Logic Masters Deutschland e.V.

Gift Delivery in Overlapsville

Year in and year out the sharpest logistics gnomes get together to plan out the delivery of gifts to Overlapsville, a tiny, relatively unknown community behind the Ringkogel. The head logistician Hirdrud carefully unrolls the three maps which several surveyor gnomes have been working on since the spring, and the whole gathering gasps in wonder.

"As we feared, "began Hirdrud, "the Overlapsvilleans have once again rearranged themselves this year, which we can't really hold against them. Ever since they, nearly 400 years ago, elected to build their settlements to overlap one another rather than side-by-side, it is quite understandable that they need to experiment quite a bit with the setup to find something comfortable."

"But hopefully the settlements are all still rectangular in shape?" asks the tiniest logistics gnome, with a trembling voice.

"Yes, my friend. The first community, which back then went aground, consisted of 12 orthogonally oriented, individually unique pentomino shapes which didn't overlap at all – I'll return to that later – but since the second community (the first successful one) they have always been rectangular."

"To begin, the current settlements of Overlapsville need to be placed. Each is surrounded by a rectangular-shaped fence, whose four corners all fall on the centers of cells. However," – and here Hirdrud pointed to map A – "first we must sketch in the diagonally positioned rectangles (meaning those whose fences are oriented at a 45° angle to the grid and then," – indicating map B – "the orthogonally positioned ones. Having them all on a single map would be hopelessly inscrutable to non-natives like us. The shaded squares may not be crossed by any stretches of fence, and all white squares must be traversed."

"For each map, be sure that no two settlements share a stretch of fence, and that no corners coincide. A diagonal and an orthogonal settlement may well share a corner; luckily we don't need to be concerned about that. By their nature, the for us particularly critical center points of the settlements all land on cell midpoints, cell border midpoints, or cell corners." (See Figs. 1 & 2)

"Once the 23 settlements and their midpoints have been determined, the latter need to be transferred into map C. In the gift delivery process we have an agreement with the Overlapsvilleans that these center points will function as gift depots, (so we don't have to locate - let alone visit - every individual home). Now a loop needs to be established which visits every depot exactly once, and never crosses nor touches itself. Each depot must lie on the midpoint of a stretch of loop, and both before and after each stretch a turn of 45° or 135° must take place [i.e., never 90°]. The white dots which have already been positioned in map C are also centers of loop segments."(Fig. 3) Even after the settlement midpoints have been entered, not all loop stretch midpoints are given.

Fig. 1	Fig. 2

Fig. 3	Fig. 4

"It's now that the historical pentomino settlements become pertinent... Oh! Will you look at that... they have once again almost completely vanished. Well, they'll have to be shaded in once again. Each of the well-known pentomino shapes FILNPTUVWXYZ (see Fig. 4) gets used exactly once. They may NOT overlap nor share edges. They may, however, touch one another at corners, which in map C has unfortunately also been marked with black ink (the surveyor gnomes didn't have red ink handy. Sorry). The loop may never cross these historical settlements, but can strafe their edges, even at those points where the shapes touch one another."

Can you help the logistics gnomes locate the modern settlements in maps A and B, then place the historical settlements & sketch in the complete loop path in map C so Overlapsville children can enjoy their Christmas presents?