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The 10,000 Puzzle Square

ANSWER: RHONE VALLEY

Author: Charles Steinhardt

The 10000 individual sets of clues can be uniquely solved with the additional constraint that every number from to 9999 appears exactly once. The final grid is available here.

As suggested by the individual puzzles, it is useful to check which entries are triangular numbers. There are 141 of them, and this yields the following grid:




















              XXXXX  X   X                   XXXXX     X   X
                X    X   X                     X       XX  X
                X    X   X                     X       XXX X
                X    X X X                     X       X X X
                X    X X X                     X       X XXX
                X    X X X                     X       X  XX
                X     X X                    XXXXX     X   X









































                                        XXX              XXX            XXXXX        X   X   XXXX
                                        X  X             X  X             X          XX XX   X
                                        X  X             X  X             X          X X X   X
                                        XXX              XXX              X          X X X   XXX
                                        X                X  X             X          X   X   X
                                        X                X  X             X          X   X   X
                                        X                X  X           XXXXX        X   X   XXXX

























Doing the same thing for the 409 twin primes which appear in the grid (205 pairs, but 5 appears in two pairs) yields:


























          XXXXXXX                  XXXX                     XXXXX                  XXXXX
          XXXXXXX                  XXXX                     XXXXXX                XXXXXXX
          XX   XX                   XX                      XX  XXX               XX   XX
          XX                        XX                      XX   XX               XX   XX
          XX                        XX                      XX  XXX               XX   XX
          XXXX                      XX                      XXXXXXX               XX   XX
          XXXX                      XX                      XXXXXX                XX   XX
          XX                        XX                      XX  XXX               XX   XX
          XX                        XX                      XX   XX               XX   XX
          XX                        XX                      XX  XXX               XX   XX
          XX                       XXXX                     XXXXXX                XXXXXXX
          XX                       XXXX                     XXXXX                  XXXXX







 XX    XX             XXX                           XXXXX                    XXXXX            XXXX
 XXX   XX            XXXXX                          XXXXX                    XXXXX            XXXX
 XXXX  XX           XX   XX                         XX                       XX                XX
 XXXXX XX           XX   XX                         XX                       XX                XX
 XX XXXXX           XX   XX                         XX                       XX                XX
 XX  XXXX           XXXXXXX                         XX                       XX                XX
 XX   XXX           XXXXXXX                         XX                       XX                XX
 XX    XX           XX   XX                         XX                       XX                XX
 XX    XX           XX   XX                         XX                       XX                XX
 XX    XX           XX   XX                         XX                       XX                XX
 XX    XX           XX   XX                         XXXXX                    XXXXX            XXXX
 XX    XX           XX   XX                         XXXXX                    XXXXX            XXXX











































There are 20 Fibonacci numbers in the grid, taking f(0) = 0. Doing the same produces a square:
























                XXXXXX
                X    X
                X    X
                X    X
                X    X
                XXXXXX






































































Finally, taking the 100 square numbers which appear produces the answer:





















              XX    X X                 XXX                    XXX
              X X   X X                 X X                    X
              XX    XXX                 X X        XXX         XX
              X X   X X                 X X        X X         X
              X X   X X                 XXX        X X         XXX


































































          X X                X                 X               X                XXX             X X
          X X               X X                X               X                X               XXX
          X X               XXX                X               X                XX               X
           X                X X                X               X                X                X
           X                X X                XXX             XXX              XXX              X




The final answer is RHONE VALLEY.