6. Integers can be written in bases other than the usual base 10 . For example, the notation $(235)_{7}$ stands for the base 7 representation of the integer $2 \times 7^{2}+3 \times 7+5$ (which equals 124 in base 10). In general, if $x, y$ and $z$ are integers between 0 and $b-1$, inclusive, then $(x y z)_{b}=x b^{2}+y b+z$.
Find all triples $(x, y, z)$ for which $(x y z)_{10}=2(x y z)_{7}$, where each of $x, y$ and $z$ comes from the list $1,2,3,4,5,6$.
6. Since we are told that $(x y z)_{b}=x b^{2}+y b+z$, then $(x y z)_{10}=10^{2} x+10 y+z=100 x+10 y+z$ and $(x y z)_{7}=7^{2} x+7 y+z=49 x+7 y+z$.
From the given information,
$$
\begin{aligned}
(x y z)_{10} & =2(x y z)_{7} \\
100 x+10 y+z & =2(49 x+7 y+z) \\
100 x+10 y+z & =98 x+14 y+2 z \\
2 x & =4 y+z
\end{aligned}
$$

Since the left side of this equation is an even integer $(2 x)$, then the right side must also be an even integer.
Since $4 y$ is an even integer, then for $4 y+z$ to be an even integer, it must be the case that $z$ is an even integer.
This gives us three possibilities: $z=2, z=4$ and $z=6$.
Case 1: $z=2$
Here, $2 x=4 y+2$ or $x=2 y+1$.
We try the possible values for $y$ :
- $y=1$ gives $x=2(1)+1=3$; this gives the triple $(x, y, z)=(3,1,2)$
- $y=2$ gives $x=2(2)+1=5$; this gives the triple $(x, y, z)=(5,2,2)$
- If $y$ is at least 3 , then $x=2 y+1$ is at least 7 , which is impossible.

Therefore, there are two triples that work when $z=2$.
Case 2: $z=4$
Here, $2 x=4 y+4$ or $x=2 y+2$.
We try the possible values for $y$ :
- $y=1$ gives $x=2(1)+2=4$; this gives the triple $(x, y, z)=(4,1,4)$
- $y=2$ gives $x=2(2)+2=6$; this gives the triple $(x, y, z)=(6,2,4)$
- If $y$ is at least 3 , then $x=2 y+2$ is at least 8 , which is impossible.

Therefore, there are two triples that work when $z=4$.
Case 3: $z=6$
Here, $2 x=4 y+6$ or $x=2 y+3$.
We try the possible values for $y$ :
- $y=1$ gives $x=2(1)+3=5$; this gives the triple $(x, y, z)=(5,1,6)$
- If $y$ is at least 2 , then $x=2 y+3$ is at least 7 , which is impossible.

Therefore, there is one triple that works when $z=6$.
Finally, the triples that satisfy the equation are $(x, y, z)=(3,1,2),(5,2,2),(4,1,4),(6,2,4),(5,1,6)$.
ANSWER: $(x, y, z)=(3,1,2),(5,2,2),(4,1,4),(6,2,4),(5,1,6)$