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'The Clue Is in the Question' printed from http://nrich.maths.org/
This problem is in two parts. The
first part provides some building blocks which will help you to
solve the final challenge. These can be attempted in any order.
This problem can also test your powers of conjecture and discovery:
As you start from one of the minichallenges, how many of the other
related minichallenges will you invent for yourself?
This challenge involves building up a set $F$ of fractions using a
starting fraction and two operations which you use to generate new
fractions from any member of $F$.
Rule 1: $F$ contains the fraction $\frac{1}{2}$.
Rule 2: If $\frac{p}{q}$ is in $F$ then $\frac{p}{p+q}$ is also in
$F$.
Rule 3: If $\frac{p}{q}$ is in $F$ then $\frac{q}{p+q}$ is also in
$F$.
Choose a minichallenge from below to get started. There is a lot
to think about in each of these minichallenges, so as you think
about them, continually ask yourself: Do I have any other thoughts?
Do any other questions arise for me? Make a note of these, as they
might help when you consider other parts of the problem.
Minichallenge A 
Which of these fractions
can I reach? $$ \frac{1}{2}\,, \frac{1}{7}\,, \frac{2}{7}\,,
\frac{5}{9}\,, \frac{11}{13}\,, \frac{17}{16}\,, \frac{19}{8}\,,
\frac{2}{1}\,, $$

Minichallenge B 
What is the
biggest/smallest fraction you can make? What is the biggest
numerator/denominator you can make?

Minichallenge C 
Is it true that the
numerators never decrease?

Minichallenge D 
Can I make a fraction for
which the numerator and denominator have a common factor?

Minichallenge E 
Can I make a 'closed
loop': a sequence of transformations which end up back at the
starting point?

Minichallenge F 
Can you make sense of the
process of working backwards from various fractions?

FINAL CHALLENGE 
Show that every rational
number between $0$ and $1$ is in $F$
