the bottom right of the nonet is part of a 3-cage of 6; it can therefore only contain 1, 2 or 3. If it contained 1 or 2, the other cell would have to contain 11 or 10 respectively; this is impossible. It must, therefore, contain 3, and the other cell 9. Complements With 6-cell, 7-cell or 8-cell cages, correlating the combinations with their 3-cell, 2-cell, or 1-cell complements usually simplifies things. The table for 6 cell cages is the complement of the 3 cell table adding up to 45 minus the listed value; similarly, the 7 cell table complements the 2 cell table. An 8-cell cage is of course missing only one digit (45 minus the sum of the cage). For example, the complement of a 7-cell cage totalling 41 is a 2-cell cage totalling 4 (because 9–7=2 and 45–41=4). As a 2-cell cage totalling 4 can contain only 1 and 3, we deduce that a 7-cell cage totalling 41 contains neither 1 nor 3. Cage total tables The following tables list the possible combinations for various sums. 1 cell 1: 1 2: 2 3: 3 4: 4 5: 5 6: 6 7: 7 8: 8 9: 9 2 cells 3: 12 4: 13 5: 14 23 6: 15 24 7: 16 25 34 8: 17 26 35 9: 18 27 36 45 10: 19 28 37 46 11: 29 38 47 56 12: 39 48 57 13: 49 58 67 14: 59 68 15: 69 78 16: 79 17: 89 3 cells 6: 123 7: 124 8: 125 134 9: 126 135 234 10: 127 136 145 235 11: 128 137 146 236 245 12: 129 138 147 156 237 246 345 13: 139 148 157 238 247 256 346 14: 149 158 167 239 248 257 347 356 15: 159 168 249 258 267 348 357 456 16: 169 178 259 268 349 358 367 457 17: 179 269 278 359 368 458 467 18: 189 279 369 378 459 468 567 19: 289 379 469 478 568 20: 389 479 569 578 21: 489 579 678 22: 589 679 23: 689 24: 789 4 cells 10: 1234 11: 1235 12: 1236 1245 13: 1237 1246 1345 14: 1238 1247 1256 1346 2345 15: 1239 1248 1257 1347 1356 2346 16: 1249 1258 1267 1348 1357 1456 2347 2356 17: 1259 1268 1349 1358 1367 1457 2348