The Simplest Impossible Problem

A 7-year-old can understand this problem which completely baffles mathematicians.

Collatz calculator: http://www.nitrxgen.net/collatz/

https://twitter.com/TPointMath

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Pure mathematics, that is, math for its own
sake,
has produced fascinating patterns, such as
erie
strange attractors, or tables of knots. Applied
mathematics
has been used in many areas, such as heat
flow or turbulence.
There's one problem, however, which leaves
mathematicians utterly defeated.
And it only involves simple arithmetic that
a seven-year-old can follow.
This is definitely the simplest impossible
problem.
Starting with a positive whole number n, let's
produce a new number
according to the following rule: if n is even,
divide it by 2. If it's odd,
multiply by 3, then add 1. For example, let's
start with 10.
Since 10 is even, divide it by 2 to get 5.
5 is odd, so multiply it by 3 and add 1 to
get 16.
Keep going to produce 8, 4, 2, 1, 4, 2, 1,
etc.
This pattern of 4,2,1 repeats forever.
Well, this isn't hard. So what's the problem?
Try other starting numbers, like 11, 23, or
29.
They all eventually reach one.
This is the challenge; show that no matter
which
starting number you choose, the numbers will
always reach one.
This problem drives mathematicians crazy because
there don't
seem to be any clear patterns. Sure, some
special numbers, such as 8192,
which is a power of 2, collapse down to 1
pretty quickly; it takes
only 13 steps to get there. However, if you
start with 27,
it takes 110 steps to reach the number 1.
A graph of the points
when we start at 27 shows the erratic nature
of these numbers.
The graph reaches its peak at 9232.
Of course researchers have used computers
to help out.
You can click on this box to enter your own
starting number
and explore what happens.
To date, all starting numbers less than 5x2^60
have been checked
and they all eventually reach one. Of course
this doesn't
prove the conjecture for larger starting numbers,
but it does mean
that working by hand is not a good idea.
This impossible problem is usually called
the 3x+1 problem, but it's
also known as the Collatz Conjecture, named
after Lothar
Collatz who invented the problem
back in the 1930s. Other mathematicians who
were intrigued by the problem
mentioned it in their lectures, so this conjecture
also became known as
Hasse's problem, Kakutani’s Problem, and
Ulam's problem.
With all this interest, it was joked in 1960
that the 3x+1 problem
was part of a conspiracy to slow down mathematical
research in the U.S.
But getting back to the problem, what could
happen if
a starting number doesn't reach the cycle
{4,2,1}?
One possiblity is that it approaches some
other cycle.
Advanced theory shows that any cycle
besides {4,2,1} must have at least 10 billion
numbers.
The only other possibility is that the numbers
would get arbitrarily
large and approach infinity. But both of these
scenarios are highly unlikely.
Over time, mathematicians have built complex
theories
to try to understand the 3x+1 problem, but
they've made little progress.
Even the 20th century genius Paul Erdos said
about this
challenge, "Mathematics is not yet ready for
such problems".
But hey, but don't let me or Paul discourage
you. What can
you see in this problem?