# ROUNDTRIP PUZZLES. Extended Introduction with solving strategies. by Glenn A. Iba

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1 Introduction & Background ROUNDTRIP PUZZLES Extended Introduction with solving strategies by Glenn A. Iba RoundTrip puzzles first appeared in Dell Magazines in the early 1990's. They were invented by a puzzler named Stitch (that's his nom in NPL, the National Puzzlersʼ League). His puzzles were presented as an 8x8 square grid of 64 dots with just enough links provided to make the solution unique. Links can only connect neighboring dots (horizontally or vertically, but not diagonally). The puzzle objective is always to form a closed path (one that returns to its starting point) that connects all the dots in a single large loop that visits each dot exactly once. The solution path is formed by drawing in additional links connecting neighboring dots. It is possible to solve RoundTrip puzzles using logical reasoning. Random guessing is not likely to succeed with any but the simplest puzzles. I enjoyed solving Stitch's RoundTrip puzzles so much that I decided to write a computer program to generate more of them. I started by creating a program to solve the puzzles, and then used that to generate uniquely solvable puzzles. I also extended the program to generate puzzles using a triangular grid of dots (in which case interior dots have 6 neighbors compared to the 4 neighbors of square grids). I laid out the dots in a hexagon-shaped grid. I submitted puzzles of both shapes (square and hex) and various sized grids to Dell Puzzle Magazines. The first of my puzzles appeared in Dell Champion Variety Puzzles November 1992, and have appeared in various Dell Magazines since then. Some of my puzzles have also appeared (under the name "GrandTour") in a Page-a-Day Calendar edited by Scott Kim for Workman Publishing. Additionally, I have posted Grand Tour puzzles on my web site: These puzzles can be solved interactively in a GrandTour Java Applet I wrote. This book contains many examples of both square and hex RoundTrip puzzles. They range in difficulty from quite easy to very challenging. In a following section I include discussion of various solving techniques, both elementary and advanced. I hope you will have as much fun with these puzzles as I have.

2 The Elements of RoundTrip Puzzles Here is a simple example of a square RoundTrip puzzle and its solution: Example Square 6x6 Puzzle Solution to Example Square 6x6 Puzzle A RoundTrip puzzle consists of a grid of dots and a number of required links connecting neighboring dots. A solution is a single looping path connecting all of the dots, and closing on itself so that there are no endpoints. The links in the path must include all the required links given in the initial puzzle presentation, and each additional link must connect 2 neighboring dots. In this square 6x6 example, there are 36 dots and 9 required links given initially. Observe that the solution has 36 links including those given, and that every dot has exactly 2 links connecting to it. In this book you will find Square puzzles of sizes: 6x6, 8x8, 10x10, and 12x12. This book also contains Hex RoundTrip puzzles of various sizes: 5x5, 6x6, 7x7, 8x8, 9x9, 10x10, and 11x11. Here is an example of a 6x6 Hex puzzle and its solution: Example Hex 6x6 Puzzle Solution to Example Hex 6x6 Puzzle

3 A Hex puzzle has its dots laid out in a grid shaped like a hexagon. In this Hex example there are 27 dots and 9 required links given. The solution consists of a total of 27 links. It will always be true that the number of links in a solution is the same as the number of dots in the puzzle grid. This is because every dot must be traversed exactly once. Neighboring Dots Links can only connect a dot with a neighbor dot. Informally, two distinct dots are neighbors if they are as close as possible in the grid. Here are sample square and hex grids with all the possible neighbors drawn in: (Fig. N1) Square 6x6 with all neighbors (Fig. N2) Hex 5x5 with all neighbors There are 2 types of grid layout used in the RoundTrip puzzles of this book. They are illustrated in figures N1 and N2. The two types of grid are called Square and Hex respectively, due to the shape of the grid outline. Note that when all neighboring links are drawn, the square grid consists of square spaces, and in the hex grid consists of triangles. The following figures (N3 and N4) illustrate the fact that dots can have different numbers of neighbors depending on their position in the grid: (Fig. N3) Neighbors of different Dot types: Interior (A), Border (B), and Corner (C) Note that dots in the square grid can have 2, 3, or 4 neighbors. Diagonal links are not allowed. Interior dots will have 4 neighbors, dots at the edge or border of the grid have

5 Chains and Loops A chain is a connected sequence of links connecting 2 endpoints. Except for the endpoints, each dot connected by the chain must have exactly 2 links. Every chain will have exactly 2 endpoints. Chains are illustrated in Figure N-chains. Figure N-chains: Examples of chains in both square and hex grids The simplest example of a chain is a single link, such as A-B in the grids of figure N- chains. Dots A and B are the 2 endpoints. Longer chains are illustrated by the sequence of links connecting C and D and the dots labelled X. Here C and D are the endpoints of the chain, and all the dots labelled X form the interior of the chain. Note that each of the X dots is filled since they each have 2 links, and are therefore unavailable. Chains can be extended by adding links to their endpoints. If a link is added from an endpoint to an open dot, then this extends the chain and the open dot now becomes a new endpoint of the chain. If a link connects 2 endpoints of distinct chains, this merges the 2 chains into a single longer chain.

12 Region Analysis This section introduces an advanced, non-local, set of reasoning strategies for solving RoundTrip puzzles. These methods revolve around the analysis of regions, which are areas of the grid separated from other areas by walls of filled dots. Figure Regions- Analysis illustrates this concept. We say 2 unfilled dots are in the same region if there is some sequence of neighboring dots leading from one to the other without crossing any filled dots. Two dots are said to be in different regions if the only way to get from one to the other is by crossing over 1 or more filled dots. In Figure Region-Analysis, part (a) shows the grid and the unlabeled links, part (b) shows how the filled dots (F) divide the un-filled dots into 3 regions: A, B, and C. (a) Unlabeled Links (b) Regions (c) Chain types Figure: Region Analysis We now look at the chains in the grid and how they relate to the regions. This is shown in part (c) of the figure. Chains can be of 2 types with respect to regions. We say a chain is an internal chain if both its endpoints lie in the same region. Chain K in the figure part c, is an internal chain. A crossing chain is a chain whose 2 endpoints lie in different regions. Such a chain crosses through a wall from one region to another. The remaining chains (J, L, M, and N) are all crossing chains. Chains J and L cross between regions A and B. Chains M and N cross between regions B and C. We make two observations about solvable link patterns: 1. Every region must have an even number of crossing chains. 2. Every region must contain an even number of endpoints These statements are actually equivalent, but letʼs see why they are true. The first must be true because the solution path has to cross into and out of the region an equal number of times, for a total number of crossings that is even. A region having an odd number of crossing paths is therefore another type of forbidden pattern and this can be used to rule out some links and force others as we will soon see.

14 Can a region have 0 crossing chains? The answer is no, as long as the region is not the only region. If there are 2 or more regions, they all must have crossing chains connecting them, otherwise it would be impossible to ever connect dots of one region with any dots of another (as is required in a legal solution). Can a region have zero endpoints? Once again, the answer is no, as long there are 2 or more regions. If there are no endpoints, then there can be no crossing chains, since a crossing chain contributes one endpoint to each of the two regions it connects. Since zero crossing chains is forbidden (as we have just seen) it must be that a region (that is not the only region) with zero endpoints is also forbidden. Summary of forbidden patterns involving regions The following are all forbidden: 1. A region with an odd number of crossing chains 2. A region with an odd number of endpoints 3. A region with zero crossing chains (unless it is the only region) 4. A region with zero endpoints (unless it is the only region) Remember that any link which would create or lead to a forbidden pattern is thus a forbidden link. Dividing Regions into Sub-Regions If we consider endpoints as well as filled dots, we can further divide regions into subregions, consisting of contiguous open dots. Two open dots are in the same subregion if you can get from one dot to the other stepping neighbor to neighbor using only open dots. Two open dots are in different subregions if every path from one to the other (stepping from neighbor to neighbor) must necessarily go through an endpoint or a filled dot. Figure Sub-Regions illustrates this in a sample 8x8 puzzle.

16 Figure: Sub-Region (A) with only 1 border endpoint (X) Figure Sub-Region-1-border-endpoint illustrates this forbidden pattern. Sub-region A has exactly 1 bordering endpoint (X). Any link connecting to dot X will turn A into a region, and create a forbidden region pattern. 2. A chain that caps a sub-region is forbidden We want to look closely at the endpoints that border a given sub-region. Each endpoint has a matching endpoint at the other end of the chain it belongs to. We define an endpoint to be internal (to the sub-region) if both it and its matching endpoint border the given sub-region and no other sub-region. Any other border endpoints of the sub-region are called external. There are 2 ways a bordering endpoint can be external: either by itself bordering on an additional sub-region, or by having its matching endpoint border a different sub-region. We define a capping chain of a sub-region to be a chain whose 2 endpoints are the only external endpoints of the sub-region. This means that at least one of the endpoints of the capping chain must border on an additional sub-region (otherwise they would be internal endpoints). We say that such a chain caps the sub-region (it isolates it from all other sub-regions). Why is a capping chain forbidden? Because no matter how we fill the 2 endpoints of the capping chain, we create a forbidden pattern. Observe that filling both endpoints of the capping chain will complete a wall defining a new region including all of the original sub-region plus its internal endpoints. The endpoints of the capping chain could each be filled by linking either into or away from the original subregion. If both endpoints connect the same way this does not create a crossing chain for the new region, and if they link in opposite ways (one into an the other away from the sub-region) then it becomes a single crossing chain contributing one endpoint to the new region. The key point is that there can be no other crossing chains for the new region, otherwise there would have been additional external endpoints of the original sub-region. But this

18 SOLVING A COMPLETE PUZZLE USING THESE STRATEGIES In this section we will walk through the complete solution of a square 8x8 puzzle. This will illustrate many of the concepts and strategies discussed above. Figure Start presents the puzzle we will solve: Start Step 1 In Step 1 we look for simple forced links. There are 5 open dots (labelled O) which have only 2 available neighbors, so have 2 forced links at each of these dots. These forced links are shown in gray. Step 2 Step 3 In Step 2 we now find two new open dots (labeled O) with just 2 available neighbors, so each of these has 2 forced links (shown in gray). In Step 3 we employ the 3-way

20 Step 6 Step 7 In Step 6 we add four more forced links. Dot A must connect to its only two available neighbors B and C. Then C is filled, so dot D must connect to its only 2 remaining available neighbors E and F. In Step 7 it is hard to make progress without using Region Analysis. We consider what the possibility of link A-B, and observe that this link would create a region (dots C, D, and E) with 3 endpoints. Since an odd number of endpoints in a region is forbidden, we conclude that link A-B is a forbidden link, and mark it with X. Now endpoint B can only extend to F, so link B-F is forced. Step 8 Step 9

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