3. Possible pattern of 3 digits

Observing the lower digits of squares, we can see the following results.

lower first digit (one : 10⁰)

1, 4, 5, 6, 9

lower second digit (ten : 10¹)

first	second
1, 4, 9	even (0, 2, 4, 6 ,8)
5	2
6	odd (1, 3, 5, 7, 9)

lower third digit (hundred : 10²)

first	second	third
1, 9	0, 4, 8	even
1, 9	2, 6	odd
4, 6	----	any
5	----	0, 2, 6

lower fourth digit (thousand : 10³)

Any digit.

Observing the above results, we can get the possible / impossible pattern of digits for squares.

possible pattern : 92 cases

012, 013, 014, 015, 016, 017, 018, 019, 024, 025, 029, 034, 036, 039, 045, 046, 047, 048, 049, 056, 059, 067, 069, 079, 089, 123, 124, 125, 126, 127, 128, 129, 134, 136, 138, 145, 146, 147, 148, 149, 156, 158, 167, 168, 169, 178, 189, 234, 235, 236, 239, 245, 246, 247, 248, 249, 256, 257, 258, 259, 267, 269, 279, 289, 345, 346, 347, 348, 349, 356, 367, 368, 369, 389, 456, 457, 458, 459, 467, 468, 469, 478, 479, 489, 567, 568, 569, 589, 678, 679, 689, 789

impossible pattern : 28 cases

023, 026, 027, 028, 035, 037, 038, 057, 058, 068, 078, 135, 137, 139, 157, 159, 179, 237, 238, 268, 278, 357, 358, 359, 378, 379, 578, 579