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 Adjacently Greater Than One (Posted on 2010-01-01)
Determine all possible values of a 6-digit positive integer N having 6 different digits from 0 to 9, with the first digit being nonzero, such that:

(i) Either, the digits at the even positions are even and the digits at the odd positions are odd – or, the digits at the even positions are odd and the digits at the odd positions are even. Zero is regarded as an even digit.

(ii) The absolute difference between two adjacent digits is always greater than one.

(iii) The 2-digit number formed by the first and second digit as well as the 2-digit number formed by the third and fourth digit is divisible by the 2-digit number formed by the fifth and sixth digit.

 No Solution Yet Submitted by K Sengupta Rating: 2.5000 (2 votes)

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 A computer program listing and spoiler Comment 6 of 6 |

DEFDBL A-Z
FOR a = 1 TO 9
used(a) = 1
FOR b = 0 TO 9
IF used(b) = 0 AND (a + b) MOD 2 = 1 AND ABS(a - b) > 1 THEN
used(b) = 1
FOR c = 0 TO 9
IF used(c) = 0 AND (c + b) MOD 2 = 1 AND ABS(b - c) > 1 THEN
used(c) = 1
FOR d = 0 TO 9
IF used(d) = 0 AND (c + d) MOD 2 = 1 AND ABS(c - d) > 1 THEN
used(d) = 1
FOR e = 0 TO 9
IF used(e) = 0 AND (d + e) MOD 2 = 1 AND ABS(d - e) > 1 THEN
used(e) = 1
FOR f = 0 TO 9
IF used(f) = 0 AND (e + f) MOD 2 = 1 AND ABS(e - f) > 1 THEN
used(f) = 1

ab = 10 * a + b
cd = 10 * c + d
ef = 10 * e + f
IF ab MOD ef = 0 AND cd MOD ef = 0 THEN
PRINT a; b; c; d; e; f
END IF

used(f) = 0
END IF
NEXT
used(e) = 0
END IF
NEXT
used(d) = 0
END IF
NEXT
used(c) = 0
END IF
NEXT
used(b) = 0
END IF
NEXT
used(a) = 0
NEXT

finds the previously given

`4  9  6  3  0  76  9  2  7  0  38  1  6  3  0  98  1  6  9  0  39  0  3  6  1  8`

I note that KS did not specifically forbid leading zeros from the 2-digit numbers (whereas he had from the 6-digit number), so there could be a difference of opinion about the validity of the first four solutions.

 Posted by Charlie on 2010-01-01 14:01:25

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