N=200

N=200

For which N is there a permutation of {1,...,N} s.t. termwise sum with {1,...,N} gives all squares?

Here's a solution for N=200, solved with http://sentient-lang.org/

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100,
101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120,
121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140,
141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160,
161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180,
181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200}

+

{120, 7, 78, 77, 31, 94, 42, 161, 55, 134, 70, 184, 3, 67, 49, 180, 32, 18, 45, 149,
60, 174, 2, 97, 75, 199, 142, 116, 71, 34, 90, 4, 48, 47, 14, 28, 107, 131, 157, 156,
80, 102, 57, 20, 99, 98, 122, 121, 176, 146, 93, 117, 143, 171, 141, 44, 112, 63, 22, 61,
83, 163, 106, 105, 35, 159, 129, 188, 100, 30, 29, 153, 123, 182, 25, 5, 179, 178, 65, 89,
88, 87, 86, 37, 59, 58, 82, 137, 11, 79, 53, 52, 103, 162, 130, 73, 192, 46, 1, 69,
155, 154, 41, 152, 39, 183, 118, 181, 12, 115, 10, 113, 56, 175, 54, 140, 172, 51, 50, 136,
23, 167, 21, 72, 164, 198, 17, 196, 160, 66, 158, 124, 191, 62, 189, 8, 187, 186, 185, 85,
84, 147, 26, 81, 144, 110, 109, 108, 76, 139, 173, 104, 43, 135, 101, 133, 132, 38, 165, 96,
64, 127, 33, 197, 91, 195, 194, 193, 27, 119, 190, 24, 151, 150, 114, 148, 19, 111, 145, 16,
15, 74, 13, 177, 40, 138, 9, 68, 36, 6, 170, 169, 168, 95, 166, 128, 92, 126, 125, 200}

=

{121, 9, 81, 81, 36, 100, 49, 169, 64, 144, 81, 196, 16, 81, 64, 196, 49, 36, 64, 169,
81, 196, 25, 121, 100, 225, 169, 144, 100, 64, 121, 36, 81, 81, 49, 64, 144, 169, 196, 196,
121, 144, 100, 64, 144, 144, 169, 169, 225, 196, 144, 169, 196, 225, 196, 100, 169, 121, 81, 121,
144, 225, 169, 169, 100, 225, 196, 256, 169, 100, 100, 225, 196, 256, 100, 81, 256, 256, 144, 169,
169, 169, 169, 121, 144, 144, 169, 225, 100, 169, 144, 144, 196, 256, 225, 169, 289, 144, 100, 169,
256, 256, 144, 256, 144, 289, 225, 289, 121, 225, 121, 225, 169, 289, 169, 256, 289, 169, 169, 256,
144, 289, 144, 196, 289, 324, 144, 324, 289, 196, 289, 256, 324, 196, 324, 144, 324, 324, 324, 225,
225, 289, 169, 225, 289, 256, 256, 256, 225, 289, 324, 256, 196, 289, 256, 289, 289, 196, 324, 256,
225, 289, 196, 361, 256, 361, 361, 361, 196, 289, 361, 196, 324, 324, 289, 324, 196, 289, 324, 196,
196, 256, 196, 361, 225, 324, 196, 256, 225, 196, 361, 361, 361, 289, 361, 324, 289, 324, 324, 400}