Number Link
Over the past few years I’ve made at least one example of pretty much every reasonably popular Japanese number puzzle, but with one notable exception: Number Link. In this puzzle the aim is to draw lines connecting squares with identical values in, but without these lines crossing over or visiting any square more than once. The lines can also only travel horizontally or vertically, not diagonally.
In a completed Number Link puzzle, the lines will be placed such that every square is used – and that solution is guaranteed to be the unique solution too. The rules are not, however, always stated that strongly. In fact the most common form of the rules simply states that you must connect all sets of numbers, with an implied assumption that there is a unique solution, and furthermore again only an implication that that solution uses every square. But in terms of solving the puzzles in practice, it’s fair to assume that:
- There is a unique solution, and that unique solution happens to use every square
This assumption means that whilst solving a puzzle you can make various deductions about how lines can and cannot go based on noting what would make a puzzle non-unique, or not use every square. Once you discard these assumptions (as you need to do when creating the puzzle, since you can’t assume uniqueness without proving it!) it’s typically much harder.
It stands to reason that as puzzles get larger that they can get much trickier, but that isn’t always the case: today’s first example puzzle, whilst quite large at 20×9, is actually really easy, as you’ll probably find out when you try solving it! 
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about 15 years ago
Did you mean that lines can only travel horizontally and vertically?
I looked up the rules for Numberlink elsewhere and eventually figured them out. They confused me at first because I was used to seeing the numbers as PathPix or Fillomino values. Once I understood that you needed to connect the numbers no matter the line length, it fell into place.
I admit that I am ho-hum on these types of puzzles, preferring Masyu, PathPix, or Slitherlink for line drawing puzzles.
about 15 years ago
Oops – yes! Sorry for the typo – I was thinking ‘but not diagonally’ in my head but managed to type ‘horizontally or diagonally’ when of course as you say I meant ‘horizontally or vertically’. I’ve edited the post and fixed it above. (So to be clear, the lines can only travel between squares horizontally or vertically, but not diagonally – although you could perhaps design a [much?] harder alternative version which did allow that!). And yes, you can connect the numbers no matter the length of line – the numbers are just tags for the squares, and the actual value has no meaning other than to distinguish one tag from another, as in Sudoku.