Single Square Sums

	A mathematician, named J.A.H. Hunter, once discoved an interesting
numerical pattern that involved square numbers.  Study these examples to
see what he found.

			  20 45 = 20 + 452

			494 703 = 494 + 7032

		       744 2728 = 744 + 27282

	Put into words, we could say: The number is equal to the sum of
the first part with the the square of the second part, the parts being
formed by separating the digits in half, or as nearly so as possible.

	For 4-digit numbers, this would look like:

abcd = ab + cd2

	For 6-digit numbers, this would look like:

abcdef = abc + def2

	For 7-digit numbers, this would look like:

abcdefg = abc + defg2

	Of course, Mr. Hunter found many more cases such as these, 59
in fact, the largest such case printed in his magazine article being

9999998 9999999 = 9999998 + 99999992

	That last one was especially interesting as it was part of a
set of numbers with a unique extra pattern.  Notice:

8 9 = 8 + 92
98 99 = 98 + 992
998 999 = 998 + 9992
9998 9999 = 9998 + 99992
99998 99999 = 99998 + 999992
999998 999999 = 999998 + 9999992


	How far do you think you could continue this pattern?  I mean,
really prove it by multiplying the squared part long hand, then adding
the other part.  Try it; it isn't hard at all.




	Mr. Hunter probably used a computer program to find these.  But
you and I don't have that program at hand (or do you?).  So I propose
a little challenge for you that you should be able to do.  Below is a
list of numbers, which came from Hunter's list.  Some should be cut at
the "half-way" spot, a few do not.  Can you find the proper separation
point for these "Hunter" numbers?

		3055		88297		4942223

		23804879	25505050	23804879

		28005292	52887272	493822222




	Mr. Hunter also discovered a subtraction variation of this
"single square" theme, which he called, naturally enough, Single
Square Differences.  He gave a mere 14 instances, but the 14th one
is a big, BIG number: 52892567272728.  So they must be rare and hard
to find.  These fascinating numbers take the following form (for a
5-digit instance): 

abcde = cde2 - ab

	The only 5-digit case presented by Hunter was 82287.  So I
assume that there are no more.  Please notice the reversal of the
first part with that of the second part.

	He also only showed one 4-digit case.  In order to give you
something to put in your calculators, I will only tell you one clue:

6070  <  N  <  6080

	To give you some fun in discovering the six-digit cases, we
will give you some numbers that are real and some that are "fakes".
Can you separate the "wheat from the chaff"?

		132364		245646		406638

		416637		510715		852924





References:  J.A.H.Hunter.  Single Square Sums.  J. of Recreational
Mathematics.  4(1) Jan 1971.  pp. 64-65.

--. Single Square Differences.  JRM. (same issue) p. 65.