This is just a mathematical curiosity, not anything that will help you predict Pick 2/2/3/4/5/6. For each game I have listed all of the box combos where both the sum of the digits and the product of the digits is a square number. It's not the same thing as Pick-n digit strings that represent square numbers. In parentheses below each list is the number of box combos.
Pick 2
00, 01, 04, 09, 22, 88
(6)
Pick 3
000, 001, 004, 009, 013, 018, 022, 027, 036, 045, 079, 088, 144, 466, 889
(15)
Pick 4
0000, 0001, 0004, 0009, 0013, 0018, 0022, 0027, 0036, 0045, 0079, 0088, 0112, 0117, 0126, 0135, 0144, 0169, 0178, 0225, 0234, 0259, 0268, 0277, 0333, 0349, 0358, 0367, 0448, 0457, 0466, 0556, 0799, 0889, 1111, 1177, 1224, 1339, 2266, 2338, 3355, 4444, 4669, 9999
(44)
Pick 5
00000, 00001, 00004, 00009, 00013, 00018, 00022, 00027, 00036, 00045, 00079, 00088, 00112, 00117, 00126, 00135, 00144, 00169, 00178, 00225, 00234, 00259, 00268, 00277, 00333, 00349, 00358, 00367, 00448, 00457, 00466, 00556, 00799, 00889, 01111, 01116, 01125, 01134, 01159, 01168, 01177, 01224, 01233, 01249, 01258, 01267, 01339, 01348, 01357, 01366, 01447, 01456, 01555, 01699, 01789, 01888, 02223, 02239, 02248, 02257, 02266, 02338, 02347, 02356, 02446, 02455, 02599, 02689, 02779, 02788, 03337, 03346, 03355, 03445, 03499, 03589, 03679, 03688, 03778, 04444, 04489, 04579, 04588, 04669, 04678, 04777, 05569, 05578, 05668, 05677, 06667, 09999, 11133, 11149, 11248, 11455, 11779, 12222, 12346, 12778, 13399, 14488, 15577, 16666, 22444, 22669, 23389, 23677, 33334, 33559, 34468, 44449, 47799, 48888
(114)
Pick 6
000000, 000001, 000004, 000009, 000013, 000018, 000022, 000027, 000036, 000045, 000079, 000088, 000112, 000117, 000126, 000135, 000144, 000169, 000178, 000225, 000234, 000259, 000268, 000277, 000333, 000349, 000358, 000367, 000448, 000457, 000466, 000556, 000799, 000889, 001111, 001116, 001125, 001134, 001159, 001168, 001177, 001224, 001233, 001249, 001258, 001267, 001339, 001348, 001357, 001366, 001447, 001456, 001555, 001699, 001789, 001888, 002223, 002239, 002248, 002257, 002266, 002338, 002347, 002356, 002446, 002455, 002599, 002689, 002779, 002788, 003337, 003346, 003355, 003445, 003499, 003589, 003679, 003688, 003778, 004444, 004489, 004579, 004588, 004669, 004678, 004777, 005569, 005578, 005668, 005677, 006667, 009999, 011115, 011124, 011133, 011149, 011158, 011167, 011223, 011239, 011248, 011257, 011266, 011338, 011347, 011356, 011446, 011455, 011599, 011689, 011779, 011788, 012222, 012229, 012238, 012247, 012256, 012337, 012346, 012355, 012445, 012499, 012589, 012679, 012688, 012778, 013336, 013345, 013399, 013444, 013489, 013579, 013588, 013669, 013678, 013777, 014479, 014488, 014569, 014578, 014668, 014677, 015559, 015568, 015577, 015667, 016666, 018999, 022228, 022237, 022246, 022255, 022336, 022345, 022399, 022444, 022489, 022579, 022588, 022669, 022678, 022777, 023335, 023344, 023389, 023479, 023488, 023569, 023578, 023668, 023677, 024469, 024478, 024559, 024568, 024577, 024667, 025558, 025567, 025666, 027999, 028899, 033334, 033379, 033388, 033469, 033478, 033559, 033568, 033577, 033667, 034459, 034468, 034477, 034558, 034567, 034666, 035557, 035566, 036999, 037899, 038889, 044449, 044458, 044467, 044557, 044566, 045556, 045999, 046899, 047799, 047889, 048888, 055555, 055899, 056799, 056889, 057789, 057888, 066699, 066789, 066888, 067779, 067788, 077778, 111114, 111166, 111229, 111499, 112228, 112255, 112489, 113344, 114559, 118899, 122236, 122488, 123469, 124558, 128889, 133477, 136899, 144466, 144999, 145555, 155889, 166779, 222244, 223333, 223468, 224449, 227799, 228888, 234556, 236889, 244899, 255888, 266778, 333349, 336699, 337788, 355689, 445599, 446688, 447777, 499999, 555588, 556677, 666666
(269)
(I don't know of any jurisdiction that offers this kind of digit pick 6 game.)