Solving Hidoku and Numbrix

Hidoku and Numbrix are both simple logic puzzles that rely on some basic constraints to find the solution. There are no complex solving techniques as the relationship between the numbers and the grid is very simple.

Constraints:

  • The numbers must form a continuous path from 1 to X where X is the number of cells.
    * Numbrix - the path can only be formed with numbers to the left/right or above/below - 4 options.
    * Hidoku - the path can be formed with numbers to the left/right, above/below, and diagonally - 8 options.

  • All cells in the grid must be filled.

Numbrix

A simple tip to get started is to scan from the lowest number to the highest and see if there are any cells that must contain a single number - a unique placement. Once you have started filling in an area concentrate on that until you can progress no further and then move on to another starting point. As you fill areas keep re-scanning the grid from lowest number to highest to find out if you have created any new unique placements.

Because all cells must be filled, always beware of creating a path that cuts off cells and would prevent them being filled. In harder puzzles you can test different paths to see if there is a unique path that would not cut off cells. Also the path can never cross itself, adding a constraint that will help when testing paths.

The grids on the right give examples of unique placements, isolated cells and a unique path.

The two green cells are unique placements as only 17 and 26 can be placed there. There is no other way to join 16 to 18 and 25 to 27.

When joining 3 to 5, 4 can be placed in either of the blue cells. However, since 49 is the last number on this 7 x 7 board, if 4 is placed in the red position the cell between 49 and 5 would be isolated and could not be filled. This is also true when joining 27 and 29 - if 28 is placed in the red position the cell between 27 and 35 would be isolated.

When considering how to join 5 to 10 there are three possible paths that could contain the four numbers needed. However, only the path marked with the green line would not isolate 10 and at least 1 other cell.

Hidoku

Exactly the same logic for Numbrix above can be applied to Hidoku. However, diagonal paths are allowed so the path can cross, unlike Numbrix where the path will never cross itself. Despite the added complexity of a path that can cross itself, solving Hidoku is the same iterative process of finding unique placements and evaluating possible paths.

Here are examples of the kind of logic you should be using. There are two unique placements that can be spotted immediately. There's only one possible position for the 4 and the 20 to continue the path 3 to 5 and 19 to 21. The grey square is also a unique placement for 14, but is less obvious until the 20 is placed due to the convolutions possible with diagonal paths.

With the 4 in place there are three possible locations for 2, the blue cells. However, any other placement than in the green numbered cell would leave that cell isolated and empty, preventing all cells being filled.

There is only one possible path from 41 to 44: the green line. But then we need a path from 44 to 48. There are two possibilities shown by the blue lines. The placements of 45 and 46 are unique, it just remains to determine where 47 should go. With 45 and 46 in place, the cell to the left of 35 must be 36, to provide a path to 37. If 47 is now placed to the right of 48, the cell to the right of 49 would be blank and could not be filled - if 32 is placed there 33 cannot be placed anywhere, and if 34 is placed there, again there is nowhere for 33. So 47 must be placed in the cell to the right of 49.