I will explain what the links are in a logic puzzle. Simply put, all of the relationships between two nouns in a puzzle must be expressed as verbs and links. If you are familiar with facts, then you know that a fact has the form "noun 1 verb link noun 2". In the logic puzzle "Five Houses", there are three links: "with", "directly to the right of", and "next to". All of the links use the noun type House.
To define your links, you may want to draw a picture of the five houses in this puzzle. You don't have to be an artist, but you need to be accurate. You want a picture of five numbered houses in a row, where house 1 is on the far left, and house 5 is on the far right. Use crayons and paper if you don't have a 3D printer. Below is my modest rendition. Assume each house is initially colorless.
The first clue in our logic puzzle states: "The Englishman lives in the red house." If a clue states that one noun is or is not with another noun, then the default link "with" is used. In less-than-stellar English, the first clue tells us "The Englishman is with the red house." The "with" link is a one-to-one relationship because it means that a noun is with itself. This link is automatically defined for you, and it defaults to the first noun type of the puzzle. For our puzzle, here are the statements that are true:
House 1 is with house 1, House 2 is with house 2, House 3 is with house 3, House 4 is with house 4, House 5 is with house 5
The second link given by the clues is "directly to the right of". This link is in clue 5 "The green house is directly to the right of the white one." Going back to our picture, this means the following statements are true.
For this link, all other comparisons between two houses are false. This type of link has a "one-to-one" relationship because exactly one house is directly to the right of another house.
The third link given by the clues is "next to". This link is in clues 10, 12, and 14. While some houses are only next to one house, other houses are between two houses. Therefore, this link is not "one-to-one". Can you determine all of the statements that are true for this link?
The "directly to the right of" link is a "more than" comparison because we want to see if noun A is higher than noun B. This comparison is done using the one-based number of the noun. To reduce the number of links in a logic puzzle, use either "more than" or "less than" comparisons, but not both. For example, if a fact stated "A is less than B", you can also say "B is more than A", and vice versa.
Links, along with rules, are the hardest things to work out in a logic puzzle.