## Wednesday, 11 July 2018

### Calendar of Mathsy Moments

The calendar of mathsy 'moments' (downloads provided below) is a calendar marking at least one mathsy 'moment' (or event) of interest for each day from 1 August 2018 to 31 December 2019.  It is primarily intended for schools and teachers to use in gentle cultivation of a respectful yet intellectually playful approach to mathematics in their students, in support moreover of the development of students' deeper and perhaps wider appreciation and framing of mathematics in the historical, cultural and intellectual sense.

The calendar is shared to help teachers establish and cultivate an ethos of mathematics as progress in the classroom, to help teachers in other words help their students develop and sustain happy and fulfilling relationships with their mathematics, encouraging the desire to explore further, deeper, and more independently — fuelling students' curiosity and their capacity for such.  It is hoped that in sharing these brief moments of mathematics with students, in unabashedly celebrating the joy that is to be found in mathematics, teachers will fulfil their roles as enthusiasts, advocates and celebrators of mathematics, both as an intellectual discipline and cultural artefact, and students will in turn stand a greater chance that the subject will inspire — rather than beguile — them.

The calendar is provided below for access from this page.  Downloads in a range of formats are also provided for easy import.  Each 'moment' is outlined in more detail within the description / notes section of each calendar entry, which may serve as a prompt for the teacher with respect to how s/he may use / share it, if at all.  Where applicable, links to sources, and other sources of information — including online articles, historical artefacts, videos, podcasts, etc. — are provided within each description as a start for further exploration, as desired.

The moments are categorised within the calendar as follows:
• Moments / Events occurring at various times throughout the year (note that a 'Day' relates to the day of the year, e.g. 14 March 2019 is the 73rd day of the year, and so a curio about 73 may be given.  A 'Date' relates to the date as a concatenated number, in dd/mm/yyyy format unless otherise indicated, e.g. 14 March 2019, the 14th day of the 3rd month in the 2019th year, 14/3/2019, which concatenates to 1432019).  These moments include:
• Prime Number Dates, Mathematicians' Birthdays, Historical Dates to note in Mathematics, Circular Prime Number Dates, Emirp Dates, Frustrating Prime Number Dates, Days for Curious Calculations, Days for Number Facts, Palindromic Number Dates, Palindromic Prime Number Days, Pi Dates, et al.
• The 'Day for a Number Fact' days aims to feed our natural love of numbers (or perhaps revitalise it), exposing students to — or maybe tantalising them with — a wide range of more exotic sounding mathematics from number theory that they may not ordinarily come across in their day to day curriculum.  For example, in sharing such moments students will know of: good primes, home primes, long primes, Sophie Germain primes, Ramanujan primes, Fibonacci primes, near-repdigit primes, quadruprimes, prime triplets, prime sums, primes that remain prime when added to their reverse, circular primes, emirps, permutable primes, near primorial primes, palindromic primes, dihedral primes, twin primes, sexy primes, numbers commonly assumed incorrectly to be prime, semiprimes, sphenic numbers, other prime products, unprimeable numbers, happy numbers, sad numbers, distinctly happy numbers, perfect numbers, superperfect numbers, pseudoperfect numbers, weird numbers, abundant numbers, deficient numbers, almost integers, duck-duck-goose numbers, Mersenne numbers, lucky numbers, amicable numbers, polite numbers, sociable numbers, automorphic numbers, narcissistic numbers, odious numbers, pernicious numbers, evil numbers, practical numbers, arithmetic numbers, polygonal numbers, undulating numbers, telephone numbers, antisigma numbers, highly composite numbers (or anti-primes), refactorable numbers, taxicab numbers, tetranacci numbers, vampire numbers, untouchable numbers, Bell numbers, Euclid numbers, Franel numbers, Frobenius numbers, mountain numbers, star numbers, Smith / Joke numbers, Friedman numbers, the divisor (factor) function, the partition function, Jacobi's formula, Aronson's sequence, Fibonacci factorials, Euler's 6n+1 theorem, repdigits, primorials, subfactorials, double factorials, superfactorials, hyperfactorials, trees, Golomb's sequence, numbers in different bases, etc.
• Moments / Events occurring once in the year (including some national or globally recognised awareness days with particular reference to mathematics).  These moments include:
• A Perfect Day, An Anti-Perfectt Day, Ada Lovelace Day, e day (Europe and US), Fibonacci Day (US), Half Year Day, Halloween (Vampire Numbers), Mathematics and Statistics Awareness month, McNugget Numbers Day, National Numeracy Day, Phi Day, Phi Day (US), Pi Approximation Day, Pi Day, Root-2 Date, Spreadsheet Day, The Golden Ratio Moment, The Oddest Prime Day, et al.
• Other Moments / Events not directly related to mathematics (that schools / teachers may nonetheless wish to mark, either because of the mathematics latent within them, or for other more school-centred reasons around values, community and relationships).  These moments include:
• Ask A Stupid Question Day, Autumnal Equinox, Black History Month, Book Lovers' Day, European Day of Languages, International Joke Day, International Women's Day, Make Up Your Own Holiday Day, National Coding week, National Poetry Day, National Read a Book Day, National School Nurse Day, National Teaching Assistant's Day, Origami Day, Random Act of Kindness Day, Roald Dahl Day, Space Day, Summer Solstice, Vernal Equinox, Winter Solstice, World Book Day, World Kindness Day, World Music Day, World Teachers' Day.

Mathsy Moments calendar
There are and will be, of course, many other 'moments' that could and maybe should be included in such a calendar.  I make no claim that the calendar offers anything comprehensive.  If there are other 'moments' that you think other teachers may enjoy sharing with students, please let me know in the comments and I will update in subsequent versions.  Equally, whilst every effort has been taken to eliminate errors, if users notice any error in the content of the calendar, I would be very grateful if you could again let me know in the comments.

GCSE Maths 2019 Countdown calendar
This calendar gives a daily countdown to GCSE mathematics exams, both in terms of days left and school days left.  It is intended as a support for teachers to help students (and their families) manage their own study time.

Further sources
Sources of information (or explanation), where applicable, have been provided in link form in the description of each moment / event.  If you wish to play further, explore the following sites:

## Tuesday, 1 May 2018

### On Mountain Numbers

Cairn Toul, a beautifully wild and remote mountain in the Cairngorms massif of the eastern Scottish highlands, is notable not only for being the fourth highest mountain in the UK, but also for being the only mountain in the UK whose elevation in metres — 1291m above sea level — is also a 'mountain'.  Großglockner, by way of a further example, the highest mountain in Austria, is a mountain whose elevation in feet — 12461ft above sea level — is also a 'mountain'.

To elaborate: A number is a 'mountain' if its decimal digits start with 1, i.e. 'base camp', ascend continuously to a unique summit, i.e. to one largest digit, then descend continuously back to 1, i.e. back to base camp.  Such mountain numbers are recorded as sequence A134941 (itself a mountain) in the On-line Encyclopedia of Integer Sequences (OEIS), from which you can also find this table of all possible 21,846 mountains (including 1 itself).

Mountain primes, as the name suggests, are prime mountain numbers, prime numbers in other words whose decimal digits start with 1, ascend continuously to the summit of a single largest digit, then descend continuously back to 1.  (The area chart image above shows a mountain range of the first 17 such numbers.)  Mountain primes are recorded in the OEIS as sequence A134951 (yes, also a mountain prime), from which you can find this table of all 2620 such primes.

When climbing a mountain, it is typical to descend back to base camp, i.e. back to where you started your ascent, but it is not always the case.  Mountain numbers that ascend from one location (as an elevation) but descend to another — e.g. 3,598,432 — are considered to be generalised mountain numbers, recorded as sequence A134853 in the OEIS (there are 173,247 such mountains).

• Explain why there must be a finite number of mountain numbers.
• Construct a mountain range diagram for mountains of your choice.
• What is the Everest of mountain numbers?
• How will you define 'Everest'; how will you define 'elevation'?  (In the image above, the 'height' of mountain 1291, for example, is greater than the 'height' of mountain 1571.)
• Explore the distribution of mountain number digits, i.e. how many mountains have 1 digit, 2 digits, 3 digits, etc. (OEIS sequence A135417).
• If we regard the number of digits in a mountain number as the horizontal distance travelled when climbing such a mountain:
• What mountain(s) has (have) the shallowest ascent?
• What mountain has the steepest ascent and descent?
• Many people find the most beautiful mountains to be those most pyramidal in shape, such as for example the Matterhorn in the Alps and Machapuchare in the Himalaya.
• Define and find your most beautiful mountain numbers:
• Palindromic mountain numbers perhaps (OEIS sequence A173070),
• Or palindromic mountain prime numbers (the largest is 123467898764321).
• Find all possible — i.e. 45 — Giza numbers.
• Giza numbers are so-called because they represent the pyramids of Giza in the sense that their first digits increase in consecutive order to a largest central digit, and their last digits decrease in the same consecutive order as they increased.
• The largest Giza number is 12,345,678,987,654,321 (OEIS sequence A134810).
• Of two mountain numbers chosen at random, what is the probability that their numerical heights — i.e. the magnitude of the number — will be in alignment with the elevation of actual mountains?
• For example, mountain number 1291 is 'higher' than mountain number 1571, but as actual elevations, 1291m < 1591m, so these two mountains are not in 'alignment'.
• Find as many actual mountains whose heights are mountain numbers, or mountain primes.
• What is the highest mountain in the world whose height is also a mountain (use metres and/or feet for the height)?

## Friday, 27 April 2018

### On Overheard Conversations About Maths #1

S
SCENE:
The floor.  Just before bedtime.
Three siblings are islands in a sea of Lego.
The eldest is trying to make a Lego Penrose triangle.

3 YEAR OLD:

(Silently counting the studs on a red 2 × 4 Lego brick,
mouthing each number until, to no-one in particular...)

Eight!

(He opens up his thumb and fingers successively
from the fist of one hand as he counts, this time out loud.)

One, two, three, four, five...

(Now to the other hand.)

...six, seven, eight.

(Some time passes as he collects all the 2 × 4 bricks
he can find, piles them together, and pretends to count them all.)

A hundred!  A hundred is bigger than four.  Or five.  Or eight.

6 YEAR OLD:

It’s not as big as a million though.

YEAR OLD:

(With a nod to his Numberphile education...)

Or Graham.

YEAR OLD:

Or ten.  Ten's bigger.

YEAR OLD:

No, a million's bigger than ten.  A hundred is bigger than ten.  A million is really big.

YEAR OLD:

Who's Graham?

YEAR OLD:

Infinity is bigger than Graham.

YEAR OLD:

Nifity?

YEAR OLD:

No, infinity.  In-fin-ity.

YEAR OLD:

(Straining to speak; arms outstretched as wide as he can make them go.)

Is Nifity big like this?

YEAR OLD:

That’s not even a number.

YEAR OLD:

Nifity?

YEAR OLD:

Infinity.  In-fi-nity.  In-finity.  No, it’s not a number it just means the biggest thing.  People say it's a number but it's not because you could just say infinity plus one but that's still just infinity because...

YEAR OLD:

Infinity plus a hundred then.

YEAR OLD:

(Sarcastically.)

Infinity plus infinity.

YEAR OLD:

Two times infinity.

YEAR OLD:

Duh, that's the same.  And it's still just infinity.

YEAR OLD:

(Quietly, in the ear of his 6 year old sibling.)

Is Graham friendly?

YEAR OLD:

Infinity times infinity then.

YEAR OLD:

Do you even know what that means?  OK then, infinity times infinity infinity times.  It's still just infinity!  It's still...

(Establishing the correct perspective
from which to see the Lego Penrose triangle
doesn't seem to matter quite as much right now.)

Isn't it...?  Erm...

YEAR OLD:

Nifity?

CURTAIN.

What is infinity?
• Share this typically accessible introduction to the idea of infinity from MathsIsFun.com.  (Play, for example, with the provocation 'An infinite series of A's followed by a B will NEVER have a B'.)  The page gently introduces infinity as 'the idea that something has no end', that doesn't grow, and subsequently moves into arithmetic properties and the exciting idea that there are different sizes of infinity.

Why does $$\infty + 1 = \infty + 100 = \infty + \infty = \infty \times \infty = \infty$$?
• This article by Robert Crowston in NRICH maths is a thoughtful introduction to the countably infinite via (David) Hilbert's hotel — to the idea that adding a finite number to an infinite set is countably infinite (moving guests from room $$n \to n + 1$$), as is adding an infinite number to an infinite set (moving guests from room $$n \to 2n$$).
• For more on infinity paradoxes, including Hilbert's Hotel, watch this video from the always inspiring Numberphile.
• Introduce students to the idea of $${\aleph _0}$$, ("Aleph-Null" or "Aleph-Nought").
• Segue then into the uncountably infinite, why they are bigger infinities than the countably infinite, intuitively through comparing the cardinality of real numbers $$\left| \mathbb{R} \right|$$ on the number line to the cardinality of the naturals $${\aleph _0}$$, i.e. $$\left| \mathbb{R} \right| > {\aleph _0}$$, and then perhaps via Cantor's diagonal argument, with this article from Plus Magazine.
• For more on these different infinities, watch these superb videos:
• In addition, this article from Katherine Körner in NRICH maths and this from Peter Macgregor on 'Cantor's paradise' in Plus Magazine, both provide more depth, and this, on raising infinity to the power of infinity, from Reginald Braithwaite via GitHub.

We can (sometimes) add an infinity of numbers.

Play around with infinite sums, starting perhaps with this article by Luciano Rila from Plus Magazine.   And consider, for example:
$\underbrace {1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...}_\infty$Or, to put it another way:
\begin{array}{c}\begin{align}\sum\limits_{k = 0}^\infty {\left( {\frac{1}{{{2^k}}}} \right)} &= \frac{1}{{{2^0}}} + \frac{1}{{{2^1}}} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + ...\\ &= \frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...\end{align}\end{array}
If we add up the first few terms (i.e. find the first few partial sums):
\begin{array}{c}\begin{align}\frac{1}{1} + \frac{1}{2} &= 1\frac{1}{2}\\\frac{1}{1} + \frac{1}{2} + \frac{1}{4} &= 1\frac{3}{4}\\\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} &= 1\frac{7}{8}\\\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} &= 1\frac{{15}}{{16}}\\etc.\end{align}\end{array}
We can see that each iteration produces a number that gets closer and closer to 2.  We produce another infinite sequence in others words with a limit of 2, thus:
$\left\{ {1,\;1\frac{1}{2},\;1\frac{3}{4},\;1\frac{7}{8},\;1\frac{5}{{16}},\;...} \right\}$
And we can therefore show that the sum of infinite series is 2.  Say:
$\sum\limits_{k = 0}^\infty {\left( {\frac{1}{{{2^k}}}} \right)} = s = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...$
Then:
\begin{array}{c}\begin{align}\ 2s &= 2\left( 1 \right) + 2\left( {\frac{1}{2}} \right) + 2\left( {\frac{1}{4}} \right) + 2\left( {\frac{1}{8}} \right) + ...\\2s &= 2 + 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...\end{align}\end{array}
And so:
\begin{array}{c}\begin{align}\ 2s - s &= \left( {2 + 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...} \right) - \left( {1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...} \right)\\s &= 2\end{align}\end{array}
Thus:
$\sum\limits_{k = 0}^\infty {\left( {\frac{1}{{{2^k}}}} \right)} = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ... = 2$