Fill in Numbers | Lines and Paths

ByJacob Valdez

Numbrix, created by Marilyn vos Savant, is a puzzle in which you fill in the grid with numbers so that you can follow a single path from 1 to the maximum number possible in the grid. In a 7×7 grid, for example, it would be 1-49. Our example puzzle is 10×10, so we will be filling in numbers for a sequence from 1-100.

This path may only move orthogonally when tracing the sequence of numbers. Each number will be in an adjacent cell to the previous and following numbers in a horizontal or vertical direction.

The general technique suggested by the inventor is to “scan from the lowest number to the highest one and fill in any missing numbers where the placement is certain.”

So let’s start near the beginning. There will only be one square between the 1 and 3, so it is either the purple or red square here.

Between the 3 and 6, you need two squares, so our sequence will be either (3, red, green, 6) or (3, yellow, blue, 6).

Now, look at the 9. There is only one path that can be taken from the 6 – the blue, then yellow squares. That means red and green must come after the 3, and leaves the purple cell holding our 2.

Our example puzzle is made to be an easy one, since I’m mostly demonstrating principles of how the puzzle works. As a result, our next several steps are pretty obvious, and don’t require much deduction. Printed puzzles will usually be a bit more challenging.

What you’re looking for is a pair of numbers that require a certain number of cells between them, and that ideally only have one possible path.

Following the blue path, we have 9, (10, 11, 12), 13. There is no other set of 3 empty squares to get from 9 to 13, so this is the only possible way.

Then, the green cell is the only blank space between 13 and 15. once that’s filled, only the yellow square is between 15 and 17, and then the red path is the only path possible for 17 (18, 19, 20, 21), 22.

Continuing that path, blue is the only square between 22 and 24, then green is the only space for 25, red must hold a 27, and yellow is the only cell between 28 and 30. So that whole set is easy to fill in.

However, between 30 and 33, we have two possible paths, either orange then purple, or orange then pink.

This means we’re going to need to do some deduction by looking a little ahead.

Let’s look first between 33 and 36. We need two cells, which could either be pink then yellow, or green then blue.

Moving ahead, we see 36 and 38, but again, there are two options for the single space between them, either the blue or red.

One more time, and we have only one option for the single cell between 38 and 40.So that means the red cell hold the 39, leaving only the blue cell for 37.

Now there’s only one option left for the two squares between 33 and 36, which must be pink then yellow. Once we fill that in, we now know that orange then purple holds 31 and 32.

Also, because all the cells in the grid must be used, it means the green cell has to hold 52, because otherwise it will be a gap in the grid if the 51 goes the other direction to get to 53.

At first, it looks like there are two possible paths between 40 and 43, but you can quickly realize that the next number after 43 cannot be beneath it, because from there, you don’t have any possible path to the next given number, which is 50, near the upper right of the grid.

Therefore, 41 and 42 must be along this blue path.

Next, we have a fairly long sequence and a pretty open area. Between 43 and 50 are (44, 45, 46, 47, 48, 49), six numbers. Let’s zoom in on possible paths below.

Now, the blue path can be eliminated as an option immediately, because it would leave two open cells to the right of the 43 that are unlikely to be filled by anything else, and we can’t have gaps in the finished puzzle.

We don’t have any immediate clues to easily choose between the green or red paths, but they do share that the first two places would be the same. So we can fill in 44 and 45 to the right of the 43, and then we will move on for now and come back to figure out where 46 through 49 will go later.

The blue cell must hold 54, and the green cell must be 56, because they are the only spaces between the 53 and 55, and then the 55 and 57.

Next, we only have one possible two-space path between 57 and 60, so the yellow squares must hold 58 and 59.

This cuts off the top two cells of the red path we looked at for 46-49, so we now know that the 46 must be to the left of the 45, then up for 47, and then right two cells until we reach 50.

Next, we have another easy sequence. The blue cells are the only two-space path between 60 and 63, so they must contain 61 and 62.Next, the green cell is the only option for 64.The red space has to hold 68, leaving the yellow space for 66.

Finally, the purple cell must be for 70. Looking ahead, it seems we may have more than one option for the path between 72 and 78. To aid in our deduction, let’s solve some of the rest of the puzzle to eliminate path options.

It is important to remember that you are not required to solve the puzzle in sequence. It’s often useful to solve other areas first to eliminate options elsewhere in the grid.

So let’s start with 82 to 85. We need two cells, and the blue path is the only option which doesn’t leave a gap.

Next, at the top, greenis the only square between 93 and 95, leaving yellow as the only possibility from 97 to 99.

The red square is the only possibility to hold 100, the highest number in this grid. If we instead put it to the right of 99, we would be blocking access to 93.

Now we get into our final moves. From 87 to 90, we follow the green path, because if we went upward, we’d leave a gap below the 90.

Same for 90 to 93. We take the blue path, because if we’d gone up and then left, we’d leave a gap next to 100.

Again, to fill a gap, 78 must first go down, then jog to the right on its path down to 82.

And last, but not least, our long winding red path takes us from 72, two steps to the left, then down, then right, then down again, and finally left to reach 78.

And here, we have our completed puzzle!