## Thursday, 17 February 2022

### On Twos for Twosday

At 22 seconds past 22 minutes past ten in the evening of the 22nd day of the 2nd month of the 22nd year this century (which falls on a Tuesday, or Twosday), the time and date will consist of a single repeating digit: the last occasion this will happen in our lifetimes.  In the date format dd/mm/yyyy, 22/02/2022 is also a palindromic date, i.e., is read the same backwards as it is forwards.  To mark and celebrate Twosday, here are a few curios about twos...

## 2...

2 is the first (and only even) prime.

2 is the only number which, when added to itself, gives the same result as when multiplied by itself (2+2 = 2×2).

2 is the only difference between two consecutive primes that is prime  (3 and 5, 5 and 7, 11 and 13...).

2 is the smallest prime that produces another prime seven consecutive times by adding a digit (2, 29, 293, 2939, 29399, 293999, 2939999, 29399999).

2 has a square root that is probably the first number known to be irrational, known as Pythagoras' Constant (watch this Numberphile video, and this from VSauce).

2 is the number most feared by primonumerophobics.

2 is strobogrammatic (the same upside-down) in an 8-segment display.

2 has 2 homphones (to and too).

2 is a child's age on 22/2/22 if they were born on 22/2/2020 (children born at 20:20 on 6/12/2019 will be 2 years, 2 months, 2 weeks, 2 days, 2 hours and 2 minutes old at 22:22 on 22/2/22).

Listen to 'Two - At the double' from Simon Singh's 'Numbers'.

Listen to 'Funbers 2' from Tom Rocks Maths Funbers broadcast.

## 22...

22 is the first self-describing number (it has 'two twos").

22 is the smallest number that can be expressed as the sum of 2 primes in more than 2 ways (3+19 = 5+17 = 11+11).

22 is the number of ways 23 can be expressed as the sum of positive integers (8, 7+1, 6+2, 6+1+1, 5+3, 5+2+1, 5+1+1+1, 4+4, 4+3+1, 4+2+2, 4+2+1+1, 4+1+1+1+1, 3+3+2, 3+3+1+1, 3+2+2+1, 3+2+1+1+1, 3+1+1+1+1+1, 2+2+2+2, 2+2+2+1+1, 2+2+1+1+1+1, 2+1+1+1+1+1+1, 1+1+1+1+1+1+1+1).

22 is the largest known arithmetic sequence of primes, starting with 11,410,337,850,553 and ending with 108,201,410,428,753, with a common difference of 4,609,098,694,200.

22 is the number of different ways five pentagons can be linked together (i.e., there are 22 pentahexes).

22 has a digit sum (2+2 = 4) equal to the sum of the digits of its prime factors (2, 11... 2+1+1).

22 has a digit sum (2+2 = 4) equal to the product of its digits (2×2 = 4).

22 is the maximum number of pieces that can be created when cutting a circle with six lines (i.e., 22 is the 6th term of the lazy caterer's sequence).

22 is a person's age on 22/2/22 if they were born on 22/2/2000 (people born at 00:00 on 06/12/1999 will be 22 years, 2 months, 2 weeks, 2 days, 22 hours and 22 minutes old at 22:22 on 22/2/22).

## 222...

222 is the product of exactly three distinct primes (2×3×37).

222 is the sum of 2 successive primes (109+113).

222 is the sum of three squares in at least three ways (1+25+196, 1+100+121, 4+49+169).

222 is the sum of all 2-digit primes formed from consecutive digits (23+43+67+89).

222 is the 22nd number that produces a prime when divided by the sum of its digits [222÷(2+2+2) = 37].

222 has factors that can be partitioned into 2 sets with equal sums (1+2+3+222 = 6+37+74+111).

222 cannot be written as a number + the sum of the number's digits, for any number.

## 2222...

2222 is the smallest number divisible by a 1-digit, 2-digit and 3-digit prime (2222÷2 = 1111, 2222÷11 = 202, 2222÷101 = 22).

2222 has factors that are all palindromic (1, 2, 11, 22, 101, 202, 1111, 2222).

2222 squared is the sum of 88 consecutive squares (22222 = 4,937,284 = 1922+1932+...+2792).

2222 is the sum of six cubes in exactly six ways.

## 22222...

22222 is a Kaprekar number because 222222 = 493,817,284 and 4,938+17,284 = 22,222.

22222 is the smallest palindromic multiple of 41 with even digits (22222 = 41×542).

## Wednesday, 2 December 2020

### On the Colour of the Third Card

Three cards are dealt from a normal deck.  You don't see them.  You are told that the first two are the same colour, but not what colour they are.  What is the probability that the next card is the same colour?

$\frac{1}{2},\;\frac{1}{4},\;\frac{{12}}{{25}},\;or\;\frac{4}{{17}}\;?$

This beautiful, onthefaceofit innocuous little problem comes from the great Martin Gardner's 'Modelling Mathematics with Playing Cards'.  It's a problem that never fails to invoke heated discussion and vehement argument when I share it with students to play with.  The four solutions and reasoning sketched out below are those typically proffered (and invariably staunchly defended) by students.  Which solution would you go with, and why?  Do you have a different solution?  And what argument would you give to those who firmly hold the solution to be one of the others to show them that they are wrong and you are right?  (Note that Gardner's solution was solution 2 [1])

## Solution 1

The last card could either be 1) the same colour by being red, or 2) a different colour by being red, or 3) the same colour by being black, or 4) a different colour by being black.  So there are four possibilities and two of these result in the final card being the same colour as the previous two.  So:

$\frac{2}{4} = \frac{1}{2}$

## Solution 2

There are eight ways that the arrangement of the colours of the three cards can occur, namely RRR, RRB, RBR, BRR, RBB, BRB, BBR, or BBB (where R = Red and B = Black), and two of these arrangements have the final card as the same colour as the previous two.  So:

$\frac{2}{8} = \frac{1}{4}$

## Solution 3

After the first two cards, fifty cards remain in the deck.  If the first two cards are both red, there remain twenty-four cards that are red.  Similarly, if the first two cards are both black, there remain twenty-four cards that are black.  So:

$\frac{24}{50} = \frac{12}{25}$

## Solution 4

For the last card to be the same colour as the first two, the colours of all three cards are either RRR or BBB.  You have fifty-two cards to choose from for your first card, and twenty-six of these are red and twenty-six are black.  After taking the first card from the deck, you have fifty-one cards left to choose from for your second card, and assuming that the first card was red, twenty-five of the fifty-one cards left are also red.  After taking the second card from the deck, you have fifty cards left to choose from for your third and final card, and assuming that the first two cards were red, twenty-four of the fifty cards left are also red.  The same would be true if the cards pulled from the deck were black.  So:

$\left( {\frac{{26}}{{52}} \times \frac{{25}}{{51}} \times \frac{{24}}{{50}}} \right) \times 2 = \frac{{31200}}{{132600}} = \frac{4}{{17}}$

## Wednesday, 17 July 2019

### 42 Mathsy Things to do

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

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.)

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## 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?