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Adjacent Square Sum (Posted on 2013-05-20) Difficulty: 2 of 5
Arrange the numbers 1 to 15 inclusively in a straight line in such a way that any two adjacent numbers add up to a perfect square.

*** Disregard reflections.

No Solution Yet Submitted by K Sengupta    
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Solution P 'n P and then ... | Comment 2 of 3 |
After playing around I decided to begin with 1. I chose 15 to make my next square. That set up a chain whereby I could only choose one number at a time tomake my next square.  Eventually I arrived at 9 with 8 unplaced.  Then came realisation, 8 could be the beginning of the string.


Now KS did say to ignore reflections.  I wondered about any other possiblities.  This program listing gives just:
8 1 15 10 6 3 13 12 4 5 11 14 2 7 9 and
9 7 2 14 11 5 4 12 13 3 6 10 15 1 8.

DIM st(15)
DIM used(15)
FOR a = 1 TO 15
 IF used(a) = 0 THEN
  used(a) = 1
  st(1) = a
  FOR b = 1 TO 15
   IF used(b) = 0 THEN
    used(b) = 1
    st(2) = b
    IF SQR(a + b) = INT(SQR(a + b)) THEN
     FOR c = 1 TO 15
      IF used(c) = 0 THEN
       used(c) = 1
       st(3) = c
       IF SQR(b + c) = INT(SQR(b + c)) THEN
        FOR d = 1 TO 15
         IF used(d) = 0 THEN
          used(d) = 1
          st(4) = d
          IF SQR(c + d) = INT(SQR(c + d)) THEN
           FOR e = 1 TO 15
            IF used(e) = 0 THEN
             used(e) = 1
             st(5) = e
             IF SQR(d + e) = INT(SQR(d + e)) THEN
              FOR f = 1 TO 15
               IF used(f) = 0 THEN
                used(f) = 1
                st(6) = f
                IF SQR(e + f) = INT(SQR(e + f)) THEN
                 FOR g = 1 TO 15
                  IF used(g) = 0 THEN
                   used(g) = 1
                   st(7) = g
                   IF SQR(f + g) = INT(SQR(f + g)) THEN
                    FOR h = 1 TO 15
                     IF used(h) = 0 THEN
                      used(h) = 1
                      st(8) = h
                      IF SQR(g + h) = INT(SQR(g + h)) THEN
                       FOR i = 1 TO 15
                        IF used(i) = 0 THEN
                         used(i) = 1
                         st(9) = i
                         IF SQR(h + i) = INT(SQR(h + i)) THEN
                          FOR j = 1 TO 15
                           IF used(j) = 0 THEN
                            used(j) = 1
                            st(10) = j
                            IF SQR(i + j) = INT(SQR(i + j)) THEN
                             FOR k = 1 TO 15
                              IF used(k) = 0 THEN
                               used(k) = 1
                               st(11) = k
                               IF SQR(j + k) = INT(SQR(j + k)) THEN
                                FOR l = 1 TO 15
                                 IF used(l) = 0 THEN
                                  used(l) = 1
                                  st(12) = l
                                  IF SQR(k + l) = INT(SQR(k + l)) THEN
                                   FOR m = 1 TO 15
                                    IF used(m) = 0 THEN
                                     used(m) = 1
                                     st(13) = m
                                     IF SQR(l + m) = INT(SQR(l + m)) THEN
                                      FOR n = 1 TO 15
                                       IF used(n) = 0 THEN
                                        used(n) = 1
                                        st(14) = n
                                        IF SQR(m + n) = INT(SQR(m + n)) THEN
                                         FOR o = 1 TO 15
                                          IF used(o) = 0 THEN
                                           used(o) = 1
                                           st(15) = o
                                           IF SQR(n + o) = INT(SQR(n + o)) THEN
                                            FOR z = 1 TO 15
                                             PRINT st(z);
                                            NEXT: PRINT
                                           END IF
                                           used(o) = 0
                                          END IF
                                         NEXT

                                        END IF
                                        used(n) = 0
                                       END IF
                                      NEXT

                                     END IF
                                     used(m) = 0
                                    END IF
                                   NEXT

                                  END IF
                                  used(l) = 0
                                 END IF
                                NEXT

                               END IF
                               used(k) = 0
                              END IF
                             NEXT

                            END IF
                            used(j) = 0
                           END IF
                          NEXT

                         END IF
                         used(i) = 0
                        END IF
                       NEXT

                      END IF
                      used(h) = 0
                     END IF
                    NEXT

                   END IF
                   used(g) = 0
                  END IF
                 NEXT

                END IF
                used(f) = 0
               END IF
              NEXT

             END IF
             used(e) = 0
            END IF
           NEXT

          END IF
          used(d) = 0
         END IF
        NEXT

       END IF
       used(c) = 0
      END IF
     NEXT

    END IF
    used(b) = 0
   END IF
  NEXT
  used(a) = 0
 END IF
NEXT


  Posted by brianjn on 2013-05-21 02:18:24
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