I liked your problem from Jan. '03. I too wondered why it worked,
  and came across the following idea.If a number is one less than a
  perfect square (N^2) then this number will always be the product
  of (N+1)(N-1)...as this equals N^2 - 1. If you examine each of
  the numbers that were used it turns out that 24 will be 4x6; 48 =
  6x8; 120 = 10x12; 168 = 12x14; 288 = 16x18; 360 = 18x20. Each of
  these products, when the prime factorization is completed, will
  always contain 2x2x3 somewhere in the expression. Because the two
  numbers that produce N^2-1 will always be the product of 2 even
  numbers it is easy to get a two from each number and have the
  2x2. It also appears that one of the numbers will always be a
  multiple of 3, and thus the 2x2x3. This is certainly no proof,
  but a partial explanation<br />
  <br />
  Sent by George Miller (teacher) via email to Liz<br />
  <br />