We show in this post that an elegant continued fraction for 
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Archive for December, 2020
Derangements and Continued Fractions for e
Published December 31, 2020 Occasional ClosedTags: Combinatorics, Number Theory
Arrangements and Derangements
Published December 24, 2020 Irish Times ClosedTags: Combinatorics, Number Theory
Six students entering an examination hall place their cell-phones in a box. After the exam, they each grab a phone at random as they rush out. What is the likelihood that none of them gets their own phone? The surprising answer — about 37% whatever the number of students — emerges from the theory of derangements.
On what Weekday is Christmas? Use the Doomsday Rule
Published December 17, 2020 Irish Times ClosedTags: Algorithms
An old nursery rhyme begins “Monday’s child is fair of face / Tuesday’s child is full of grace”. Perhaps character and personality were determined by the weekday of birth. More likely, the rhyme was to help children learn the days of the week. But how can we determine the day on which we were born without the aid of computers or calendars? Is there an algorithm – a recipe or rule – giving the answer? [TM201 or search for “thatsmaths” at irishtimes.com].
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Will RH be Proved by a Physicist?
Published December 10, 2020 Occasional ClosedTags: Analysis, Number Theory
The Riemann Hypothesis (RH) states that all the non-trivial (non-real) zeros of the zeta function lie on a line, the critical line, 
The non-trivial zeros of 
eigenvalues of a hermitian operator.
Is there a Riemann operator? What could this operater be? What dynamical system would it represent? Are prime numbers and quantum mechanics linked? Will RH be proved by a physicist?
This last question might make a purest-of-the-pure number theorist squirm. But it is salutary to recall that, of the nine papers that Riemann published during his lifetime, four were on physics!
Decorating Christmas Trees with the Four Colour Theorem
Published December 3, 2020 Irish Times ClosedTags: Geometry, Topology
When decorating our Christmas trees, we aim to achieve an aesthetic balance. Let’s suppose that there is a plenitude of baubles, but that their colour range is limited. We could cover the tree with bright shiny balls, but to have two baubles of the same colour touching might be considered garish. How many colours are required to avoid such a catastrophe? [TM200 or search for “thatsmaths” at irishtimes.com].
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