Wednesday, 17 July 2019

42 Mathsy Things to do this Summer



42 is an admirable, pentadecagonal, interprime cake number; a pernicious, odious and wasteful yet polite, primary pseudoperfect number.  It is also the result of two consecutive sums of consecutive integers (9 + 10 + 11 + 12, and 13 + 14 + 15), the area of a Heronian triangle, and of course, the 'Answer to the Ultimate Question of Life, the Universe, and Everything' as calculated by the Super computer Deep Thought in Douglas Adams' 'The Hitchhiker's Guide to the Galaxy' [1].  Most importantly, perhaps, 42 is also the number of days in 6 weeks — the typical length of the summer holiday from school for children in England and Wales.  This post suggests some mathsy play, activities, and games for each day of the school summer holidays.  Enjoy!


1

Play a game of Hex

The Game of Hex is a phenomenal, beautiful mathematical game.  This post gives some historical background to the game, and an introduction to the mathematics underpinning it and, of course, instructions on how to play it.  This link will open up a folder of blank boards for you to use up to 11 by 11.


2

Explore Mountain Numbers

A number is a 'mountain' if its 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.  For example 1291, 1571, 1821, etc.  This post on Mountain Numbers provides more ideas for more mathematical play and diversions with these intriguing numbers. 


3

Sum a finite series of consecutive numbers

Learn how to add up a series of consecutive numbers efficiently (i.e. really quickly) by thinking mathematically.  For example, 1 + 2 + 3 + ... + 99 + 100 = (100 + 1)(100 ¸ 2) = 101 Í 50 = 5050.  This is because the first and last numbers added (1 and 100), is the same as the 2nd and 2nd to last numbers being added (2 and 99), which is the same as the 3rd and 3rd to last numbers being added (3 and 98), and so on.  And of course, if there are 100 numbers, there must be half the amount of pairs of numbers.  Read this story from NCTM about one of our greatest ever mathematicians, Carl Friedrich Gauss, to help you learn, and this brilliant article about the story by Brian Hayes in American Scientist.
\[\sum\limits_{i = 1}^n {i = 1 + 2 + 3 +  ...  + \left( {n-2} \right) + \left( {n-1} \right) + n =\frac{{n\left( {n + 1} \right)}}{2}} \]

4

Play someone at Connect 4

This Numberphile video exposes a little of the mathematics behind the game.  It has been proven that if you play first and drop your counter in the center column, and play perfectly from then on, you are guaranteed a win within 41 moves.  What if you change the rules so that both players play to lose?  (You can play online here at www.mathsisfun.com.)


5

Play Misère Os and Xs but with Xs only and on 3 boards

Watch this Numberphile video with Thane Plambeck explaining the game.  The Misère version of Os and Xs is simply the normal version but with the rules changed such that both players play X, with the winner being the person who avoids getting three Xs in a row (see this video).  Trying to avoid making three in a row on any one of three boards simultaneously is just another way of making the game trickier and more interesting, mathematically!


6

Make — and amaze your friends with — a Hexaflexagon

A Hexaflexagon is a hexagonal paper toy (a three-faced hexagon) which can be folded and unfolded and opened out to reveal hidden faces.  Chalkdust magazine provides a helpful set of instructions to get you started, with a range of templates and a page from which you can upload your own pictures/photos onto a template.  This page from Vi Hart is a treasure trove for all things Hexaflexagon.


7

Explore the beautiful Fold and Cut Theorem

The incredible fold-and-cut theorem proven by Erik Demaine simply states that any shape with straight sides can be cut from a single sheet of paper by folding it flat, and making one straight line cut across the fold(s) you have made.  This Numberphile video with Katie Steckles will inspire you to play.  You could maybe start by learning the renowned magician Harry Houdini's 'trick' of folding a piece of paper and cutting through it with one cut to make a 5-pointed star.


8

Listen to a mathsy radio program (or podcast) or two

There are many radio programs and podcasts out there with a mathematics bent.  I have collated a few here and have listed some podcasts for you to while away a mathsy day.  Maybe pick a topic to become an 'expert' on and astound your teacher with a snippet or two of your newly discovered knowledge when you return to school in September. 


9

Play Dots and Boxes

Dots and Boxes, or 'squares' as my school friends and I called it, was first published in the 19th century by French mathematician Édouard Lucas, who called it la pipopipette.  It is a lovely device for children to develop their ability to strategize and provides a nice exposure to arrays and multiplication.  (You can play online here at www.mathsisfun.com.)


10

Play a game of Play Your Cards Right

Take a normal deck of cards, shuffled, and turn over each card having predicted whether the card will be of a higher or lower value than the preceding one.  How far can you go?  What arrangements of cards in the deck would guarantee you go through correctly all the way?  What are the chances of a normal deck of cards being shuffled into one of these arrangements?  What if you combine more than one deck of cards?  (You can also play online here at www.mathsisfun.com.)


11

Make a 3-set Venn diagram

This isn't as straightforward as you might think.  Try and find three groups of items that you can categorize into three sets that intersect in the classical fashion (as in these examples from Passy's World of mathematics).  If you want a real challenge, try and make a 4-set Venn diagram.  This article from the always engaging Mental Floss about 'funny and delicious Venn diagrams' might inspire your thoughts.


12

Find a spurious correlation or two

A spurious correlation is a relationship between two variables that appear to be associated but actually are not, or are caused by a third, hidden factor.  Spurious correlations can help us learn how to treat statistics with care.  This website by Tyler Vigen is a curation of such lovely, fun, spurious correlations.


13

Make a mind-reading trick

Play this famous trick on as many people as you can, and figure out why it works, before creating one of your own:
  • Think of a number between 1 and 9.  
  • Multiply it by 9.  
  • If you have a 2-digit number, add the digits (e.g. if you have 34, 3 + 4 = 7).  Otherwise, if you have a one-digit number, just keep that.
  • Now subtract 4 from your result.  
  • Now turn this number into a letter of the alphabet, where A is 1, B is 2, C is 3, etc.  So if your number was 11, say, this would be the 11th letter of the alphabet, K.  
  • Now I want you to think of an animal that begins with this letter.  Don't change your mind!  
  • Now think of a country beginning with the letter immediately before the letter you just found, so if you had a K before, your new letter is J.  
  • You are thinking of an elephant in Denmark.


14

Play the Countdown Numbers game

In case you are not aware of the Countdown 'numbers' problem: Choose a problem set of six numbers from a set of four 'large' numbers {25, 50, 75, 100} and a set of 20 'small' numbers {1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10}.  A 3-digit target number is then randomly generated, which you must try and make using only the four basic arithmetic operations and using the 6 numbers chosen, once each — but not necessarily all of them.  My post 'On 'Incredible' Feats of Mental Arithmetic (in Countdown)' provides links to clips from the show across the World, plus a little bit about the maths behind the problem, and the calculating strategies used by contestants.


15

Perpetrate a Random Act of Maths (or two)

Find as many people as you can and carry out a random act of maths on them, i.e. share a surprising maths fact, curiosity, fascinating bit of maths history, or problem with them.  You could use one of the Random Acts of Maths prompts I've shared here.  Maybe leave one inside a library book or two, accidentally leave one on a shopping trolley, or on the seat of your bus, or distribute them to your neighbours... you get the gist!


16

Write a fib poem

A 'fib' poem is a poem is based upon the Fibonacci sequence, where the number of syllables in each line of the poem is the sum of the previous two lines: 1, 1, 2, 3, 5, 8.  The Poetry Foundation explains more and offers some examples.


17

Set a new Personal Best for doubling

Start at one and keep doubling (in your head).  How far can you go?  Can you improve?  What if you started at a number other than 1?  What if you halved rather than doubled?  Trebled?


18

Construct a Koch curve

Koch curve (or Koch Snowflake) is one of the earliest fractals to have been described (in 1904 by Swedish mathematician Helge von Koch).  This Wikihow gives an easy to follow, step by step guide to the construction of the Koch curve, and this from Datagenetics is also superb, leading to a clear appreciation of the apparent paradox that the curve has a finite area, but infinite perimeter!  You may also enjoy reading this illuminating article from Evelyn Lamb.


19

Construct a Sierpinski carpet

The Sierpinski carpet is a fractal described by Wacław Sierpiński in 1916, that is a 2D representation of the Cantor Set. This Fractals Activity sheet from ThinkMaths describes how you can draw the Sierpinski Carpet, and encourages you to explore other fractals including the Cantor Set, Sierpinski Triangle, Koch Curve (see 18 above), and the Menger sponge and Sierpinski tetrahedron 3D fractals.  This page provides you with instructions and printable nets to help you build the 3D fractals and this page from the wonderful Mathigon helps you explore fractals in a little more depth.


20

Explore the aliquot sequences

The aliquot sum s(n) of a positive whole number n is the sum of all the divisors of the number other than itself, for example s(16) = 1 + 2 + 4 + 8 = 15.  An aliquot sequence is a sequence of numbers in which each term is the aliquot sum of the previous term.  Many aliquot sequences end in 0, such as the sequence starting with 16: 16, 15, 9, 4, 3, 1, 0.  Find all starting numbers less than 100 whose aliquot sequence does not end.


21

Take the 100 to 999 challenge

All the numbers from 100 to 999 can be made using the numbers 1, 2, 7, 9, 10 and 50 and the four basic arithmetic operations. For example, 153 = (7+10)×9, 799 = (2×(9×(50-7)))+25.  Make them all!


22

Etch a massive Fibonacci spiral in the sand (if you go to the beach)

A Fibonacci — or Golden — spiral, as described in this Quora article, 'is a series of connected quarter-circles drawn inside an array of squares with Fibonacci numbers for dimensions [that] fit perfectly together because of the nature of the sequence, where the next number is equal to the sum of the two before it'.  This article will take you through how to draw a Fibonacci spiral step by step.  Play around with it on paper and then, if you go to the beach, find a good spot and etch a massive one in the sand.


23

Race someone to a sum and always win

Ask someone to write down two numbers, without you seeing, and then to write eight more so that every number after the first two is the sum of the two preceding ones, e.g. 4, 9, 13, 22, 35, 57, 92, 149, 241, 390.  Now look at the numbers together, and race to sum all ten numbers.  You will win because all you need to do is take the 7th term in their sequence and multiply it by 11.  (Try and work out why this is the case, before you look at what I've shown below!)
\[\begin{array}{c}\begin{align}{i_1} &= a\\{i_2} &= b\\{i_3} &= a + b\\{i_4} &= a + 2b\\{i_5} &= 2a + 3b\\{i_6} &= 3a + 5b\\{i_7} &= 5a + 8b\\{i_8} &= 8a + 13b\\{i_9} &= 13a + 21b\\{i_{10}} &= 21a + 34b\\\\\sum\limits_{n = 1}^{n = 10} {i = \;} 55a + 88b &= 11\left( {5a + 8b} \right)\end{align}\end{array}\]

24

Go and visit something mathsy

If you're in London, for example, maybe visit the Science Museum Mathematics Winton Gallery, and while you're there, seek out Charles Babbage’s Difference Engine number 1 and his Trial Analytical Engine and the jacquard loom in the 'Making the Modern World' gallery that all inspired Ada Lovelace to write the world's first computer program.  Or pilgrimage to Colinette road in Putney and stand at the door of the once hospital where Hardy and Ramanujan had their famous conversation about the number 1729.  Or take some inspiration from the University of Oxford's Mathematical Institute's London Tour.  Or visit the home of the Codebreakers, Bletchley Park.  Or find your inspiration from Resourceaholic's 'Maths School Trips' page.  Or make your own 'Maths Tour' itinerary for where you live, and take people on it.  Maybe map out a route that takes you from one geometrical pattern in your environs to another.  


25

Borrow a mathsy book from the library

Once you've read it, why not write a little mathsy problem on a little note and slip it inside the book when you return it so that the next reader will get a nice surprise.  Or leave a Random Act of Maths inside.


26

Find some Happy Numbers

19 is a 'happy number', because if you sum the square of its digits, and repeat with the result, and keep repeating, it ends at 1 (see the detail below).  All numbers with this property, i.e. that 'end' at 1, are called 'happy'.  (In technical terms we say that happy numbers are 'numbers whose trajectory under iteration of sum of squares of digits map includes 1'.)  Pick a number and see if it's happy.  Find as many happy numbers as you can.  (This link opens the Happy Numbers sequence on the Online Encyclopedia of Integer Sequences.)  Create your own 'rule' to generate a sequence (the weirder the rule the better) — what will you call your sequence?
\[\begin{array}{c}\begin{align}{1^2} + {9^2} &= 1 + 81 = 82\\{8^2} + {2^2} &= 64 + 4 = 68\\{6^2} + {8^2} &= 36 + 64 = 100\\{1^2} + {0^2} + {0^2} &= 1 + 0 + 0 = 1\end{align}\end{array}\]


27

Learn the number 27 card trick

Watch this Numberphile video with Matt Parker to learn the trick and the beautiful mathematics underpinning it, then play the trick on as many people as you can, and most importantly, reveal and explain your secret to them — namely the beautiful mathematics that makes it work.


28

Play the Mystery Calculator trick on someone

My post 'On the Mystery Calculator Trick' talks you through the 'trick', provides you with the cards to download and use, and takes you into the mathematics of why it works.  Learn how to play the trick on someone, learn why it works (i.e. learn about binary numbers), then play the trick (or maybe one you make up yourself) on someone, and once you've befuddled them, explain the beautiful mathematics underpinning the 'trick' to them.


29

Get lost in an Arithmetic Maze

I tweeted about these lovely arithmetical mazes created by Dan Finkel here in the Hindu Times.  Have a go at them and then maybe make up your own.


30

Watch Donald Duck in Mathmagic Land

No matter how old or how young you are, you're never too old or too young to watch this.


31

Play a game of Sprouts

Sprouts is a two-player game invented in 1967 by mathematicians John Conway and Michael Paterson.  It's a game with simple rules that can become remarkably complicated to play.  You start by drawing two or more spots on a piece of paper, before players take turns to make a move, according to the following rules: The first player draws a line joining two spots, or a single spot to itself, and then draws a spot on the new line. (The line must not cross another line or pass through another spot and no more than three lines can emerge from any spot.)  The second player then makes their move, and play continues until one of the players is unable to move.  The last player to be able to move wins.  This article by NRich further explains the rules, and this goes into the fascinating maths a little deeper, with lots of wonderful questions to consider and proofs to explore!


32

Work out how long your hair would be had it never been cut

The average speed of human hair growth is, according to a range of sources, around 1.08029879 × 10\(^{ - 8}\) (or 0.0000000108029879) mph.  (This by Joseph Castro in Live Science gives a nice little precis of the complexities in coming to a single statistic for the speed of human hair growth).  How long would your parents' hair be if they had never had it cut?  What about your grandparents?


33

Make a planar network

Write the numbers 2 to 10 on a piece of paper.  Connect a number with its factors by drawing a line from the number to its factor.  Do this for every number, without any of your lines crossing.  We call this a plane graph.  Keep adding a number to the graph (i.e the next number would be 11).  What's the highest number you can reach before lines have to cross?  (See the problem here.  The problem was adapted from Matt Parker's wonderful book 'Things to Make and Do in the Fourth Dimension'.)


34

Play 'What's My Rule?'

Watch this wonderful video by the always captivating Veritasium, and then repeat the experiment yourselves with as many friends and family as you can.  How do you investigate hypotheses?  Do you seek to confirm your theory, do you look for 'white swans' or do you look for 'black swans'?


35

Sum an infinite series

Amaze people by showing them that you can take an infinite amount of numbers and add them up.  For example, start with a number and keep halving, say 2, 1, 0.5, 0.25, 0.125, etc.  If you kept going to infinity and wanted to add all of the numbers up (to infinity remember), the answer is 4.  Let me try and explain: Call the sum of numbers n, now double each number to get 2n and write each number in 2n underneath the numbers in the set of n.  You will see that all of the numbers in n repeat in 2n, with the exception of the very first number in 2n, 4.  Now 2n - n = n, so if we take each number in the n set and subtract it from each number in the 2n set, we can see that every number that occurs in the n set cancels out with every number in the 2n set, with the exception of the first number in 2n, in this case, 4.  This shows that the sum of the infinite series, 2, 1, 0,5, 0,25, ... is 4.  Make sure gthat you practice before you astound anyone.  Maybe make you own infinitie series and sum all of its numbers.  How complex can you make your series?
\[\begin{array}{c}\begin{align}n &= 2 + 1 + \frac{1}{2} + \frac{1}{4} + ...\\2n &= 4 + 2 + 1 + \frac{1}{2} + \frac{1}{4} + ...\\2n - n &= 4 + \left( {2 - 2} \right) + \left( {1 - 1} \right) + \left( {\frac{1}{2} - \frac{1}{2}} \right) + \left( {\frac{1}{4} - \frac{1}{4}} \right)\\n &= 4\end{align}\end{array}\]

36

Counting to one trillion

Estimate how long it would take you to one trillion.  How accurate can you make your estimation?  Are you going to take into account when you wouldn't be able to count, i.e. when you're sleeping?  Are you going to count out loud and say the actual number in words, if so, would it take you the same length of time to count from say three to four as it would to count from say seven hundred and eighty-nine thousand seven hundred and forty-three to seven hundred and eighty-nine thousand seven hundred and forty-four?


37

Avoid making a square

On a 5 by 5 grid of 25 points, a maximum of 17 points can be marked so that no four of them are vertices of a square with sides parallel to the grid.  Explore what happens with different sized grids.  What is the maximum number of points that can be marked on a 2 by 2 grid, a 3 by 3 grid, etc., so that no four of them are vertices of a square with sides parallel to the grid?  Can you describe what would happen on a grid of size n by n, of n² points?  This link will open up blank grids from 2 by 2 to 14 by 14, plus a larger 30 by 30 if you want a big challenge!  This link will take you to the page for the sequence of known results from the Online Encyclopedia of Integer Sequences.


38

Learn Lewis Caroll's Amazing Number-Guessing game

Read this article from Futility Closet, and practice and learn the fascinating number-guessing game devised by mathematician and author of Alice's Adventures in Wonderland Charles Lutwidge Dodgson (aka Lewis Caroll).  After asking someone to think of a number, and then applying a long series of convoluted arithmetic manipulations, you will always be able to find the original number after asking just three deceptively and disarmingly simple questions — 1) Is the result odd or even?  2) Is the result odd or even?  3) How often does it go?


39

Prove to someone that angles in a triangle do not always sum to 180°

Inflate a balloon.  Get a marker and draw a straight line across part of the balloon.  Go to one end of this line and draw another straight line at 90° to the first line.   Go to the other end of this line and draw another straight line at 90° to the first line.  The last two lines you have drawn will meet/cross at some point on the balloon.  This means that you will have a triangle with two 90° angles at its base, which means in turn that the sum of the angles inside the triangle you have drawn is greater than 180°.  Maybe read this brief article on Non-Euclidean Geometry.


40

Play with 'Look and Say' sequences

Start with number 1.  The next number is 11 because when we describe the preceding number we can see that 1 is just 'one', i.e. we have 'one one', i.e. 11.  The next number describes what we now have, i.e. 'two ones', or 21.  The next number describes what we now have, i.e. 'one two and one one', 1211.  The next number describes what we now have, i.e. 'one one one two and two ones', 111221.  And so on.  Generate this sequence with someone for a few terms and ask them if they can find the next number in the sequence.  They invariably will not be able to but savour their response when you explain to them the rule.  Watch this lovely Numberphile video with mathematician John Conway, who gloriously describes the sequence as 'the stupidest problem you could conceivably imagine that led to the most complicated answer you could conceivably imagine'.  (You might also enjoy this lovely little course on 'sequences and patterns' from Mathigon.)


41

Do some calculator magic

Write the numbers 3, 7, 11, 13 and 37 on 5 separate cards or pieces of paper.  Turn them upside down.  Give someone (your dupe) a calculator and tell them to key in any single-digit number they wish.  Tell your dupe to choose any of the five cards and turn it over, then instruct your dupe to multiply their number by the number on the card, and press equals.  Tell your dupe to choose and turn over another card, and then multiply the answer they have in their calculator by this number, and press equals.  Repeat this with the remaining three cards, but on the last one, slow your dupe down and tell them to only press a button when you tell them to.  Tell them to press multiply, then to key in the number on the last card, and then tell them that when they press the equals sign you will make the number they originally chose reappear lots of times.  (This works because 3, 7, 11, 13 and 37 are the prime factors of 111111, and of course, when you multiply a single digit say 5 by 111111, you get that digit repeated 6 times, i.e 555555.)


42

Discover emirps and bemirps

An emirp is a prime number whose reversal is a different prime, like 37, because 37 is prime and 37 reversed is 73, which is also prime!  Find as many other emirps as you can.  (Note that the word emirp is prime in reverse.)

Interestingly, a ‘cyclic emirp’ is an emirp that generates only more emirps when you move the 1st digit to the rear, e.g. 11,939 → 19,391 → 93,911 → 39,119 → 91,193 → 11,939...  and 193,939 → 939,391 → 393,919 → 939,193 → 391,939 → 919,393 → 193,939...

You might want to play with the bemirps as well.  A bemirp is a 'bi-directional emirp', which means that it is a prime that yields a different prime when turned upside down with reversals of each being two more different primes.  For example, 168601 produces 106861, 198901 and 109891.  (Here is the Bemirps entry in the Online Encyclopedia of Integer Sequences.)  Every number greater than 40258 can be written as the sum of bemirps: Choose some random numbers greater than 40258 and see if you can write then as the sum of bemirps; can you see (or find out) why the number 40258 is the point from which every number becomes expressible as the sum of bemirps?




Notes, References & Links:

Tuesday, 14 August 2018

On Descartes Numbers



A number is said to be Perfect if it is equal to the sum of its proper positive factors, i.e. the sum of its factors excluding the number itself.  Summing all of the factors of a number \(n\) — i.e. including the number \(n\) itself — is known as the sum-of-divisors function:

\[\sigma \left( n \right) = \sum\limits_{d\left| n \right.} d \]
A number \(n\) is thus said to be Perfect when \(\sigma \left( n \right)\) = 2n.  For example, the factors of 28 — 1, 2, 4, 7, 14 and 28 — sum to 28 + 28 = 56, thus \(\sigma \left( {28} \right)\) = 2 \(\times \) 28 and so 28 is a Perfect number.  In contrast, and by way of illustration, the factors of 42 — 1, 2, 3, 6, 7, 14, 21 and 42 — sum to 96, or 2.2857... \(\times \) 42, thus \(\sigma \left( {42} \right) \ne\) 2 \(\times \) 42 and so 42 is not a Perfect number.  In fact, 42 is said to be an Abundant number because the sum of its proper positive divisors is greater than the number itself, i.e. \(\sigma \left( n \right) > 2n\).  When the sum of a number's proper positive divisors is less than the number itself, i.e. \(\sigma \left( n \right) < 2n\), the number is said to be Deficient.  For example, the factors of 50 — 1, 2, 5, 10, 25 and 50 — sum to 93, or 1.86 \(\times \) 50, and thus \(\sigma \left( {50} \right)\) < 2 \(\times \) 42.  As such:

\[\begin{array}{l}\sigma \left( n \right) < 2n \Rightarrow n\;{\rm{is}}\;{\rm{deficient}}\\\sigma \left( n \right) = 2n \Rightarrow n\;{\rm{is}}\;{\rm{perfect}}\\\sigma \left( n \right) > 2n \Rightarrow n\;{\rm{is}}\;{\rm{abundant}}\end{array}\]
Perfect numbers have intrigued us since Euclid, who observed via Proposition 36 in Book IX of his Elements that a number of the form (2\(^{p - 1}\))(2\(^p\) − 1) is a perfect number whenever 2\(^p\) − 1 is prime (i.e. what subsequently became known as a Mersenne prime) [1].  The fact that every even perfect number is of this type, i.e. that every even perfect number can be written in the form (2\(^{p - 1}\))(2\(^p\) − 1) whenever 2\(^p\) − 1 is prime, was proposed by René Descartes in his famed correspondence with Marin Mersenne (specifically in his letter to Mersenne of 15 November 1638) [2], and later proven in 1849 by Leonhard Euler [3].  Descartes was, indeed, 'among the first to consider the existence of odd perfect numbers' (Greathouse and Weisstein, 2012).

As of writing, of the fifty perfect numbers that have been found, all are even, and it is not known if there are infinitely many or, indeed, whether any odd perfect numbers exist — although Pascal Ochem and Michaël Rao showed (2012) that no number up to 10\(^{1500}\) is an odd perfect number.

One number has been found, however, that would have been an odd perfect number if only one of its factors was prime rather than a 'spoof prime' (i.e. a composite number wrongly assumed to be prime).  This number — 198,585,576,189 — was found in 1638 by Descartes, documented in his 15 November letter to Mersenne, and is known as a Descartes Number, or, as an 'odd spoof perfect number' (note that the spoof prime in question is 22021, i.e. 19\(^2\) \(\times \) 61).

\[198,585,576,189 = {3^2} \times {7^2} \times {11^2} \times {13^2} \times 22021\]
Descartes showed that if 22021 was prime, the proper factors of 198,585,576,189 [4] would sum to 198,585,576,189 — making it the only odd perfect number ever found.  As it is, because 22021 is not prime, or because 22021 is a spoof prime, the proper factors of 198,585,576,189 do not sum to 198,585,576,189 (they actually sum to 227,441,894,589).  This makes 198,585,576,189 an odd spoof perfect number, which in itself is the only such number ever found!

To elaborate: Descartes showed that an equivalence to the sum-of-divisors function using the prime factorisation of a number is the following [5]:

\[\sigma \left( n \right) = \prod\limits_{{p^a}\left\|{\;n} \right.} {\frac{{{p^{a + 1}} - 1}}{{p - 1}}} {\rm{ }}\]
where \(p\) is a distinct prime factor.  This means that a number's factors can be summed using only the number's prime factors.  Taking 28 again as an example, we have seen that it is a perfect number because

\[\sigma \left( 28 \right) = \sum\limits_{d\left| 28 \right.} d  = 1 + 2 + 4 + 7 + 14 + 28 = 56 = 2n\]
We can also observe this result using the prime factors of 28, i.e. 2\(^2 \times \) 7:

\[\prod\limits_{{p^a}\left\|{\;n} \right.} {\frac{{{p^{a + 1}} - 1}}{{p - 1}}}  = \frac{{{2^{2 + 1}} - 1}}{{2 - 1}} \times \frac{{{7^{1 + 1}} - 1}}{{7 - 1}} = \frac{7}{1} \times \frac{{48}}{6} = 56 = 2n\]
And so, similarly, the sum of the divisors of Descartes' number:

\[\begin{array}{c}\begin{align}\sigma \left( 198,585,576,189 \right) &= \prod\limits_{{p^a}\left\| {\;n} \right.} {\frac{{{p^{a + 1}} - 1}}{{p - 1}}} \\ \\&= \frac{{{3^{2 + 1}} - 1}}{{3 - 1}} \times \frac{{{7^{2 + 1}} - 1}}{{7 - 1}} \times \frac{{{{11}^{2 + 1}} - 1}}{{11 - 1}} \times \frac{{{{13}^{2 + 1}} - 1}}{{13 - 1}} \times \frac{{{{22021}^{1 + 1}} - 1}}{{22021 - 1}}\\  \\&= 13 \times 57 \times 133 \times 183 \times 22022\\ &= 13 \times \left( {3 \times 19} \right) \times \left( {7 \times 19} \right) \times \left( {3 \times 61} \right) \times \left( {2 \times 7 \times 11 \times 11 \times 13} \right)\\ &=2 \times {3^2} \times {7^2} \times {11^2} \times {13^2} \times \left( {{{19}^2} \times 61} \right)\\ &= 2 \times {3^2} \times {7^2} \times {11^2} \times {13^2} \times 22021\\ &= 2 \times 198,585,576,189\end{align}\end{array}\]
Hence, if 22021 were prime rather than 19\(^2\) \(\times \) 61, Descartes' number 198,585,576,189 would  be an odd perfect number because \(\sigma \left( n \right)\) = 2n.



Notes, References & Links:

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
    (Click here to go back to the top)
    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
    (Click here to go back to the top)
    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: