# 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
• 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? Jun 4, 2016 at 9:43
• @Aurey I have tested my clues. Yes, those are the digits. Jun 4, 2016 at 9:44
• Ah, whoops. I had mod 6 instead of mod 4. My bad. Jun 4, 2016 at 16:14

## 2 Answers

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 $733489$

$2x_4=x_5 \to x_4\lt5$
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