#
# "Re-basing" a number is the process of taking a number in one base (in this
# case, base 10) and expressing the same value in another numeric base. 
# In all cases, numerical representations express *power* and *terms*.
# Whenever you have a number, <1><2><3><4>bX, the actual value of that 
# number is:
#		  <4> * X ** 0
#		+ <3> * X ** 1
#		+ <2> * X ** 2
#		+ <1> * X ** 3
#
# Thus, the process of converting one number into another base is simply
# a matter of calculating the values of <1> <2> <3> <4> and so on, until
# the number is exhausted.
#
# There are two ways to calculate this value.  You can go forwards, or
# backwards.  The way you go doesn't matter much, but it may to your teacher.
# Since the new representation of the number is a determinite value, it 
# does not matter how you arrive at the result.  Let's convert 500 into
# base 3 on paper:
#
# To begin, it helps to calculate how many digits will be in the new number.
# There exists exactly one number X such that:
#
#		base ** X <= num	and		base ** (X + 1) > num
#
# (X + 1) is the number of digits in the new number.  In this case...
#	3 ** 1 -> 3		3 ** 2 -> 9		3 ** 3 -> 81
#	3 ** 4 -> 243		3 ** 5 -> 729
#
# Well, 243 < 500 < 729, so X is 4 -- and there will be five digits in the
# answer.  Now we just need to find what the five values are.
#
# ### Going forward ###
#
# There exists one set of numbers [a, b, c, d, e] such that:
#	(a * 3 ** 4) + (b * 3 ** 3) + (c * 3 ** 2) + (d * 3 ** 3) + e == 500
# such that [a, b, c, d] are integers.
#
#		  a * 243
#		+ b *  81
#		+ c *   9
#		+ d *   3
#		+ e
#		=========
#		     500
#
# So what is the value of 'a'?  Well, it is simply the floor of 
# 500 / 243 -- 2.  Now we know what 'a' is, we can remove it from the 
# equation:
#
#		  2 * 243 (486)
#		+ b * 81
#		+ c *  9
#		+ d *  3
#		+ e
#		=========
#		    500
# Subtracting 486 from both sides, we're left with:
#
#		  b * 81
#		+ c *  9
#		+ d *  3
#		+ e
#		========
#		      14
#
# What is the value of 'b'?  Why it floor(14 / 81) -- 0.
#
#		  c * 9
#		+ d * 3
#		+ e
#		=======
#		     14
#
# What is the value of 'c'?  It is floor(14 / 9) -- 1.  That leaves us with:
#
#		  d * 3 + e == 5
#
# So we know that d is 1.  That leaves us with
#
#		e = 2
#
# So now we have our result:
#
#	[2, 0, 1, 1, 2]
#
# Thus, 500b10 == 20112b3
#
# Viola!
#
# ### Going backward ###
#
# There exists one set of numbers [a, b, c, d, e] such that:
#	(a * 3 ** 4) + (b * 3 ** 3) + (c * 3 ** 2) + (d * 3 ** 3) + e == 500
# such that [a, b, c, d] are integers.
#
#		  a * 243
#		+ b *  81
#		+ c *   9
#		+ d *   3
#		+ e (*  1)
#		=========
#		     500
#
# So what is the value of 'e'?  This is more complicated -- it is the 
# number of times that this multiplier (1) goes into the remainder (modulus)
# of the previous multiplier (3), in this case, (500 % 3) / 1 == 2
#
# Now that we know what 'e' is, we can remove it from the equation:
#
#		  a * 243
#		+ b * 81
#		+ c *  9
#		+ d *  3
#		+ 2
#		=========
#		    500
# Subtracting 2 from both sides, we're left with:
#
#		  a * 243
#		+ b * 81
#		+ c *  9
#		+ d *  3
#		=========
#		    498
# What is the value of 'd'?  It is (498 % 9) / 3 == 1
#
#		  a * 243
#		+ b * 81
#		+ c *  9
#		=========
#		    495
# Calculating for 'c', (495 % 81) / 9 == 1
#
#		  a * 243
#		+ b * 81
#		=========
#		    486
# Calcuating for 'b', (486 % 243) / 81 == 0
#
# So a * 243 == 486, and it doesn't take a rocket scientist to see that a == 2.
#
# So our result is [2, 0, 1, 1, 2], therefore 500b10 == 20112b3.
#