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 Perplexus Cryptarithm Puzzle (Posted on 2007-03-31)
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 No Rating

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 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:

`6255028791128 * 27837284`
`4399732605532 * 36216328`
`4177319682219 * 46906195`
`1499246307746 * 73653468`

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

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