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Perplexus Cryptarithm Puzzle (Posted on 2007-03-31) Difficulty: 3 of 5
The cryptarithm FLOOBLE + PUZZLE = PERPLEX has no solutions. But if the word 'PUZZLE' is included twice, then there is a solution.
How many more times can the word 'PUZZLE' appear in the addition and the cryptarithm still have a solution?

See The Solution Submitted by Brian Smith    
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Solution computer solution | Comment 1 of 4

First, we assume neither F nor P can be zero.

Then, if there were 10 occurrences of PUZZLE, you'd get a 7-digit number that would have at least F added to the leading P (and possibly a carry), making the sum a number that's larger than a 7-digit one beginning with P. So any more than 9 occurrences of PUZZLE are ruled out. The program checks one through nine occurrences of PUZZLE:

DEFDBL A-Z
FOR n = 1 TO 9

 PRINT n
 
 FOR f = 1 TO 9
  PRINT f;
  taken(f) = 1
 FOR l = 0 TO 9
  IF taken(l) = 0 THEN
    taken(l) = 1

 FOR o = 0 TO 9
  IF taken(o) = 0 THEN
    taken(o) = 1

 FOR b = 0 TO 9
  IF taken(b) = 0 THEN
    taken(b) = 1

 FOR e = 0 TO 9
  IF taken(e) = 0 THEN
    taken(e) = 1

    flooble = f * 1000000 + l * 100010 + o * 11000 + b * 100 + e

 FOR p = 1 TO 9
  IF taken(p) = 0 THEN
    taken(p) = 1
    p0 = p * 1001000 + e * 100010 + l * 100

 FOR u = 0 TO 9
  IF taken(u) = 0 THEN
    taken(u) = 1

 FOR z = 0 TO 9
  IF taken(z) = 0 THEN
    taken(z) = 1

  puzzle = n * (p * 100000 + u * 10000 + z * 1100 + l * 10 + e)

 FOR r = 0 TO 9
  IF taken(r) = 0 THEN
    taken(r) = 1
    p1 = p0 + r * 10000

 FOR x = 0 TO 9
  IF taken(x) = 0 THEN
    taken(x) = 1
    perplex = p1 + x

    IF flooble + puzzle = perplex THEN
      PRINT
      PRINT flooble: PRINT puzzle / n; "*"; n
      PRINT perplex: PRINT
    END IF

    taken(x) = 0
  END IF
 NEXT x

    taken(r) = 0
  END IF
 NEXT r

    taken(z) = 0
  END IF
 NEXT z

    taken(u) = 0
  END IF
 NEXT u

    taken(p) = 0
  END IF
 NEXT p

    taken(e) = 0
  END IF
 NEXT e

    taken(b) = 0
  END IF
 NEXT b

    taken(o) = 0
  END IF
 NEXT o

    taken(l) = 0
  END IF
 NEXT l
  taken(f) = 0
 NEXT
 PRINT
NEXT

The solutions have 2, 3, 4 or 7 occurrences of PUZZLE, the number of times marked after the * in the following:

6255028
791128 * 2
7837284
4399732
605532 * 3
6216328
4177319
682219 * 4
6906195

1499246
307746 * 7
3653468

  Posted by Charlie on 2007-03-31 16:25:48
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