The Tetractys™ Number Puzzle is based on sets of four digits, each from 0 to 9 and related to one another by a simple, arbitrary rule. It is this rule which defines every puzzle and makes each digit in a set of four significant. In fact, the rule of the Tetractys™ Number Puzzle is so simple, that all that is required to understand it is a basic knowledge of the multiplication table up to 9 × 9 and the decimal place value system.
Once a student has sufficiently memorized the multiplication table up to 9 × 9 and has a reasonably good grasp of place value notation, this puzzle can be introduced to the student not only to deepen that knowledge but to show that this knowledge is just the starting point to even deeper knowledge about these same numbers. We believe that if you can accomplish this much early on in a student’s learning of mathematics, you are well on your way to preparing that student for more advanced topics, especially those that invariably involve learning more complex symbolism at greater levels of abstraction.
Many people have a negative view of rules and rule-following, likening these things to blind obedience, mechanical actions, or simply to something lifeless or 'deadening' or perhaps even 'boring'. But rule-following is actually something entirely different: it is the outward manifestation of someone's understanding of a rule or how someone grasps the concept behind a rule. To grasp a concept or follow a rule is no mean thing, no trivial skill. In mathematics in particular rule-following gets to the heart of doing mathematics and above all of creating new mathematical concepts.
The rule underlying the Tetractys Number Puzzle provides yet one way in which a student can demonstrate his or her mastery of basic mathematical facts concerning multiplication and place value notation. But this rule, once understood, is also one way in which an original concept can additionally be formed out of these same basic facts. A rule, once understood and followed, becomes a new mathematical object in its own right, one that can be studied further, explored, and mined for new information. Moreover, since a rule always governs or rules over something, the behavior exhibited in following a rule is normative, that is, a rule distinguishes between correct and incorrect behavior. Without rule-following there would be no correct from incorrect behavior in mathematics, and hence no proof in mathematics. Without rule-following what would be left of mathematics?
The Tetractys Number Puzzle aims to teach one such rule in order to show not only how well students have grasped basic facts of arithmetic and mathematical notation but also how well they are able to conceive these basic facts in a novel form or 'rule'.