Enter numbers using the steppers or enter a number by clicking on a box and typing 0-9. Swapping a pair of numbers will always create a change in the check digit. This was inspired by: https://www.youtube.com/watch?v=yaoSdFAL4UY
Rediscover and explore the Verhoeff-Gumm algorithm, a check digit formula which is more reslient to common errors than the Luhn algorithm, which is widely used in credit card numbers, IMEI numbers and more. Historical Notes: "Error Detecting Decimal Codes" [1], a PhD thesis by J Verhoeff was published in 1969. It showed how the vast majority of digit typos were single digit or transposition errors, traditional "modulo 10" algorithms always missed some transposition errors, and introduced a novel class of algorithms based on "the dihedral group of order 10" (pentagon flips and rotations). In section 4.4, Verhoeff outlines using "search program" to find a permutation function that is optimal for detecting errors. A proof of its correctness is omitted. I found Verhoeff's writing to be difficult to approach, so I recommend a section in "Contemporary Abstract Algebra" (seventh edition) by Joseph A Gallian (pg 111-114) for a clearer write-up. A witty quote from Verhoeff in the introduction made me chuckle: "[I believe] that the codes explained in chapter 4 provide the first practical application of the dihedral group. This would illustrate the old saying that all beautiful mathematics will find an application, sooner or later." In 1985, H. Peter Gumm published "A new class of check-digit methods for arbitrary number systems" [2]. It starts with a dense proof that "modulo 10" (indeed modulo 2k) formulas will always be flawed. The paper then justifies the use of the dihedral group, which to me sounded like a mathematician walking around a store looking for the right outfit ("needs cancellation", "should be associative", "finite members", "can generalize for any even number"). Gumm then proves an algorithm using D_s works, using the number pair notation and a permutation function tau. Gumm claims to have been unaware of Verhoeff's work. Additionally, Gumm adds a proof and a way to scale it beyond 10 digits, so I decided to credit them both with discovering the algorithm in this video. Felix Klein (same as the Klein bottle) was an important contributor to group theory [3], and was chosen to be the cardholder in the intro. Likewise, Évariste Galois coined the term "group" and was thus chosen to be the online vendor.
