1. The two-digit square numbers are 16, 25, 36, 49, 64, and 81. So, a two-digit square must end in 1, 4, 5, 6, or 9. The squares 25, 36, and 81 are therefore eliminated as possibilities for the lower number in Part A.

2. The lower number in Part B must consist only of the digits 1, 4, 5, 6, or 9. There are only five three-digit squares which meet this criterion: 144, 169, 196, 441, and 961.

3. No square number can end in 3 or 8. This means that 441 and 961 cannot be the lower number. For 441 or 961 to be the lower number, the upper number must end in 8. Having narrowed down the possibilities for the lower number to 144, 169, and 196, there are four possibilities that show three two-digit squares:
   

   866	814	834	841
   144	169	169	196

   Of the four possibilities, the number that is a square is 841.

4. Here are the thirteen solutions to the problem in the last hint. It will help students who try this problem to investigate properties of the middle digit. For example, if a square ends in 0 or 5, its second-last digit must be 0 or 2, respectively.
   

   144	529	729	169	441	841	144
   484	256	256	676	400	400	400
   441	961	961	961	100	100	100
   441	841	121	121	361	961
   484	484	289	256	676	676
   144	144	196	169	169	169

81

16

841 written over 196 gives the two-digit squares 81, 49, and 16. This is the only solution to the problem.