We show in this post that an elegant continued fraction for $latex {e}&fg=000000$ can be found using derangement numbers. Recall from last week's post that we call any permutation of the elements of a set an arrangement. A derangement is an arrangement for which every element is moved from its original position. The number of … Continue reading Derangements and Continued Fractions for e
Month: December 2020
Arrangements and Derangements
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 … Continue reading Arrangements and Derangements
On what Weekday is Christmas? Use the Doomsday Rule
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 … Continue reading On what Weekday is Christmas? Use the Doomsday Rule
Will RH be Proved by a Physicist?
The Riemann Hypothesis (RH) states that all the non-trivial (non-real) zeros of the zeta function lie on a line, the critical line, $latex {\Re(s) = 1/2}&fg=000000$. By a simple change of variable, we can have them lying on the real axis. But the eigenvalues of any hermitian matrix are real. This led to the Hilbert-Polya … Continue reading Will RH be Proved by a Physicist?
Decorating Christmas Trees with the Four Colour Theorem
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 … Continue reading Decorating Christmas Trees with the Four Colour Theorem
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