`x -- x -- x -- x`
`| | | |`
`x -- x -- x -- x`
`| | | |`
`x -- x -- x -- x`
` | |`
` x -- x``
`

`The conditions are that one edge cannot be used and one other one
must`
`be used twice. Note that it is impossible to find a path using
every`
`edge exactly once since there are four vertices of degree 3. All
we need`
`to do is remove one edge and "double" another and end up with only
two`
`vertices of odd degree. There are a few ways to do this (most,
if not`
`all, having multiple solutions even then), such as:``
`

`20 -- 1,19 -- 2,8 -- 9`
`| |
| |`
`17,21 -- 4,18 -- 3,7 -- 10`
`| |
| |`
`16 -- 5,15 -- 6,12 -- 11`
` |
|`
` 14
-- 13``
`

`If you have to make a series of only 19 jumps (you asked for 20),
then`
`the series of jumps is not unique but the starred square which
must be`
`unused is unique as is the pair of starting and ending square for
the`
`king. In that case, we are just removing an edge so we must remove
the`
`only edge which connects two odd vertices. This is the middle edge
of`
`the top row. There is still no unique path to jump all of the pawns,`
`however. One such path is:``
`

`2 -- 3 14
-- 13`
`| |
| |`
`1,5 -- 4,18 -- 15,19 -- 12,20`
`| |
| |`
`6 -- 7,17 -- 10,16 -- 11`
` |
|`
` 8 --
9``
`

`Dave`