actually xD i didn't use any of those things i said... what i did was just take numbers and randomly use process of elimination
if its a number that can produce 4 digits when squared then.... by elimination 11 and 22 are out since they can only produce a 3 digit number when square
and if the last 2 digits of the squared number are exactly the same.... then 33, 44, 55, 66, 77, and 99 are out... so whats left are 0 and 8... since the value of 00 is still 0 xD i excluded that cause you said the first 2 digits are lesser numbers of the original number... then all that's left is 88
of course i checked it with the last thing you said.... 44 is half of 88 :3
hence i concluded that the answer to your problem is 88
*nice nice xD i like a few brain teaser