Logic Masters Deutschland e.V.

To Christmas 2022 some eager puzzlefriends have again arranged a group in which everybody had to make anonymously a puzzle for someone else. These puzzles are now presented one after the other here in the portal.

In search of new markets for their Christmas cookies, the gnomes gaze out from the reindeer-drawn sleigh into an unexplored valley. Some houses are scattered about, but at the edge is a free surface that seems ideally suited as a landing strip (top left corner, see fig.1). Scarcely having poked their heads out of the sleigh upon arrival, they already can see some of the residents rushing up to them. The gnomes introduce themselves politely and declare their intent. The residents - who call their valley 'Pentalia' and themselves 'Pentalians' - confer with one another briefly. Then they provide the following response to the gnomes:
„You are welcome to sell your cookies here, but we must insist on a few stipulated conditions: there's an old looping trade route through the valley that passes exactly once through each of the empty cells that you see on the old map. We made it that way so the children get lots of opportunities to marvel at the traveling salespeople. The old loop doesn't go through houses (marked as circles in the map, see fig.1) and it enters and exits every home's lot exactly once.“
The gnome's heads are already spinning from all this information, when the Pentalians begin adding further tidbits: „Each lot has a size of exactly five cells and exactly one house - oh yes, and two lots of the same shape never touch, that would be totally boring...“ Here the gnomes interrupt and say: „Just a minute - in that case there aren't enough houses shown on the map, are there?“ „Oh yeah, um..“, the Pentalians reply. „here in the corner where you have landed is where out five Pentenarians have built their homes. You need to discover where they are located - be glad that you didn't accidentally land on one of them.“ The gnomes exchange looks of contrition - that would have been embarrassing indeed! Then they say: „So far we've only heard information from you. Now what are your stipulations?“ The Pentalians declare: „We want a new trade route! Many of us are confined to our homes for health-related reasons, but nevertheless want to see the traveling merchants. The new route must therefore pass through each house. But... well, some of us are a little bit eccentric...“ (At this point the gnomes look at one another suggestively. 'Some'? Just 'a little bit'?) „Those in the black houses (see fig.2) want the route to turn in their houses, but move straight by at least one step before and after. Those in white houses want the route to travel straight through their house, but turn directly beforehand or afterward. The Pentenarians have disguised the color of their houses, of course - oh, and one of them has become such a recluse that his entire lot shouldn't be visited at all. Other than that, every cell must be visited, as before - we want to have our fun, after all!“ Inwardly, the gnomes are rolling their eyes again, as the Pentalians append further stipulations: „In those houses with a grey courtyard live our traditionalists. They prefer that the new route still pass through their house, but the rest of the route needs to remain the same as the old route. But those who have marked their white home with an additional circle, on the other hand, are particularly innovative. They want that no cell on their lot is traversed in the same way as before.“
The gnomes have already lost much of their motivation to sell cookies here, but still the Pentalians are not finished: „Oh yes, and as for those two cells in the middle: there the ground has an unusual property, in that only one path traversal is possible.“ Now the Pentalians look at their visitors expectantly - it seems that all of the conditions have been stipulated. The gnomes are unsure whether selling a handful of cookies here is really worth the effort. But fortunately, they have brought you along as a guest on their sleigh. You set to work and, with the help of the map are quickly able to figure out the original trade loop. With that, you are sufficiently equipped to designate a new route for the Pentalians, thereby helping the gnomes to expand their customer base.