In a pentagon, each vertex is assigned a real number which sum is positive. If there is a negative number $y$, use the transformation $$T:(x,y,z)\mapsto(x+y,-y,y+z)$$ where $x,y,$ and $z$ are consecutive vertices. This transformation can always used whenever there still exist a negative real number. Is it necessary after a number of transformation, all number in the pentagon will be non negative?
2026-03-25 01:24:00.1774401840
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On
Invariant principle in tranformation number
219 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
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There are 3 best solutions below
0
On
greedy algorithm for pentagon. It does appear that the sum of squares may grow, but always shrinks from its current size within five steps. If true, that is a proof.
Tue Jun 26 12:07:21 PDT 2018
-77 44 -31 28 37 sos 10979
*
-77 13 31 -3 37 step 1 sos 8437
*
-77 13 28 3 34 step 2 sos 8047
*
77 -64 28 3 -43 step 3 sos 12667
*
13 64 -36 3 -43 step 4 sos 7419
*
13 28 36 -33 -43 step 5 sos 5187
*
13 28 3 33 -76 step 6 sos 7827
*
-63 28 3 -43 76 step 7 sos 12387
*
63 -35 3 -43 13 step 8 sos 7221
*
28 35 -32 -43 13 step 9 sos 5051
*
28 3 32 -75 13 step 10 sos 7611
*
28 3 -43 75 -62 step 11 sos 12111
*
-34 3 -43 13 62 step 12 sos 7027
*
34 -31 -43 13 28 step 13 sos 4919
*
3 31 -74 13 28 step 14 sos 7399
*
3 -43 74 -61 28 step 15 sos 11839
*
3 -43 13 61 -33 step 16 sos 6837
*
-30 -43 13 28 33 step 17 sos 4791
*
30 -73 13 28 3 step 18 sos 7191
*
-43 73 -60 28 3 step 19 sos 11571
*
-43 13 60 -32 3 step 20 sos 6651
*
-43 13 28 32 -29 step 21 sos 4667
*
-72 13 28 3 29 step 22 sos 6987
*
72 -59 28 3 -43 step 23 sos 11307
*
13 59 -31 3 -43 step 24 sos 6469
*
13 28 31 -28 -43 step 25 sos 4547
*
13 28 3 28 -71 step 26 sos 6787
*
-58 28 3 -43 71 step 27 sos 11047
*
58 -30 3 -43 13 step 28 sos 6291
*
28 30 -27 -43 13 step 29 sos 4431
*
28 3 27 -70 13 step 30 sos 6591
*
28 3 -43 70 -57 step 31 sos 10791
*
-29 3 -43 13 57 step 32 sos 6117
*
29 -26 -43 13 28 step 33 sos 4319
*
3 26 -69 13 28 step 34 sos 6399
*
3 -43 69 -56 28 step 35 sos 10539
*
3 -43 13 56 -28 step 36 sos 5947
*
-25 -43 13 28 28 step 37 sos 4211
*
25 -68 13 28 3 step 38 sos 6211
*
-43 68 -55 28 3 step 39 sos 10291
*
-43 13 55 -27 3 step 40 sos 5781
*
-43 13 28 27 -24 step 41 sos 4107
*
-67 13 28 3 24 step 42 sos 6027
*
67 -54 28 3 -43 step 43 sos 10047
*
13 54 -26 3 -43 step 44 sos 5619
*
13 28 26 -23 -43 step 45 sos 4007
*
13 28 3 23 -66 step 46 sos 5847
*
-53 28 3 -43 66 step 47 sos 9807
*
53 -25 3 -43 13 step 48 sos 5461
*
28 25 -22 -43 13 step 49 sos 3911
*
28 3 22 -65 13 step 50 sos 5671
*
28 3 -43 65 -52 step 51 sos 9571
*
-24 3 -43 13 52 step 52 sos 5307
*
24 -21 -43 13 28 step 53 sos 3819
*
3 21 -64 13 28 step 54 sos 5499
*
3 -43 64 -51 28 step 55 sos 9339
*
3 -43 13 51 -23 step 56 sos 5157
*
-20 -43 13 28 23 step 57 sos 3731
*
20 -63 13 28 3 step 58 sos 5331
*
-43 63 -50 28 3 step 59 sos 9111
*
-43 13 50 -22 3 step 60 sos 5011
*
-43 13 28 22 -19 step 61 sos 3647
*
-62 13 28 3 19 step 62 sos 5167
*
62 -49 28 3 -43 step 63 sos 8887
*
13 49 -21 3 -43 step 64 sos 4869
*
13 28 21 -18 -43 step 65 sos 3567
*
13 28 3 18 -61 step 66 sos 5007
*
-48 28 3 -43 61 step 67 sos 8667
*
48 -20 3 -43 13 step 68 sos 4731
*
28 20 -17 -43 13 step 69 sos 3491
*
28 3 17 -60 13 step 70 sos 4851
*
28 3 -43 60 -47 step 71 sos 8451
*
-19 3 -43 13 47 step 72 sos 4597
*
19 -16 -43 13 28 step 73 sos 3419
*
3 16 -59 13 28 step 74 sos 4699
*
3 -43 59 -46 28 step 75 sos 8239
*
3 -43 13 46 -18 step 76 sos 4467
*
-15 -43 13 28 18 step 77 sos 3351
*
15 -58 13 28 3 step 78 sos 4551
*
-43 58 -45 28 3 step 79 sos 8031
*
-43 13 45 -17 3 step 80 sos 4341
*
-43 13 28 17 -14 step 81 sos 3287
*
-57 13 28 3 14 step 82 sos 4407
*
57 -44 28 3 -43 step 83 sos 7827
*
13 44 -16 3 -43 step 84 sos 4219
*
13 28 16 -13 -43 step 85 sos 3227
*
13 28 3 13 -56 step 86 sos 4267
*
-43 28 3 -43 56 step 87 sos 7627
*
43 -15 3 -43 13 step 88 sos 4101
*
28 15 -12 -43 13 step 89 sos 3171
*
28 3 12 -55 13 step 90 sos 4131
*
28 3 -43 55 -42 step 91 sos 7431
*
-14 3 -43 13 42 step 92 sos 3987
*
14 -11 -43 13 28 step 93 sos 3119
*
3 11 -54 13 28 step 94 sos 3999
*
3 -43 54 -41 28 step 95 sos 7239
*
3 -43 13 41 -13 step 96 sos 3877
*
-10 -43 13 28 13 step 97 sos 3071
*
10 -53 13 28 3 step 98 sos 3871
*
-43 53 -40 28 3 step 99 sos 7051
*
-43 13 40 -12 3 step 100 sos 3771
*
-43 13 28 12 -9 step 101 sos 3027
*
-52 13 28 3 9 step 102 sos 3747
*
52 -39 28 3 -43 step 103 sos 6867
*
13 39 -11 3 -43 step 104 sos 3669
*
13 28 11 -8 -43 step 105 sos 2987
*
13 28 3 8 -51 step 106 sos 3627
*
-38 28 3 -43 51 step 107 sos 6687
*
38 -10 3 -43 13 step 108 sos 3571
*
28 10 -7 -43 13 step 109 sos 2951
*
28 3 7 -50 13 step 110 sos 3511
*
28 3 -43 50 -37 step 111 sos 6511
*
-9 3 -43 13 37 step 112 sos 3477
*
9 -6 -43 13 28 step 113 sos 2919
*
3 6 -49 13 28 step 114 sos 3399
*
3 -43 49 -36 28 step 115 sos 6339
*
3 -43 13 36 -8 step 116 sos 3387
*
-5 -43 13 28 8 step 117 sos 2891
*
5 -48 13 28 3 step 118 sos 3291
*
-43 48 -35 28 3 step 119 sos 6171
*
-43 13 35 -7 3 step 120 sos 3301
*
-43 13 28 7 -4 step 121 sos 2867
*
-47 13 28 3 4 step 122 sos 3187
*
47 -34 28 3 -43 step 123 sos 6007
*
13 34 -6 3 -43 step 124 sos 3219
*
13 28 6 -3 -43 step 125 sos 2847
*
13 28 3 3 -46 step 126 sos 3087
*
-33 28 3 -43 46 step 127 sos 5847
*
33 -5 3 -43 13 step 128 sos 3141
*
28 5 -2 -43 13 step 129 sos 2831
*
28 3 2 -45 13 step 130 sos 2991
*
28 3 -43 45 -32 step 131 sos 5691
*
-4 3 -43 13 32 step 132 sos 3067
*
4 -1 -43 13 28 step 133 sos 2819
*
3 1 -44 13 28 step 134 sos 2899
*
3 -43 44 -31 28 step 135 sos 5539
*
3 -43 13 31 -3 step 136 sos 2997
*
0 -43 13 28 3 step 137 sos 2811
*
-43 43 -30 28 3 step 138 sos 5391
*
-43 13 30 -2 3 step 139 sos 2931
*
-43 13 28 2 1 step 140 sos 2807
*
43 -30 28 2 -42 step 141 sos 5301
*
13 30 -2 2 -42 step 142 sos 2841
*
13 28 2 0 -42 step 143 sos 2721
*
-29 28 2 -42 42 step 144 sos 5157
*
29 -1 2 -42 13 step 145 sos 2779
*
28 1 1 -42 13 step 146 sos 2719
*
28 1 -41 42 -29 step 147 sos 5071
*
-1 1 -41 13 29 step 148 sos 2693
*
1 0 -41 13 28 step 149 sos 2635
*
1 -41 41 -28 28 step 150 sos 4931
*
1 -41 13 28 0 step 151 sos 2635
*
-40 41 -28 28 0 step 152 sos 4849
*
-40 13 28 0 0 step 153 sos 2553
*
40 -27 28 0 -40 step 154 sos 4713
*
13 27 1 0 -40 step 155 sos 2499
*
-27 27 1 -40 40 step 156 sos 4659
*
27 0 1 -40 13 step 157 sos 2499
*
27 0 -39 40 -27 step 158 sos 4579
*
0 0 -39 13 27 step 159 sos 2419
*
0 -39 39 -26 27 step 160 sos 4447
*
0 -39 13 26 1 step 161 sos 2367
*
-39 39 -26 26 1 step 162 sos 4395
*
-39 13 26 0 1 step 163 sos 2367
*
39 -26 26 0 -38 step 164 sos 4317
*
13 26 0 0 -38 step 165 sos 2289
*
-25 26 0 -38 38 step 166 sos 4189
*
25 1 0 -38 13 step 167 sos 2239
*
25 1 -38 38 -25 step 168 sos 4139
*
0 1 -38 13 25 step 169 sos 2239
*
0 -37 38 -25 25 step 170 sos 4063
*
0 -37 13 25 0 step 171 sos 2163
*
-37 37 -24 25 0 step 172 sos 3939
*
-37 13 24 1 0 step 173 sos 2115
*
37 -24 24 1 -37 step 174 sos 3891
*
13 24 0 1 -37 step 175 sos 2115
*
-24 24 0 -36 37 step 176 sos 3817
*
24 0 0 -36 13 step 177 sos 2041
*
24 0 -36 36 -23 step 178 sos 3697
*
1 0 -36 13 23 step 179 sos 1995
*
1 -36 36 -23 23 step 180 sos 3651
*
1 -36 13 23 0 step 181 sos 1995
*
-35 36 -23 23 0 step 182 sos 3579
*
-35 13 23 0 0 step 183 sos 1923
*
35 -22 23 0 -35 step 184 sos 3463
*
13 22 1 0 -35 step 185 sos 1879
*
-22 22 1 -35 35 step 186 sos 3419
*
22 0 1 -35 13 step 187 sos 1879
*
22 0 -34 35 -22 step 188 sos 3349
*
0 0 -34 13 22 step 189 sos 1809
*
0 -34 34 -21 22 step 190 sos 3237
*
0 -34 13 21 1 step 191 sos 1767
*
-34 34 -21 21 1 step 192 sos 3195
*
-34 13 21 0 1 step 193 sos 1767
*
34 -21 21 0 -33 step 194 sos 3127
*
13 21 0 0 -33 step 195 sos 1699
*
-20 21 0 -33 33 step 196 sos 3019
*
20 1 0 -33 13 step 197 sos 1659
*
20 1 -33 33 -20 step 198 sos 2979
*
0 1 -33 13 20 step 199 sos 1659
*
0 -32 33 -20 20 step 200 sos 2913
*
0 -32 13 20 0 step 201 sos 1593
*
-32 32 -19 20 0 step 202 sos 2809
*
-32 13 19 1 0 step 203 sos 1555
*
32 -19 19 1 -32 step 204 sos 2771
*
13 19 0 1 -32 step 205 sos 1555
*
-19 19 0 -31 32 step 206 sos 2707
*
19 0 0 -31 13 step 207 sos 1491
*
19 0 -31 31 -18 step 208 sos 2607
*
1 0 -31 13 18 step 209 sos 1455
*
1 -31 31 -18 18 step 210 sos 2571
*
1 -31 13 18 0 step 211 sos 1455
*
-30 31 -18 18 0 step 212 sos 2509
*
-30 13 18 0 0 step 213 sos 1393
*
30 -17 18 0 -30 step 214 sos 2413
*
13 17 1 0 -30 step 215 sos 1359
*
-17 17 1 -30 30 step 216 sos 2379
*
17 0 1 -30 13 step 217 sos 1359
*
17 0 -29 30 -17 step 218 sos 2319
*
0 0 -29 13 17 step 219 sos 1299
*
0 -29 29 -16 17 step 220 sos 2227
*
0 -29 13 16 1 step 221 sos 1267
*
-29 29 -16 16 1 step 222 sos 2195
*
-29 13 16 0 1 step 223 sos 1267
*
29 -16 16 0 -28 step 224 sos 2137
*
13 16 0 0 -28 step 225 sos 1209
*
-15 16 0 -28 28 step 226 sos 2049
*
15 1 0 -28 13 step 227 sos 1179
*
15 1 -28 28 -15 step 228 sos 2019
*
0 1 -28 13 15 step 229 sos 1179
*
0 -27 28 -15 15 step 230 sos 1963
*
0 -27 13 15 0 step 231 sos 1123
*
-27 27 -14 15 0 step 232 sos 1879
*
-27 13 14 1 0 step 233 sos 1095
*
27 -14 14 1 -27 step 234 sos 1851
*
13 14 0 1 -27 step 235 sos 1095
*
-14 14 0 -26 27 step 236 sos 1797
*
14 0 0 -26 13 step 237 sos 1041
*
14 0 -26 26 -13 step 238 sos 1717
*
1 0 -26 13 13 step 239 sos 1015
*
1 -26 26 -13 13 step 240 sos 1691
*
1 -26 13 13 0 step 241 sos 1015
*
-25 26 -13 13 0 step 242 sos 1639
*
-25 13 13 0 0 step 243 sos 963
*
25 -12 13 0 -25 step 244 sos 1563
*
13 12 1 0 -25 step 245 sos 939
*
-12 12 1 -25 25 step 246 sos 1539
*
12 0 1 -25 13 step 247 sos 939
*
12 0 -24 25 -12 step 248 sos 1489
*
0 0 -24 13 12 step 249 sos 889
*
0 -24 24 -11 12 step 250 sos 1417
*
0 -24 13 11 1 step 251 sos 867
*
-24 24 -11 11 1 step 252 sos 1395
*
-24 13 11 0 1 step 253 sos 867
*
24 -11 11 0 -23 step 254 sos 1347
*
13 11 0 0 -23 step 255 sos 819
*
-10 11 0 -23 23 step 256 sos 1279
*
10 1 0 -23 13 step 257 sos 799
*
10 1 -23 23 -10 step 258 sos 1259
*
0 1 -23 13 10 step 259 sos 799
*
0 -22 23 -10 10 step 260 sos 1213
*
0 -22 13 10 0 step 261 sos 753
*
-22 22 -9 10 0 step 262 sos 1149
*
-22 13 9 1 0 step 263 sos 735
*
22 -9 9 1 -22 step 264 sos 1131
*
13 9 0 1 -22 step 265 sos 735
*
-9 9 0 -21 22 step 266 sos 1087
*
9 0 0 -21 13 step 267 sos 691
*
9 0 -21 21 -8 step 268 sos 1027
*
1 0 -21 13 8 step 269 sos 675
*
1 -21 21 -8 8 step 270 sos 1011
*
1 -21 13 8 0 step 271 sos 675
*
-20 21 -8 8 0 step 272 sos 969
*
-20 13 8 0 0 step 273 sos 633
*
20 -7 8 0 -20 step 274 sos 913
*
13 7 1 0 -20 step 275 sos 619
*
-7 7 1 -20 20 step 276 sos 899
*
7 0 1 -20 13 step 277 sos 619
*
7 0 -19 20 -7 step 278 sos 859
*
0 0 -19 13 7 step 279 sos 579
*
0 -19 19 -6 7 step 280 sos 807
*
0 -19 13 6 1 step 281 sos 567
*
-19 19 -6 6 1 step 282 sos 795
*
-19 13 6 0 1 step 283 sos 567
*
19 -6 6 0 -18 step 284 sos 757
*
13 6 0 0 -18 step 285 sos 529
*
-5 6 0 -18 18 step 286 sos 709
*
5 1 0 -18 13 step 287 sos 519
*
5 1 -18 18 -5 step 288 sos 699
*
0 1 -18 13 5 step 289 sos 519
*
0 -17 18 -5 5 step 290 sos 663
*
0 -17 13 5 0 step 291 sos 483
*
-17 17 -4 5 0 step 292 sos 619
*
-17 13 4 1 0 step 293 sos 475
*
17 -4 4 1 -17 step 294 sos 611
*
13 4 0 1 -17 step 295 sos 475
*
-4 4 0 -16 17 step 296 sos 577
*
4 0 0 -16 13 step 297 sos 441
*
4 0 -16 16 -3 step 298 sos 537
*
1 0 -16 13 3 step 299 sos 435
*
1 -16 16 -3 3 step 300 sos 531
*
1 -16 13 3 0 step 301 sos 435
*
-15 16 -3 3 0 step 302 sos 499
*
-15 13 3 0 0 step 303 sos 403
*
15 -2 3 0 -15 step 304 sos 463
*
13 2 1 0 -15 step 305 sos 399
*
-2 2 1 -15 15 step 306 sos 459
*
2 0 1 -15 13 step 307 sos 399
*
2 0 -14 15 -2 step 308 sos 429
*
0 0 -14 13 2 step 309 sos 369
*
0 -14 14 -1 2 step 310 sos 397
*
0 -14 13 1 1 step 311 sos 367
*
-14 14 -1 1 1 step 312 sos 395
*
14 0 -1 1 -13 step 313 sos 367
*
1 0 -1 -12 13 step 314 sos 315
*
1 0 -13 12 1 step 315 sos 315
*
1 -13 13 -1 1 step 316 sos 341
*
-12 13 0 -1 1 step 317 sos 315
*
12 1 0 -1 -11 step 318 sos 267
*
1 1 0 -12 11 step 319 sos 267
*
1 1 -12 12 -1 step 320 sos 291
*
1 -11 12 0 -1 step 321 sos 267
*
-10 11 1 0 -1 step 322 sos 223
*
10 1 1 0 -11 step 323 sos 223
*
-1 1 1 -11 11 step 324 sos 245
*
1 0 1 -11 10 step 325 sos 223
*
1 0 -10 11 -1 step 326 sos 223
*
0 0 -10 10 1 step 327 sos 201
*
0 -10 10 0 1 step 328 sos 201
*
-10 10 0 0 1 step 329 sos 201
*
10 0 0 0 -9 step 330 sos 181
*
1 0 0 -9 9 step 331 sos 163
*
1 0 -9 9 0 step 332 sos 163
*
1 -9 9 0 0 step 333 sos 163
*
-8 9 0 0 0 step 334 sos 145
*
8 1 0 0 -8 step 335 sos 129
*
0 1 0 -8 8 step 336 sos 129
*
0 1 -8 8 0 step 337 sos 129
*
0 -7 8 0 0 step 338 sos 113
*
-7 7 1 0 0 step 339 sos 99
*
7 0 1 0 -7 step 340 sos 99
*
0 0 1 -7 7 step 341 sos 99
*
0 0 -6 7 0 step 342 sos 85
*
0 -6 6 1 0 step 343 sos 73
*
-6 6 0 1 0 step 344 sos 73
*
6 0 0 1 -6 step 345 sos 73
*
0 0 0 -5 6 step 346 sos 61
*
0 0 -5 5 1 step 347 sos 51
*
0 -5 5 0 1 step 348 sos 51
*
-5 5 0 0 1 step 349 sos 51
*
5 0 0 0 -4 step 350 sos 41
*
1 0 0 -4 4 step 351 sos 33
*
1 0 -4 4 0 step 352 sos 33
*
1 -4 4 0 0 step 353 sos 33
*
-3 4 0 0 0 step 354 sos 25
*
3 1 0 0 -3 step 355 sos 19
*
0 1 0 -3 3 step 356 sos 19
*
0 1 -3 3 0 step 357 sos 19
*
0 -2 3 0 0 step 358 sos 13
*
-2 2 1 0 0 step 359 sos 9
*
2 0 1 0 -2 step 360 sos 9
*
0 0 1 -2 2 step 361 sos 9
*
0 0 -1 2 0 step 362 sos 5
*
0 -1 1 1 0 step 363 sos 3
*
-1 1 0 1 0 step 364 sos 3
*
1 0 0 1 -1 step 365 sos 3
*
0 0 0 0 1 step 366 sos 1
Tue Jun 26 12:07:21 PDT 2018
0
On
sample:
jagy@phobeusjunior:~$ ./Pentagon 1 1 -3 1 1
Tue Jun 26 18:00:48 PDT 2018
1 1 -3 1 1 sum 1 sos 13
*
1 -2 3 -2 1 step 1 sos 19
*
-1 2 1 -2 1 step 2 sos 11
*
1 1 1 -2 0 step 3 sos 7
*
1 1 -1 2 -2 step 4 sos 11
*
1 0 1 1 -2 step 5 sos 7
*
-1 0 1 -1 2 step 6 sos 7
*
-1 0 0 1 1 step 7 sos 3
*
1 -1 0 1 0 step 8 sos 3
*
0 1 -1 1 0 step 9 sos 3
*
0 0 1 0 0 step 10 sos 1
began 1 1 -3 1 1 sum 1 sos 13
Tue Jun 26 18:00:48 PDT 2018
jagy@phobeusjunior:~$
==============================================================
#include <iostream>
#include <stdlib.h>
#include <fstream>
#include <strstream>
#include <list>
#include <set>
#include <math.h>
#include <iomanip>
#include <string>
#include <algorithm>
#include <iterator>
using namespace std;
// g++ -o Pentagon Pentagon.cc -lm
int main(int argc, char *argv[])
{
if ( argc != 6) cout << "Usage: ./Pentagon w x y z t " << endl;
else {
int w,x,y,z,t ;
w = atoi(argv[1]);
x = atoi(argv[2]);
y = atoi(argv[3]);
z = atoi(argv[4]);
t = atoi(argv[5]);
int stepbound = 10000;
system("date");
int sum = w + x + y + z + t;
int sos = w * w + x * x + y * y + z * z + t * t ;
int oldw = w; int oldx = x; int oldy = y; int oldz = z; int oldt = t; int oldsos = sos;
int goon = sum > 0;
int steps = 0;
int smallsquare;
cout << setw(5) << w << setw(5) << x << setw(5) << y << setw(5) << z << setw(5) << t << " sum " << sum << " sos " << sos << endl;
while( goon)
{
int w2, x2,y2,z2,t2;
if( w >= 0) w2 = 1000000000;
if( w < 0) w2 = w * w + (t + w) * ( t + w) + ( x + w ) * ( x + w ) + y * y + z * z;
smallsquare = w2;
int use = 1;
if( x >= 0) x2 = 1000000000;
if( x < 0) x2 = x * x + (x + w) * ( x + w) + ( x + y ) * ( x + y ) + z * z + t * t;
if( smallsquare > x2)
{
smallsquare = x2; use = 2;
}
if( y >= 0) y2 = 1000000000;
if( y < 0) y2 = y * y + (y + z) * ( y + z) + ( x + y ) * ( x + y ) + w * w + t * t;
if( smallsquare > y2)
{
smallsquare = y2; use = 3;
}
if( z >= 0) z2 = 1000000000;
if( z < 0) z2 = z * z + (y + z) * ( y + z) + ( z + t ) * ( z + t ) + x * x + w * w;
if( smallsquare > z2)
{
smallsquare = z2; use = 4;
}
if( t >= 0) t2 = 1000000000;
if( t < 0) t2 = t * t + (t + z) * ( t + z) + ( w + t ) * ( w + t ) + x * x + y * y;
if( smallsquare > t2)
{
smallsquare = t2; use = 5;
}
// w x y z t
// cout << setw(8) << w2 << setw(8) << x2 << setw(8) << y2 << setw(8) << z2 << setw(8) << t2 << " smallsquare " << smallsquare << endl;
if( use == 1 ) { x += w; t += w; w *= -1; cout << " * " << endl;}
else if ( use == 2) {w += x; y += x; x *= -1; cout << " * " << endl; }
else if ( use == 3) { x += y; z += y; y *= -1; cout << " * " << endl;}
else if ( use == 4) { y += z; t += z; z *= -1; cout << " * " << endl;}
else if ( use == 5) { w += t; z += t; t *= -1; cout << " * " << endl;}
// cout << " use " << use << endl;
cout << setw(5) << w << setw(5) << x << setw(5) << y << setw(5) << z << setw(5) << t << " step " << 1 + steps << " sos " << smallsquare << endl;
goon = goon && ( w < 0 || x < 0 || y < 0 || z < 0 || t < 0 );
goon = goon && ( abs(w) <= 31622 && abs(x) <= 31622 && abs(y) <= 31622 && abs(z) <= 31622 && abs(t) <= 31622 );
++steps;
goon = goon && steps < stepbound;
} // while goon
if( w < 0 || x < 0 || y < 0 || z < 0 || t < 0 ) cout << endl << " did not finish, running with quintuple at step 10000 " << endl ;
cout << endl << " began " << setw(4) << oldw << setw(4) << oldx << setw(4) << oldy << setw(4) << oldz << setw(4) << oldt << " sum " << sum << " sos " << oldsos << endl;
system("date");
} // else argc
return 0;
}
//
// g++ -o Pentagon Pentagon.cc -lm
I wrote a greedy algorithm for a square. It looks at the negative entries that can be flipped, and uses the one the minimizes the sum of all four squares. In contrast with the triangle, the SOS occasionally increases, so this does not directly lead to a proof.
Interesting: with four vertices, it appears that one of the next three steps has a strictly smaller sum of squares. If true, that sort of thing does lead to a proof, for the case of four vertices.
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Here is one with larger numbers. As you can see, sometimes the sos increases for a few steps before shrinking again.