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Mysterious Journey II Hints

What is base 4?

  • 1 of 6: Base 4 is just like decimal (base 10) -- except that you can only count with your arms and legs, not your fingers or toes.
  • 2 of 6: When counting in decimal (base 10) -- which is our normal method of counting -- we gather the numbers in groups of ten.  When counting in base 4 -- the numbers need to be gathered in groups of four.
  • 3 of 6: To count to ten in decimal (base 10), we would count: "1, 2, 3, 4, 5, 6, 7, 8, 9, 10."  Note that we group the numbers in groups of ten by counting up to nine with single digits and creating 10 by saying that the number in the first place equals zero, and the number in the second place equals ten. 
  • 4 of 6: Counting in base four is similar -- but we do not count in single digits up to 9 -- we only count in single digits up to 3.  To count to four in base four would look like this "1, 2, 3, 10".  Note that a "10" in base 4 does not stand for a group of ten -- it stands for a group of four.  In base four, the number "10" is equal to four -- the singles digit still stands for zero -- but the next digit represents groups of four.  To count to ten in base four -- we would count "1, 2, 3, 10, 11, 12, 13, 20, 21, 22...". 
  • 5 of 6: In a base-4 number, each column represents "powers of 4" -- or, numbers that increase (to the left) by a multiple of 4.  I.e., the right-most digit represents "ones", just as it does in decimal; the next digit to the left represents "fours", the next one "sixteens" (4 times 4), the next one "sixty-fours" (4 times 4 times 4), etc.
  • 6 of 6: The number "1321" in base 4 would be 121 in decimal, because it consists of "one 'sixty-four'" plus "three 'sixteens'" plus "two 'fours'" plus "one 'one'".