4
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Feeling Puzzled

This is about finding a base-10 6-digit (numeric) password. The clues given are enough. Note that in the clues, $x_1...x_6\in\{0,\dots,9\}$. $x$ is the code.

Clues

  • $x_1 > 6$
  • $x\mod4=1$
  • $x_1+x_2+x_3=x_4+x_6$
  • $x_1-x_4=x_3$
  • $2x_4=x_5$
  • $x_3<x_4$
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  • $\begingroup$ A brute force calculation isn't giving any results. Just to clarify, the first digit is the one that corresponds to the hundred thousand, and the sixth is the ones digit, right? $\endgroup$ – Aurey Jun 4 '16 at 9:43
  • $\begingroup$ @Aurey I have tested my clues. Yes, those are the digits. $\endgroup$ – EKons Jun 4 '16 at 9:44
  • $\begingroup$ Ah, whoops. I had mod 6 instead of mod 4. My bad. $\endgroup$ – Aurey Jun 4 '16 at 16:14
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The 6- digit password is:

$733489$

Reason

$x_1 > 6$ ----> $7, 8, 9$.

$x \mod 4 = 1$ ----> $x_6 = 1, 5, 9$.

$x_1+x_2+x_3=x_4+x_6$

$x_1 - x_4 = x_3$ ----> $x_1=7$, $x_4=4$ and $x_3=3$ (all other combinations exceed 4 which is the maximum value of $x_4$. Hence we reject them)

$2x_4 = x_5$ ----> $x_5 = 0, 2, 4, 6, 8$ and $x_4 = 0, 1, 2, 3, 4$.

$x_3<x_4$ ----> $x_3 = 0, 1, 2, 3$. From $x_1+x_2+x_3=x_4+x_6$, we obtain $7+x_2+3=4+x_6$. Since $x_2$ is a positive integer, $4+x_6 > 7+3 (=10)$ So $x_6 > 6$ ----> 9.

$2x_4 = x_5$ ----> $x_5 = 8$.

$7+x_2+3=4+9$

$x_2=3$

Hence the number $733489$

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1
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$2x_4=x_5 \to x_4\lt5$
With $x_1=x_3+x_4$ along with $x_3\lt x_4$ and $x_1\gt6$ we get $x_1=7, x_3=3, x_4=4, x_5=8$
$x_5=8$ and $x \mod4\equiv1$ gives $x_6=1,5,9$
$x_1+x_2+x_3=x_4+x_6$ reduces to $6+x_2=x_6$ so $x_2=3,x_6=9$.
Password: 733489

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