Connect-the-Dots for Grown-Ups

The Contest Center
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Wappingers Falls, NY 12590

       Yashi is a new line-drawing logic puzzle from the Contest Center. Yashi was invented by Frank Rubin in June 2012. The object is to connect all of the dots together.

       Yashi looks very difficult when you first see one. Bewildering, confusing, impossible. You have no idea where to start, or what to do. We are going to show you a few tricks to cut the problem down to size. Once you learn these techniques you will always know how to proceed. We can't promise the puzzles will become easy, but you will know how to handle them with confidence.

       Once you learn these tricks, you will look like a genius. Anybody who sees you solving a Yashi puzzle will be astounded that you can solve such a mind-boggling puzzle. Let's begin at the beginning, with the rules.


       You can connect any two dots by using a single horizontal or single vertical line. You cannot use any diagonal lines or curved lines, and you cannot connect two dots with a path that has angles or corners. No L-shapes, no Z-shapes, just single straight lines. Each dot may be connected to 1, 2, 3 or 4 other dots, either horizontally, vertically or both. Your lines cannot cross each other, and they cannot pass through other dots, or go on top of other lines.

       When you have solved the puzzle, there will be exactly one path (which may pass through any number of intermediate dots) from any dot to any other dot. This means that your lines cannot form any closed loops. For example, there could not be a path that goes from Dot A to Dot B to Dot C to Dot D, then back to Dot A.

       There is an easy check you can make to verify that you have solved the puzzle correctly. The number of lines must always be 1 less than the number of dots. If there are 30 dots then a valid solution must have 29 lines. If you have more lines, then you must have formed some closed loops. If you have fewer lines, that means your network has several unconnected pieces. (A network is just a set of connected dots and lines.)
Two unconnected pieces
A closed loop
       Every Yashi puzzle on this website has a unique solution. A network that contains a closed loop can never produce a unique solution. You could remove any one of the links in that loop and the same dots would still be connected.


       Let's introduce some notation to make it easier to talk about the steps in a solution. Locations in the puzzle grid will be numbered, as shown below. Rows will have letters, A, B, C, etc. and columns will have numbers 1, 2, 3, and so forth. A letter plus a number will identify one location in the puzzle, like this:

A1 A2 A3 A4 A5 A6 A7 A8 A9
B1 B2 B3 B4 B5 B6 B7 B8 B9
C1 C2 C3 C4 C5 C6 C7 C8 C9
D1 D2 D3 D4 D5 D6 D7 D8 D9
E1 E2 E3 E4 E5 E6 E7 E8 E9
F1 F2 F3 F4 F5 F6 F7 F8 F9

and similarly for larger or smaller puzzles. A dot in location A2 would simply be called A2. If the dot at location A2 were connected to a dot at A4, the connection, or link, would be called A2A4.


       A good way to begin your solution is to lightly draw in all of the possible links between the dots. This will be shown with dashed lines. You will need a pencil with a good eraser because you are going to use that eraser a lot more than the pencil lead.

       As you solve the puzzle, you will decide which of these lines to use in your solution, and which ones don't belong. You will change the lines you use from light to heavy, and erase the lines you don't use. Start with this example.


When all of the possible links are drawn, the grid looks like this:


       Look at link A2A4 in this diagram, near the upper left corner. There are no lines crossing this link. A link with no crossings is called a FREE link. You can immediately include any free link in your solution.

       Here is the reason: Suppose that there were a solution that did not include the free link. You could add the free link to the solution. That would make two solutions. Since every puzzle has only one solution, every free link must be part of the solution.

       This small example has lots of free links. Change those lines from light to heavy to show that they are part of the puzzle answer. When you add all of them to the puzzle solution, the example will look like this:

       Notice that all of the links at the four edges of the puzzle are always free links. You can always begin solving by drawing all of the links around the edges with heavy lines.


       Once you have added all of the free links to your solution, you extend the solution by reasoning. Look at the link B1B5. This link crosses two other links, A2C2 and A4C4. Suppose that you did not use link B1B5. Then links A2C2 and A4C4 would both be free links, like this:

       Since A2C2 and A4C4 would be free links, you would have to include both of them in your solution, like this:

       That forms a closed loop, A2-A4-C4-C2-A2. A valid solution cannot contain a closed loop. This means that the link B1B5 must be part of the solution, like this:


       Since the links A2C2 and A4C4 have been eliminated, there is now only one possible connection between the group of dots A2-A4-A8 and the rest of the dots. This link is FORCED. You have to use it. When there is only one connection to a dot, or to a group of dots, that connection must be part of the solution. Otherwise, you could not connect all of the dots together. The puzzle now looks like this:


       Look at the connection G1G5. If this connection were made, then the group of dots C2-C4 would become an ISLAND, separated from all of the other dots, with no way to connect them. This means that the link G1G5 cannot be used. You cannot use any link which isolates a group of dots from the rest of the network. You must not create islands. After removing G1G5 the grid now looks like this:

       The links C2I2 and D3H3 are now free. They must be part of the solution, like this:

       This creates a new, bigger island C4-C2-I2-I6-I9-B9. There is only one way to connect this island to the rest of the network, namely the link F6I6. Using Tip #2, this link must be part of the solution, like this:

This completes the solution.


       As you solve the puzzle, you may need to try out several different alternatives, whether to use a link or delete it, whether to use Link 1 or Link 2. Sometimes you will guess wrong, and you will need to backtrack. That will be easier if you keep track of your guesses.

       One way is to make a list of your choices. For example, when you guess that link C4C8 is part of the solution, you might write +C4C8 alongside of the puzzle before you darken that line. Similarly, if you guess that link C4C8 is not part of the solution, you might write -C4C8 alongside of the puzzle before you erase that line. If you make further guesses you add them to the list.

       The job of restoring the puzzle in case a guess doesn't work will be a lot easier if you use four different kinds of lines. Use heavy lines to indicate links that are definitely part of the solution. You might draw these in ink. Use medium lines to indicate links that you are testing to decide if they are part of the solution, or links that would become part of the solution if some earlier guess turns out to be true. Use light lines to indicate places where links could be placed. When you start a new puzzle, it is best to draw all of the light lines. Use ultralight lines to indicate links that could not be used if your last guess were correct.

       There are two situations where you would abandon a guess. First, it might lead to a failure, where no solution is possible. In that case you draw the link the opposite way from your guess. Second, it might not give you any useful information. In that case, you redraw both the medium lines and the ultralight lines as light lines.

       You can develop your own system using dotted lines, dashed lines, colored lines, wavy lines, double lines or any combinations you can dream up. These will help you deal with situations where you need to make a second or even third guess.

       Over time, you will develop the ability to picture the results of your guesses in your head, just as a chess player can see several moves ahead. You will need lists and coded lines less and less. Or, you could combine lists, coded lines and mental images to solve really huge and complex puzzles.


       Let's recap this how-to section with a little list of do's and don'ts.

» DO use all free links.

» DO use all forced links.

» DO keep track of your guesses.

» DON'T form any closed loops.

» DON'T create separate islands.

       If you get stuck, REASON your way forward by asking "What happens if I include this link? What happens if I remove this link?" Does that create free or forced links? Does that create closed loops or islands?

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