Challenge Level

Bob has sent us one solution:

$6\quad\quad 4\quad\quad 9$

$\ 5\quad 7\quad 8\quad 3$

$\quad 2\quad \quad 1$

George used algebra to tackle the
problem:

Call the numbers in the different dots $a, b, \ldots, i$, so
the W looks like

$a\quad\quad e\quad\quad i$

$\ b\quad d\quad f\quad h$

$\quad c\quad \quad g$

Then $a+b+c=13$, $c+d+e$=13, $e+f+g=13$, and $g+h+i=13$.
Adding these equations together, we get $a+b+2c+d+2e+f+2g+h+i=52$.
But we know that the letters are just the numbers $1, 2, \ldots, 9$
in some order, so $a+b+c+d+e+f+g+h+i=45$, so $c+e+g=7$. Now $c$,
$e$ and $g$ are all different, so the only possibility is that they
are 1, 2 and 4 in some order. If $c$=1 and $e=2$, then we'd have
$d=10$, and that's not possible. So of the three dots $c$, $e$ and
$g$, the 1 and the 2 must be in ones that aren't next to each
other. So we could have $c=1$, $e=4$ and $g=2$. Now we know that
$d=8$ and $f=7$, and $a+b=12$ and $h+i=11$, and $a$, $b$, $h$ and
$i$ are $3, 5, 6$ and $9$ in some order. So $a$ and $b$ are $3$ and
$9$, and $h$ and $i$ are $5$ and $6$, but it could be either way
round. So there are four different solutions, not counting
reflections. One is shown below, and you get the others by
switching the two numbers on one end or the other (or both!).

$3\quad\quad 4\quad\quad 5$

$\ 9\quad 8\quad 7\quad 6$

$\quad 1\quad \quad 2$

We can do something very similar if we want the sums to be
$14$. This time, we get $c+e+g+45=4\times 14=56$, so $c+e+g=11$.
There are several possibilities, which we'll consider
separately.

Case $1$: ($1$,$2$,$8$). As above, we can't have $1$ and $2$
next to each other, so the only possibility (apart from reflection)
is $c=1$, $e=8$, $g=2$. So $d=5$ and $f=4$, and $a+b=13$ and
$h+i=12$, with $\{a,b,h,i\}=\{3,6,7,9\}$. So we get
$\{a,b\}=\{6,7\}$ and $\{h,i\}=\{3,9\}$.

Case $2$: ($1$,$3$,$7$). This time, we can't have $1$ and $3$
next to each other either (as $14-1-3=10$), so the only possibility
(again not counting reflection) is $c=1$, $d=6$, $e=7$, $f=4$,
$g=3$, and we have $a+b=13$, $h+i=11$, with
$\{a,b,h,i\}=\{2,5,8,9\}$. So $\{a,b\}=\{5,8\}$ and
$\{h,i\}=\{2,9\}$, and there are four possibilities (by swapping
ends), of which one is

$5\quad\quad 7\quad\quad 2$

$\ 8\quad 6\quad 4\quad 9$

$\quad 1\quad \quad 3$

Case $3$: ($1$,$4$,$6$). Now we can't have $4$ and $6$ next to
each other, or we'd need another $4$ in the gap between them (to
make $14$), so the only possibility is $c=4$, $d=9$, $e=1$, $f=7$,
$g=6$, $a+b=10$, $h+i=8$, and so we must have $\{a,b\}=\{2,8\}$ and
$\{h,i\}=\{3,5\}$.

Case $4$: ($2$,$3$,$6$). This time we can't have $2$ and $6$
next to each other, so the only possibilitiy is $c=2$, $d=9$,
$e=3$, $f=5$, $g=6$, $a+b=12$, $h+i=8$, and so we get
$\{a,b\}=\{4,8\}$ and $\{h,i\}=\{1,7\}$.

Case $5$: ($2$,$4$,$5$). Now we can't have $4$ and $5$ next to
each other, so the only possibility is $c=5$, $d=7$, $e=2$, $f=8$,
$g=4$, $a+b=9$, $h+i=10$, and we get the only solution as
$\{a,b\}=\{3,6\}$, $\{h,i\}=\{1,9\}$.