Flips With Bases Other Than 10

Derek Holton^*, Paul Ulrick# and Alex Holton#

^ University of Otago,
New Zealand

* University of Melbourne,
Australia

# Indepent Researcher

Published in The Mathematician Educator, 2024, Vol 5, No. 2, pp 63-90.

Abstract

In 1987 Gardiner wrote a book that contains work on what he called ‘flips’. He considered all flips in base 10. In addition, he made a few considerations to other basses. As far as we know, this has been the only time flips in bases other than base 10 have been discussed. In this paper, we take Gardiner’s work further, considerably aided by the digital production of large numbers of flips. But the result is a little about a lot and the lot covers four areas: the structure of flips, extension of flips, enumeration of flips of a certain kind, and the relation between the numbers of the base, the number of digits and the multiplier of the flip. Because we have considered such a wide range of flips, there are many unresolved questions. We note that this paper, and Gardiner’s work, can be considered to be mathematical research as they both aim to produce new results. Because of this we have added some consideration of the motivation of the different sections of the work and their relation to Gardiner’s work. So, we have tried to give an impression of what aims mathematicians have when doing research and what questions they ask themselves in the process.

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