GearBox

PROBLEM STATEMENT

When designing vehicles and LEGO robots, it is important to construct gear
boxes tailored to a desired power and speed.  One of the most useful pieces of
information about a gearbox is its gear-reduction ratio... that is, the number
of times the motor turns for each time the wheel turns.

Below is a String representation of a gearbox.  The rows each represent
individual axles.  The columns are positions on the axles that might contain
gears (if not, then just the metal rod from the axle).  If there is a "-", that
signifies no gear.  If there is an integer, n, that means there is a gear which
has a n:1 gear reduction with respect to the motor. Assume the motor is
attached to the top axle, and the wheel is always attached to the bottom axle.
There is no such thing as a 0:1 gear reduction.

Gears on the same axle turn at the same rate, if one gear turns one full time,
all gears on that axle turn one full time.

---5-
-2-3-
-5--3
--3-5
--5--

The first and second axle are connected by a 5/3 ratio.
The second and third axle are connected by a 2/5 ratio.
The third and fourth by a 3/5 ratio.
The fourth and fifth by a 3/5 ratio.
Then, the first axle to the fifth axle has a ratio of (5/3 * 2/5 * 3/5 * 3/5) =
6/25.

Remember gear reduction is the number of times the MOTOR turns for each time
the WHEEL turns, so the actual gear-reduction of the box is:

"25:6"

If there exist two adjacent axles where none of the gears "mesh", that is,
there is no column where both axles have a gear, then the gearbox is broken
since it won't allow the motor to drive the wheel.  In that case, return "0"
See the example below:

---2-
-3-5-
-2--2
-54--
---3-

On the last two axles, there is no "meshing" since the only gear on the last
axle is a 3, and there is no gear in that position in the previous axle.  The
reduction is "0". It is possible for two axles to mesh at multiple places, but
one must be careful that the proportions are the same in that case, or else the
gearbox will jam up since different sets of gears are trying to turn at
different rates.

-2-3-
-4-6-
--31-
--5--

This box should be fine, the first and second axle mesh at multiple spots, but
their ratios are the same, 1/2.
The reduction is "5:9"

-2-3-
-4-4-
--31-
--5--

This box above will jam, so the reduction is "0".

DEFINITION
Class: GearBox
Parameters: String[]
Returns: String
Method Signature:  String reduction (String[] axles)
Be sure to declare your method public

NOTES
- Gear reductions are always in simplified form. For example, "9:3" is NOT a
valid gear reduction, because it can be simplified to "3:1"

TopCoder will check the validity of the inputs.  Inputs are valid if all of the
following criteria are met:
- All axles must be the same length.
- axles must contain between 2 and 10 elements, inclusive.
- elements of axles must be between 2 and 10 characters in length, inclusive.
- Axle representation must only consist of digits [1-9] or '-'.


EXAMPLES

---5-
-2-3-
-5--3
--3-5
--5--

Returns: "25:6"

---2-
-3-5-
-2--2
-54--
---3-

Returns: "0"

-2-3-
-4-6-
--31-
--5--

Returns: "5:9"

-2-3-
-4-4-
--31-
--5--

Returns: "0"