Ben Cartford

The concept of “bracelets” have become the primary intersection of combinatorial mathematics and card magic in recent years. The new type of bracelets this research investigates are what we have called Summation Sequences. A Summation Sequence is a cycle of alphabet a and window size w, where all windows contain a unique sum between min {a} and (a x w) inclusive. While other bracelets, such as the well-known De Bruijn Sequences, are defined by their appearance, Summation Sequences are defined by their sums: a second, deeper, data structure which we have called a Derived Sequence; this allows for double the content to investigate. This research explores the properties of Summation Sequences and their subsequent Derived Sequences, as well as methods for generating these sequences efficiently. The concept of Summation Sequences is nicely applied to the realm of card magic as the numbers in the sequence can be represented by card values in a deck, where the properties of Summation Sequences are preserved. Additionally, the movement of cards, such as shuffling or mixing, is akin to the movement of numbers in a sequence. Thus, this research aims to explore the properties of Summation Sequences through the lenses of the fields of Graph Theory and Combinatorics, and directly apply the results to the motivation of new card effects.