One of the rounds in the BBC quiz show Only Connect is the "Connecting Wall". Teams are presented with a grid of sixteen clues, and they then have two-and-a-half minutes to arranged them into four connected groups of four.

In the show, this is achieved by the team selecting four unmatched clues and then being told whether they form a group or not. If they are the matching clues are taken out and they continue to try to find the next group. If not, they have to try another set of four. Once two groups have been found, the team is limited to only a further three guess, to prevent them simply going through all 35 possibilities.

One point is awarded for each ground found in the time. Additional points are awarded for spotting the connections for each of the four groups, whether or not they were found. And finally two bonus points are awarded is everything has been done correctly.

In the usual instructions to the teams, the host usually says something along the lines of "… there's only one correct solution". So I wondered if it would be possible to have a wall with more than one solution. We might call a wall with two distinct solutions a "Doubly-Connected Wall".

Ideally, the second solution should be maximally different, so one would have no pairs of clues belonging to the same group in the two solutions. In other words, the 16 clues could be arranged in a grid so that each of the rows is a group and each of the columns is a group. Clearly there are trivial walls like this, such as this one:

- One
- Two
- Three
- Four
- Un
- Deux
- Trois
- Quatre
- Eins
- Zwei
- Drei
- Vier
- Uno
- Dos
- Tres
- Cuatro

But would it be possible to have a non-trivial wall like this? The connections should all be distinct, and not rely on the same or similar ideas.

I made good use of a 10 hour delay at Barcelona airport, and managed to come up with such a wall. The clues presented below are in a random order, so you can try solving it for yourself. Can you arrange the clues and identify the eight intersecting groups of four?

- Black
- Fillet
- Square
- Yellow
- Check Mate
- Parallelogram
- Pavilion
- Oval
- Villa
- Kite
- White
- Castle
- Ground
- Hall
- Green
- Land

I'm reasonably pleased with the wall above. One of the groups is slightly tenuous, but the others all work well. There are also a few instances of clues that would also fit in one of the other groups, which is good as it makes it more of a challenge. Nevertheless, I think it's probably quite easy to solve. Only one group has a really hard-to-spot connection.

In a game-show set-up I would image that you would need to select sets of four and then be told whether they are a group or not. Groups that have been found would be arranged in the top rows and left-most columns as they are found. I'm not sure at what point you would need to restrict the team to a specific number of guesses though. It's more complicated here, since there are the numbers of both horizontal and vertical groups found/left to be considered.

Can anyone else come up with their own doubly-connected wall? Can
you do better than the example above? How do you think the guess
restriction at the end should work? Is it possible to construct a
non-trivial triply-connected wall? Let me know via
email: `puzzles@(nospam)mathmos.net`

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