An Exploding Dots Machine is an array of
boxes, any of which may contain a number of dots, plus a rule
that describes how to move dots between the boxes.
The rule is most frequently of the form 1 ← n, meaning any group of n dots in a box may be exploded and be replaced by a single dot in the next box to the left. More generally, the rule may be m ← n, in which case the n dots become m dots in the next box. The first figure below shows an example of this process.
Additionally, anti-dots, indicated by open dots, behave in the same way as normal dots but are opposite in value, and so a dot and anti-dot in the same box will cancel each other out. The second figure below shows an example of this process.
A crucial insight is the realisation that applying the rule or cancelling dot/anti-dot pairs does not change the value contained in the machine, only the representation of that value.
Exploding Dots machines, with dots and anti-dots, support the visualisation and modelling of the basic arithmetic operations (addition, subtraction, multiplication and division), as well as aspects of integer representation and base, and can even be extended to model polynomial division.