How to Solve Number Chain Puzzles

Number Chain, sometimes called Number Snake, was invented by a Russian puzzle designer named Leonid Mochalov. It contains numbers between 1 and X, where X is usually between 10 and 40, depending on the size of the grid.

Your goal is to draw a single path from the upper left cell to the bottom right cell, taking only orthogonal movements (up, down, left, or right). Along this path, all of the numbers from 1 to X will be included, but only one time – there may not be any duplicate numbers along the path, and numbers may occur in any order. This path may only visit any given cell one time, so it can’t cross itself.

A good first step to solve a Number Chain consists of two parts:

Look for duplicates of the starting and ending numbers in the path. Since we know the path starts in the top left and ends in the bottom right, those numbers anywhere else in the grid can’t be part of the path.
Look for all numbers that occur only once in the grid, because they will definitely be part of the chain.

Here, the purple highlighted cell contains the number 1, which is already in the top left, so we know it’s not part of the solution. All of the circled numbers are only in the grid once, so the solution path must pass through those cells.

Next, we’ll start looking for places where we can place line segments that are definitely part of the chain. The thing to remember is that except for the beginning and end, every number will have an entry point and an exit point, so if two directions are impossible, the remaining two directions must be the correct one.

For example, look at the 2 at the top of the grid – we noted in the first stage that it doesn’t have a duplicate anywhere, so it is definitely part of the chain. Since it’s on the top border, the path can’t go up, and we just eliminated the 1 to the left of it, because it’s a duplicate of the number in the top left corner, where the chain begins.

That means that the path going through the 2 leads down, to the 16, and then right, to the 28. This means we can eliminate any other 16s in the grid – we already know the 28 is unique.

Now, because we see a line of circles below the 28, it’s easy to assume that the path simply goes down from there, but we can’t let ourselves skip the deductive process. Because there are no restrictions stating the path can’t pass by in adjacent cells, we can’t be sure at this point that the line doesn’t go to the right after the 28, and then down and double back through the 15 and 3. Be wary, and don’t let yourself fall into that trap – make sure of your line placements.

So instead, we’re going to look at this 5 near the bottom right. We were just able to eliminate the two 16s above and to the left because we already encountered it as part of the path, so no other occurrences of 16 will be used. This leaves the 5 with only two possible sides for the path to travel, so that is the way it must go – right to the 4, and down to the 8. This allows us to mark out other 4s and 8s in the grid.

Next, we’re going to look at the 18, which we know is part of the chain because it is unique in the grid.

We can’t tell at this point whether it connects to the 4 or the 8, because either would work. But we can deduce that it can’t connect to both, because that would form a closed loop, which isn’t allowed as part of the chain. So regardless of whether it connects to the 8 or the 4, it must be the number that connects directly to the 33 that is the endpoint of the chain.

Meanwhile, at the top right corner of the grid, we can see that the 22 now has only one side open, because we just eliminated the 4 below it. Since we know it can’t be an endpoint, we can safely eliminate this copy of 22 as a possibility.

This 21 is an interesting case. It’s unique in the grid, so we know it must be part of the path. We eliminated the 4 to the right, so it only has 3 sides to choose from.

However, two of those sides are the same number, so to use either of them will eliminate the other as a potential exit. This proves that no matter what, the 21 and the 9 above it must be connected as part of the chain. And while we don’t know yet which of the 23s will be used, we know one of them must be, so we can eliminate any other appearance of 23 in the grid.

Also, because the 9 only has two available sides, that means the 31 to its left must also be part of the chain, and we can remove any other 31 or 9 in the puzzle.

Notice that with that last set of eliminations, the 19 on the right side of the grid is now left with only one side for a path, so it can’t be part of the chain. When we eliminate it as an option, the 4 below it is forced to connect downward with the 18.

Then, the 25 that we know has to be part of the chain now only has two sides available, so we know that the 17 above, and the 19 to the right of it are on the path and we can remove other instances of those numbers.

Before we move on, let’s take a look at this block of numbers on the left. Because of numbers we eliminated earlier to their right, any path leading down into this group would just dead-end. That means we can eliminate all of the numbers in these cells.

When you cross out numbers this way, it is always worthwhile to look around the rest of the grid to see if you’re down to only one possibility for each of them, because if so, you will know they must be part of the chain.

And it looks like we found a few newly unique numbers that have to be part of the solution.

We can’t do anything with them yet, but notice that we did create another corner at the 32. It only has two remaining sides, so it must connect to the 24 to the right and the 30 above it. The 24 is unique now, but we can eliminate any other appearances of a 30.

This 24 is forced to be a corner, because it only has one side left for the path to exit. Once it reaches the 22, we can also see that it won’t be able to turn left, because that would form a closed loop.

Therefore, the 30 must send the path upward, into the 6. This will eliminate the other 6 to the right of the 1 that starts the chain, and now we know the path must leave the 1 downward to connect with the 6 before moving on to the 30.

Let’s come back to the 28. The path that entered from the 2 has to exit through either the 31 or the 15.

If we turn downward into the 15, then the path coming into the 31 would also have to move down into the 23. This would eliminate the other 23 below the 21, boxing it in so it had no exit other than to form a closed loop.

Therefore, the 28 must exit toward the 31.

Next, we’re going to look at these cells where we’ve been boxed in by eliminated numbers. We need to connect the path from the 22 to the 16, and we have to include the 12.

Because of how cramped the area is, we can’t go up through the 10 and maneuver back to include the 12, so that means the path must go from the 22 into the 12, then up through 29 and finally to the right to connect with the 16.

This will leave the 15 with only two exits – to the 3 and the 23, which now eliminates the extra 23 below 21 and confirms that it will be a corner turning to the left.

As things get more closed in, the path of the chain gradually becomes more clear.

The path from the 3 has only one direction to go, down into the 20.

And we can also see that this 10 and 14 have become dead ends, so we can eliminate them, as well.

Once again, by eliminating some numbers, we discover the last instance of them, so we know they must be part of the chain.

As a result, we can now see only one path that will connect the 19 to the 8, while including all of the necessary numbers – down to 14, then right to 13, up to 11, and finally right again to the 8. That means the 27 at the bottom of the grid is eliminated.

After that elimination, we only have a single 27 left in the grid, and it only has two sides available for the path to use. Once we connect it to the 20 to the left, and then move down through the 26, it eliminates the only other 26, and the rest of the number chain becomes clear – left twice through the 7 and 10, and finally down to the 17.