Your first puzzle, step by step

This 5×5 board has four colors and exactly one solution. Below, it's solved from start to finish without a single guess. Each step names the technique it uses, so you can read the matching lesson when a step feels new. If you want to try it yourself first, copy it onto paper and come back when you're stuck.

Positions are given as row and column, counted from 1 at the top left. Blue's dots are at row 1, column 4 and row 4, column 3. Orange's are at row 2, column 2 and row 5, column 1. Green's are at row 2, column 4 and row 4, column 1. Red's are at row 3, column 3 and row 4, column 4.

Step 1: Orange's corner dot

Start with dots near the walls, because walls take away options. Orange's dot at row 5, column 1 sits in the bottom-left corner. It has two neighbors: Green's dot directly above it and the empty cell to its right. Green's dot is closed to Orange, so Orange has only one open side. It moves right into row 5, column 2.

That is a forced move. A path end with one open side has to take it.

Step 2: The top-right corner

No other path end has a single open side yet, so look at empty cells instead. The top-right corner, row 1, column 5, has only two neighbors: Blue's dot at row 1, column 4 and the empty cell below the corner. Whatever path fills the corner has to use both of them, and one of them is a Blue dot. So Blue steps right into the corner.

This is a squeeze, covered in Edges and corners. An empty cell with exactly two usable neighbors must be filled by a path that uses both.

From the corner, Blue's only way on is down to row 2, column 5. From there, the cell to the left is Green's dot, so Blue keeps going down to row 3, column 5. Both are forced moves.

Step 3: Blue's first real choice

At row 3, column 5, Blue has two open sides for the first time. It can go left into row 3, column 4 or down into row 4, column 5.

Test the left move. Row 3, column 4 is surrounded: Green's dot above, Red's dot to the left, Red's other dot below, and Blue's own path to the right. Blue would step in and have nowhere to go. A path end with no open side means the move was wrong, which is the idea behind bottlenecks.

So Blue goes down to row 4, column 5. From there the chain is forced again. Red's dot is to the left, so Blue continues down into the bottom-right corner, row 5, column 5. The corner leaves only one way out, left to row 5, column 4. From there, Red's dot is above, so Blue goes left again to row 5, column 3. That cell touches Blue's other dot at row 4, column 3. Orange already holds row 5, column 2, so joining is the only thing Blue can do, and Blue is finished.

Step 4: Orange climbs

Rescan after every finished color, because Blue's path closed off several cells. Orange's tip at row 5, column 2 now has Blue to its right and its own dot to its left. Its one open side is up, row 4, column 2. Check that cell: Green's dot is to its left and Blue's dot is to its right, so Orange is forced up again to row 3, column 2.

Now Orange's tip touches its other dot at row 2, column 2. Orange could join there, or it could move left into row 3, column 1. Red's dot is to its right. Leave Orange for a moment and look at the color whose options just shrank.

Step 5: Green goes the long way

Green's dot at row 4, column 1 started with two open sides. Orange has taken row 4, column 2, and the cell below is Orange's dot. So Green's only open side is up, row 3, column 1.

Every move from here is forced. Row 3, column 1 has Orange to its right, so Green goes up to row 2, column 1. Orange's dot is to the right of that cell, so Green goes up into the top-left corner. The corner sends it right to row 1, column 2. Orange's dot is below, so Green continues to row 1, column 3. Blue's dot is to the right, so Green turns down into row 2, column 3. That cell touches Green's other dot at row 2, column 4, and its remaining neighbors are Orange's dot and Red's dot. Green joins.

Each of those moves uses the same forced moves rule. Green's path is eight cells long even though its dots are only two rows and three columns apart. Once Orange had climbed column 2, the only route left to Green ran along the top-left walls.

Step 6: Orange and Red join

Orange's tip at row 3, column 2 has lost its last spare cell, because Green now holds row 3, column 1. The only thing left for Orange is to join its dot at row 2, column 2. When the two ends of one color sit side by side and have nothing else they can do, they connect. That is the joining case of a forced move.

One empty cell remains, row 3, column 4. Red's dot at row 3, column 3 has Green above, Orange to the left and Blue's dot below, so it moves right into that cell. Red's tip now touches its other dot at row 4, column 4, and Green and Blue block the other sides. Red joins, and the board is full.

What this puzzle teaches

Count the techniques. Almost every move was a forced move. One was a squeeze at a corner. One needed a quick test of the alternative, and that test failed after a single step. That mix is typical of a beginner board. The squeeze and the test are what get you moving again when nothing is forced.

Three habits came up again and again:

• Start at the walls. Steps 1 and 2 both happened in corners. Edge and corner cells have fewer neighbors, so they run out of options first.
• Rescan after each move. Green had two open sides at the start and only one after Orange moved. The next forced move is usually next to the last one you made.
• Test a choice by following it. At step 3 the wrong option failed the moment Blue stepped into it. On harder boards the failure takes a few moves to show, which is what thinking ahead covers.

When you're ready for a board of your own, pick any of the puzzles. Each one has three hints and a full walkthrough written the same way as this page.