enigma else- where in this issue for more about this story.) And each
time this happens, in the struggle between mathemati- cian and problem, the winner invariably is mathematics
itself; for in the encounter are born fresh concepts and
ideas, fresh ways of organizing and looking at old ideas,
fresh notation. In the case of FLT, number theory devel- oped enormously as a result of this encounter, and
a whole new field was born, now called Algebraic num- ber theory.
The problem corner is a very important compo- nent of this magazine. It comes in three parts: Fun
Problems, Problems for the Middle School, and
Problems for the Senior School. For each part,
the solutions to problems posed will appear in the
next issue.
To encourage the novice problem solver, we start
each section with a few solved problems which
convey an idea of the techniques used to under- stand and simplify problems, and the ways used
to approach them.
We hope that you will tackle the problems and
send in your solutions. We may choose your solu- tion to be the ‘official’ solution! ‘Visual proofs’ are
particularly welcome — proofs which use a mini- mum of words.
problem corner
Preamble
Problems,
The Life Blood
of Mathematics
Many mathematicians take great pleasure in problem solving, and ‘Problem Corner’ is where we
share interesting problems of mathematics with one other: talk about experiences connected
with memorable problems, show the interconnectedness of problems, and so on.
It has been said that “problems are the lifeblood of mathematics.” This short, pithy sentence contains
within it a great truth, and it needs to be understood.
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66 At Right Angles | Vol. 1, No. 1, June 2012
Another instance where this happened was in the
struggle to solve polynomial equations. Quadratic equa- tions (i.e., equations of degree 2, like x2 +3x+2 = 0) were
mastered a long time back, perhaps as early as the sev- enth century (though there was no concept of negative
numbers back then); cubic equations (degree 3) were
solved by several mathematicians independently over
the twelfth to fifteenth centuries; and biquadratic equa- tions (degree 4; also called ‘quartic equations’) were
solved soon after. Naturally, attention then turned to the
quintic equation (degree 5). Here researchers hit what
seemed to be a wall; no matter what approach was tried
they could not cross this barrier. Eventually the matter
was resolved but not in the way that everyone expected;
it was shown by a young Frenchman named Evaristé Ga- lois that in a certain sense the problem was not solvable
at all! In the process was born one of the gems of higher
algebra, now called Galois theory.
It is not difficult to see why a struggle of this kind will
bring up something new. Take any real problem, tackle
it, struggle with it and do not give up, no matter what
happens; and examine at the end how much you have
learnt in the process. What you find may surprise you ...
It is remarkable that this happens even in those instanc- es where you do not get the solution. But for that, it is
essential not to ‘give up’ ....
In the problem section of Crux Mathematicorum, which is one of the best known problem journals, there occur
these memorable words: No problem is ever closed, and the editor adds that solutions sent in late will still be con- sidered for publication, provided they yield some new insight or some new understanding of the problem. We are
happy to adopt a similar motto for our three problem sections.
Submissions to the
Problem Corner
The Problem Corner invites readers
to send in proposals for problems and
solutions to problems posed. Here are
some guidelines for the submission of
such entries.
(1) Send your problem proposals and
solutions by e-mail, typeset as a
Word file (with mathematical text
typeset using the equation editor)
or as a LaTeX file, with each prob- lem or solution started on a fresh
page. Please use the following ID:
AtRiA.editor@apu.edu.in
(2) Please include your name and con- tact details in full (mobile number,
e-mail ID and postal address) on
the solution sheet/problem sheet.
(3) If your problem proposal is based
on a problem published elsewhere,
then please indicate the source (be
it a book, journal or website; in the
last case please give the complete
URL of the website).
For convenience we list some notation and terms which
occur in many of the problems.
Coprime
Two integers which share no common factor exceeding
1 are said to be coprime.
Example: 9 and 10 are coprime, but not 9 and 12. Pairs
of consecutive integers are always coprime.
Pythagorean triple (‘PT’ for short)
A triple (a, b, c) of positive integers such that a2 +b2
= c 2
.
Primitive Pythagorean triple (‘PPT’ for short)
A triple (a, b, c) of coprime, positive integers such that
a2 +b2
= c 2
. Thus a PPT is a PT with an additional condi- tion — that of coprimeness.
Example : The triples (3, 4, 5) and (5, 12, 13). The set of
PPTs is a subset of the set of PTs.
Arithmetic Progression (AP for short)
Numbers a1, a2, a3, a4, ... are said to be in AP if
a2 –a1 = a3 –a2 = a4 –a3 = ... The number d = a2 –a1
is called the common difference of the AP.
Example : The numbers 3, 5, 7, 9 form a four term AP
with common difference 2, and 10, 13, 16 form a three
term AP with common difference 3.
Notation used in the
problem sets