## Puzzle Space

### Friday, December 30, 2016

"Where do you come up with all your crazy ideas?"

If people did ask me lots of questions about my games, I'm sure this would be one of them. Rather than reply glibly about inspiration, or process, I'd want to focus on the first word in that question: 'Where'. The answer is that all puzzles exist in puzzle space.

Mathematicians can talk about space as in the set of all possible permutations of a system. For example, a tossed coin has a simple 'space' consisting of one dimension, with two possible outcomes along that dimension. Rolling two dice, could be mapped in a two dimensional grid, with 6 positions in each dimension, we could then map a third dimension that was the sum of the dice. Dice one and two give you the X Y coordinates along the plane, then the vertical height is mapped on the Z axis, ranging from '2' in one corner, to '12' in the opposite corner. We've just mapped a simple system into a 3D 'space', and created a mental picture of a surface, with 2 input dimensions, and a third output dimension. This method of visualising permutations can be really useful.

There is no reason to stop at 3 dimensions, it just gets harder to visualise. Imagine all the ways you could design a mouse-trap. The length, height, thickness and springiness of each component could be mapped as input dimensions, and the effectiveness at catching mice could be mapped as in output dimension. Early mouse traps may have started at a coordinate in 'mouse-trap' space where the effectiveness was low, and any iteration tested that worked better was adopted as the new 'best' solution. In this sense the progression of trap design can be visualised as travelling through 'mouse-trap' space, starting at a low point on the surface, and travelling upwards towards an optimum solution. Sometimes incrementally, by small steps along the surface, sometimes by sudden inspirational jumps, breaking out of a local optimum, bypassing a valley of worse design, and landing on a new slope closer to the global optimum.

Puzzles exist in puzzle space. There are thousands of variables that you could change when creating a puzzle, and you could then measure the effectiveness by how long a player spends interacting with it, or how highly they rate it, or how 'fun' it is. Typically a game will introduce a simple puzzle and then add complications to each level, making it harder each time. Sometimes a game will abandon one direction in puzzle space, and embark in a different direction, with different ideas. A game like Tetris, takes one coordinate, (the spot where you align falling tetronimoes to clear lines), then increments just one variable (speed) until the output (difficulty) exceeds human ability. Most trivial puzzle games don't travel very far through puzzle space, and only branch along a small number of dimensions.

A game like 'The Witness' starts at a simple point (connect the start and end of a maze), then each successive puzzle travels through different dimensions in puzzle space, branching and branching, until we end up with a vast tree of space which the player has explored. Some ideas are iterated in one area until they are exhausted, and are never used again, that branch finishes. The core idea, (drawing lines in a grid that must meet conditions) stays as a constant root branch to this tree. It is the stubbornness to stick to this one concept, and push it further and further in so many directions, that gives the game its elegance.

With 'Four Floors of Doors' I've started with the concept of a branching tree. Every puzzle has this idea at heart, even if not obviously. Each puzzle room takes a different twist on the room before. I've tried to resist the temptation to just increase one dimension (eg complexity), because that would get boring. I've resisted dropping the core idea, and hopping to a fresh starting point in puzzle space, because that would detract from the overall elegance of the game, leaving it feeling disjointed. I've resisted staying in one place in puzzle space and just repeating ideas with different cosmetic permutations (like Candy Crush), because although that is very quick to write, it doesn't go anywhere.

Once a concept hooks you, like the branching trees of graph theory, you start to see it everywhere. Even when describing how I create this game I'm visualising it as a branching tree of rooms travelling through puzzle space. I'm midway through development now, and I feel I've come quite a distance on some unfamiliar paths. I don't know where the game is heading, but I'm looking forwards to finding out.

Tags: Four Floors of Doors