Haddex
A falling-hexagon arithmetic game: wherever three hexagons meet at a corner and one number is the sum of the other two, all three vanish.
Haddex began with a simple rule: if three hexagons meet at a corner, they disappear when one of the numbers equals the sum of the other two.
What interested me was the great deal of mathematical behavior that could arise from that one constraint.
Since pieces continue to fall and when hexagons are cleared the ones above them drop as well, one sum can set off another and then another. Although the maths involved is simple, the player has to think in spatial terms—not just about whether a sum exists at the moment but also about what the board will look like after a move.
I wanted the mathematical elements to form an integral part of the game rather than being added on afterwards. The player is never required to pause and solve an arithmetic problem before proceeding. Since recognizing number relationships is simply the way the game is played, continuous mental calculation takes place naturally as a result of making decisions.
A major aspect of the project involved tuning the system. To do this I carried out simulations using different versions of the rules, including bots that played blindly and others that actively searched for sums. As a result, several problems were identified which could not be anticipated simply by examining the rules. For instance, permitting too many large numbers meant that some boards became effectively unrecoverable, and so rather than altering the main mechanic I changed how those numbers could appear.
I also tried out more stringent forms of the matching rule. Although these were mathematically valid, tests involving actual play showed that they made the game seem unnecessarily restrictive, and so I decided to keep the simpler rule.
To me, Haddex is a case of designing based on a mathematical relationship and then allowing complexity to emerge from that. It involves arithmetic fluency, spatial reasoning, anticipation and pattern recognition, without treating any of these as separate exercises.
It also shows an aspect of teaching I care about greatly: it doesn't always have to seem like practice. When the mathematical decision proves useful to the learner, repetition can come from a desire to solve the problem rather than from something imposed from the outside.