History
Numstep belongs to the family of puzzles commonly associated with Hidato. Hidato was created by Gyora Benedek and was first published in the mid-2000s. The central idea is beautifully simple: place consecutive numbers into a set of cells so that each number can be reached from the previous one. A solution becomes a single continuous route through the board, making the puzzle easy to explain but surprisingly difficult to solve.
Classic Numstep sits firmly in this Hidato tradition, using a square grid and ordinary horizontal and vertical adjacency. The other Numstep variants take that familiar idea to a new dimension. Cube and Box turn the playing surface into a three-dimensional topology, while Torus removes the edges by joining opposite sides. The rules remain approachable, but the shape of the space changes what counts as a possible next step.
Maths
A useful way to think about a Numstep puzzle is as a path through a network of connected squares. In graph theory, a Hamiltonian path is a route that visits every point in a network exactly once. An n×n square lattice has n² cells, so a Hamiltonian path through the whole board would visit all n² cells without repeating any. There can be many different Hamiltonian paths on the same lattice; the exact number depends on the size and on whether the path is required to start or finish at particular cells. The important point for Numstep is not the huge numbers involved, but the idea that a grid can contain an enormous number of possible routes.
Numstep puzzles are created using random self-avoiding walks: routes are built one step at a time while refusing to visit a cell twice. The route is then turned into the numbered puzzle, with selected positions revealed as clues. Generation has to balance randomness with solvability, so the resulting puzzle should have a clear challenge rather than simply being a random scribble.
Topology is where things get especially interesting. A normal square grid has edges: step off the left and you have left the board. On a torus, the opposite edges are joined, so leaving the right side brings you back on the left, and leaving the top brings you back at the bottom. This is still based on the familiar square grid, but its connections are different.
A true sphere is harder to represent with this method because a sphere does not have a regular square grid covering it without introducing distortions, irregular cells or singular points. You cannot simply glue the four sides of a flat square together and get a sphere. The Box variant is therefore a useful approximation: its six square faces form a closed surface, giving a simple, understandable way to explore some of the ideas of moving across a three-dimensional surface without pretending that a cube has the same topology as a sphere.
FAQs
Contact
If you have a question, suggestion or issue, use the form below. Messages are directed to numstep@proton.me.