On The 'Mystery Calculator' Trick

'The Mystery Calculator' trick is a perennial, favourite 'surprise gift' found in Christmas crackers[1]  — along with the (hard) plastic moustache/slug, the fortune-telling fish, and perhaps (if you're lucky) a die.

For a maths teacher, 'The Mystery Calculator' is a potent conceit that piques students' interest and that — when used carefully[2] of course — levers and encourages strong, mathematically mature thinking.  After playing the trick out on/with students, it is demystified carefully, and collectively, before students use their newfound knowledge and understanding creatively in devising a different trick, based on the same principles.  (A ppt version of the cards, one card per slide, can be downloaded here.)

The Trick

• Ask someone to choose a number from any of the six cards.
• Show them each card in turn and ask them if their number appears on it.
• Add the numbers in the top left-hand corner of each card that contains their number.
• The total is their number.

The How / Why

The number 1 in binary is (1 x 2⁰)  → (1)₁₀ = (1)₂  → Hence the number 1 appears on the 1st card only.

The number 2 in binary is 10... (1 x 2¹) + (0 x 2⁰)  → (2)₁₀ = (10)₂  → Hence the number 2 appears on the 2nd card, but does not appear on the 1st or any other.

The number 3 in binary is 11... (1 x 2¹) + (1 x 2⁰)  → (3)₁₀ = (11)₂  → Hence the number 3 appears on the first 2 cards, but does not appear on others.

The number 4 in binary is 100... (1 x 2²) + (0 x 2¹) + (0 x 2⁰)  → (4)₁₀ = (100)₂  → Hence the number 4 appears on the 3rd card but does not appear on the 1st or 2nd, or any other.

The number 5 in binary is 101... (1 x 2²) + (0 x 2¹) + (1 x 2⁰)  → (5)₁₀ = (101)₂  → Hence the number 5 appears on the 1st and 3rd cards, but does not appear on the 2nd, or any other.

The number 6 in binary is 110... (1 x 2²) + (1 x 2¹) + (0 x 2⁰)  → (6)₁₀ = (110)₂  → Hence the number 6 appears on 2nd and 3rd cards but does not appear on the 1st, or any other.

The number 7 in binary is 111... (1 x 2²) + (1 x 2¹) + (1 x 2⁰)  → (7)₁₀ = (111)₂  → Hence the number 7 appears on on 1st, 2nd and 3rd cards, but does not appear on any other.

⋮

The number 41 in binary is 101001... (1 x 2⁵) + (0 x 2⁴) + (1 x 2³) + (0 x 2²) + (0 x 2¹) + (1 x 2⁰)  → (41)₁₀ = (101001)₂  → Hence the number 41 appears on the 1st, 4th and 6th cards but does not appear on the 2nd, 3rd or 5th.

⋮

The number 62 in binary is 111110... (1 x 2⁵) + (1 x 2⁴) + (1 x 2³) + (1 x 2²) + (1 x 2¹) + (0 x 2⁰)  → (62)₁₀ = (111110)₂  → Hence the number 62 appears on the 2nd, 3rd, 4th, 5th and 6th cards, but does not appear on the 1st.

The number 63 in binary is 111111... (1 x 2⁵) + (1 x 2⁴) + (1 x 2³) + (1 x 2²) + (1 x 2¹) + (1 x 2⁰)  → (63)₁₀ = (111111)₂  → Hence the number 63 appears on all six cards.

Possible Teaching Approaches

Thinking through some possible starting questions for students to consider (or to consider with students), before moving on perhaps with students creating their own 'Mystery Calculator' trick (see also [3]):

• What do you notice about the numbers on each card?
• Do these patterns matter — what could we do to see if they do?
• Can we have a number larger than 63 on any of the cards?
• What about all of the other numbers on the cards — can they be chosen randomly?
• What number would appear on all cards if there were four, five, seven, ten, ... cards?  
• Does the number in the top left-hand corner have to be there?
• Are there any constraints to the numbers we place on the cards?
• Would a similar trick in another base, maybe base-3, be more magical?
• How high can we count in binary on our fingers? (Watch this Ted-Ed video from James Tanton.)

---

Notes & (Select) Links:

[1]  See here for an example, or here.

[2]  See 'When Magic Fails in Mathematics,' by Junaid Mubeen.

[3]  See this by Katie Steckles in The Aperiodical, 'On Disreputable numbers'.

For other Christmathsy problems to consider with students, try the 'Santamaths' problem, the '12 Days of Christmas' problem, see this selection from @mathsjem, and have a look at my 'xmaths card'.

Problem... Santamaths

#Probability #Combinations #nCr #Binomial #Modelling

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• For a solution, see this post.

On Mathematics To Listen To

A collection of links to radio and podcast episodes (in English) with a mathematics bent — to inspire teachers to inspire, to feed the minds of our young mathematicians in the making, or, simply, to entertain.  Click here or on the 'Audio' tab above for the complete list.  Programmes have been (loosely) categorised according to themes below, and arranged alphabetically within each category by programme title.

• The History of Mathematics
• The People of Mathematics
• The Mathematics Itself
• Making Sense of things with Mathematics
• Making Sense of things with Statistics
• The Teaching of Mathematics, and the Mathematical Mind
• Other Mathematical Curios

The collection includes, in no particular order:

1. Select episodes from BBC Radio 4 In Our Time 
2. All episodes from BBC Radio 4 Marcus du Sautoy's A Brief History of mathematics
3. All episodes from BBC Radio 4 Marcus du Sautoy's Five Shapes
4. All episodes from BBC Radio 4 Simon Singh's Five Numbers
5. All episodes from BBC Radio 4 Simon Singh's Another Five Numbers
6. All episodes from BBC Radio 4 Simon Singh's A Further Five Numbers
7. Select episodes from BBC Radio 4 The Infinite Monkey Cage
8. All episodes from BBC Radio 4 A History of the Infinite
9. Select episodes from BBC Radio 4 More Or Less
10. Select episodes from BBC Radio 4 Great Lives
11. Select episodes from BBC World Service More Or Less
12. Select episodes from BBC Radio 4 Seriously
13. Select episodes from BBC Radio 4 Incarnations: India in Fifty Lives
14. Select episodes from BBC Radio 4 Start the Week
15. Select episodes from BBC Radio 4 The Life Scientific
16. Other programs from BBC Radio
17. Select episodes from The Partially Examined Life podcast
18. Select episodes from Futility Closet podcast
19. Select episodes from Freakonomics podcast
20. Select episodes from Radiolab podcast
21. Select episodes from Teach Better podcast
22. Select episodes from Modern Learners podcast
23. Select episodes from Plus Magazine podcasts
24. Select episodes from The Guardian's Science Weekly podcast
25. Select episodes from The Naked Scientists podcast
26. Select episodes from The NCETM Maths podcast
27. Select episodes from The James Altucher Show podcast
28. Select episodes from BBC Radio Cambridgeshire and BBC Radio Oxford Funbers
29. Select episodes from Things Fall Apart podcast
30. Select episodes from The Zone of Potential Construction podcast

There are, of course, many other superb podcasts from which episodes have not been included in this selection, purely because as their sole focus is on mathematics or its teaching, I would be listing links to all of their episodes.  (Episodes in this selection are from series that don’t concentrate solely on mathematics, or from shorter series that concentrate on one aspect of mathematics.)  

I have listed a selection of some other, excellent podcasts solely focused on mathematics and/or its teaching here, but of course this list is not exhaustive.  If you are aware of other podcasts or audio to point people in the direction of, do let me know. 

Problem... The 12 Days of Christmas

#Sequences, #GeneratingSequences, #TriangularNumbers
#Generalising, #WordProblem, #Series, #SummationOfSeries

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• For a solution, see this from MathsIsFun.
• To extend the activity, see this from nrich.

Problem... Countdown

#MentalArithmetic, #MentalCalculation, #OrderOfOperations, 
#AlgorithmicThinking, #ComputationalThinking

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• For a solution, see this from DataGenetics.
• To extend the activity, see this post.
• For a more in-depth algorithmic look.

On 'Incredible' Feats of Mental Arithmetic (in Countdown)

Problem... It's π O'Clock

#Fractions, #Time, #Angle, #Turn, #Conversions, #Pi #π, #PiDay, #πDay

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• For a solution, see this post.
• To extend the activity, see this problem.

On the little things in teaching #1

That beautifully judgemental silence from a class faced with your meticulously engineered pun, which, as a function of their understanding of the concept underpinning it, can only mean that they really get it...

M
ME [during a discussion about correlation]:

Yes, the more overweight men with white beards wearing red suits we see, the greater the chance it is that it's nearing Christmas.  That's a good example.  So, could we say this in a more statty way?

STUDENT:
There is a strong positive correlation between the number of overweight men with white beards and red suits and when Christmas is.

ME:
I'm not sure how we could judge the strength, but other than that yes, we can say that there is a positive correlation between the number of overweight men we observe with white beards wearing red suits and how close Christmas is.  Or maybe it's a negative correlation?  I don't know.  Does it matter?

STUDENT:

Yes, because the more men you see dressed up as Father Christmas, the less time it is until Christmas.

ANOTHER STUDENT:

But you could say that the closer Christmas is the more people we see dressed up as Santa.  And that's more positive... But I suppose we're saying the same thing.

ANOTHER STUDENT:

But if you drew a graph with the date on the x axis and the number of people dressed up as Santa on the y, it would go up.

ME [slightly panicky; the chance for the pun slipping away]:

It's a bit like the chocolate consumption and Nobel Prize data we looked at [see here].  Before I told you what the data was, you assumed that more of x would make more of y.  But when you saw what the data was...

STUDENT:

...Oh that thing where the more chocolate people ate in a country the more Nobel prizes they won...

ME:

Yep.  So when we read that graph, or misread it...

STUDENT:

It would mean that you have more chance of winning a Nobel Prize if you eat a lot of chocolate.

ANOTHER STUDENT:

Which is just dumb...

ME [quickly]:

Which is why describing it in more of a statty way is safer.  So, if we think about the men dressing up as Santa example in the same way, what would we be saying?  What would it mean if we thought about it in the wrong way?

STUDENT:

If we wanted Christmas to come quicker we would just need to see more people dressed up as Father Christmas...

ANOTHER STUDENT:

Or that we could make Christmas come quicker by getting more people to dress up as Santa.

ME [relieved]:

Exactly.  But the fact that we see an increase in overweight men with white beards wearing red suits as we get closer to Christmas, does not cause Christmas to come quicker does it?  Do you get the distinction?  If we wanted Christmas to be tomorrow would we just get millions of people to dress up as Santa?  No, of course not. The important thing to get here is that correlation may suggest a relationship between two things, but it does not imply Clausality.

AyThangYaw.

---

(See here for a selection of correlation and causation conflations.)

On the Crescent Area Problem

Problem... A circle is inscribed in a square that has a side length of one unit, which is also the radius of a quarter-circle.  Find the area of as many of the different shapes and regions that you can see.  (Which shapes, or regions, are you finding difficult?  What is it that is stopping you find their areas?)

---

Students typically define the square, the inscribed circle, the quarter circle and often a quarter of the inscribed circle as shapes and regions to find the areas of, and proceed to do so without issue.  Students also note the dark orange 'crescent' region as an area to find, and, indeed, the lighter orange regions around it, and these delineate the aspects of the problem that have the potential to develop into the most formative discussions.

Students, with the teacher's guidance / encouragement of course, can outline a strategy of how to find these more complex areas, ignoring whether or not they yet have the knowledge, skills or understanding to follow such a strategy through.  Depending on where students are in their mathematical careers, this can then lead into more direct teaching, or, moreover, can function as a segue into a body of work that will provide students with the knowledge and skills they need to apply their strategy.

To see a 'thinking through' of a typical solution / approach with students to the problem of the 'crescent' area, click on 'Read more' below.

Suggested Solution:

Problem... The Greatest π-day Birthday in History

#Time, #TimeConversion, #Conversions
#ComutationalThinking, #AlgorithmicThinking
#Pi #π, #PiDay, #πDay

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• Download png format.
• Download a solutions 'think through'

SPOILER:

Whether you take π o'clock to mean between 03:14:15 and 03:14:16, or as 3.141592653... am, people born on January 21, 1589 in the Gregorian calendar would have been π years old on π day 1592, i.e. on 3/14/1592.  One of these people was William Strode Jr, or William Strode of Barington, philanthropist and Roundhead in the English Civil War, who was born on 11 January 1589 according to the Julian calendar.  In 1589, however, the Gregorian calendar was 10 days ahead of the Julian calendar, which was still in use in Britain.

Problem... Crescent Area

#Area, #Circle, #Arc, #Sector, #Segment, #Trigonometry, #CosineRule

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• For a solution, see this post.

On 73

73 is a permutable (or anagrammatic) prime, a prime that can have its digits' rearranged (in base 10) in any permutation and still be a prime number. 73 is also an emirp, a number whose reverse, 37, is also prime — a property also evident in terms of the respective ordinal positions of 73 and its emirp partner, 73 being the 21st prime [1] while 37 is the 12th prime.  73 is a Sophie Germain prime and is palindromic in binary 1001001 (interestingly, all Fermat primes and Mersenne primes are subsets of the binary palindromic primes).  73 is also an octal palindrome 111 and the only octal prime repunit.  73 is, moreover, and for some of the reasons given here, Sheldon Cooper's favourite integer in The Big Bang Theory — as was first referenced in the show’s 73rd episode.  (Jim Parsons, incidentally, the actor who plays Sheldon Cooper, was born in 1973.)

However, after listening to Charles Yang, associate professor in the University of Pennsylvania Department of Linguistics, in this episode of the superb Tell Me Something I Don't Know podcast, hosted by Stephen Dubner of Freakonomics fame, it is another little number-73-centred-nugget-of-information that has prompted this post.  Namely, that once an English speaking child has learned to count to 73, they realise they can keep going, and thus ‘understand’ the concept of infinity (NB the 1st two comments at the end of this blog).

Having delved into Yang's fascinating research, let me try and elaborate:  If there is a linguistic rule, a generalisation in other words that can be applied to a set of N words, but within this set of N words there is a subset of words, e [2], that do not follow this rule and that must therefore be memorised, then Yang (2005, cf. 2015) has shown that:

\[e < {\theta _N}\;where\;{\theta _N}: = \frac{N}{{\ln \left( N \right)}}\]

This model, 'dubbed' the Tolerance Principle by Yang (2005, cf. 2015 pp16-17), asserts that for the generalisation to be ‘productive’, i.e. for the rule that generates words to tolerate the exceptions and thus be applied instead of the pure memorisation of all N, e must not exceed N / ln(N).  It delineates, in other words, a threshold value for e (as a function of N) at which a word-forming rule ceases to be a productive mechanism for the learner to apply, and thus provides a ‘sufficiency measure’ for such linguistic generalisation.

By way of illustration, Yang (2015, pp948-949) gave this example of regular and irregular verbs: ‘If a typical English speaker knows e = 120 irregular verbs, the productivity of the -ed rule is guaranteed [my emphasis] only if there are many more regular verbs. Specifically, there must be N verbs, including both regulars and irregulars, such that θ_N = N / ln(N) ≥ 120.  The minimum value of N is 800.  In other words, if there are at least 680 regular verbs in English, the -ed rule can tolerate 120 irregular verbs’.

In his Tell Me Something I Don’t Know appearance, Yang described applying this Tolerance Principle to how we as children learn to count.  We know how many of our counting words are exceptions to the rule, those numbers that is that ‘bear no relation to their quantity, [that are] are completely arbitrary’, those numbers to put it another way that ‘won’t tell you anything about how numbers are forming' (Yang, in Berger, 2017) — in the case of children counting in English, there are 17 of these exceptions, e = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 20, 30, 50} [3].

Once we have memorised, ‘by brute force,’ the number words one to ten, we start to observe and learn rules for continuing.  For example, we start to see that fourteen is formed from ‘four and ten’, sixteen from ‘six and ten’, twenty-four from ‘two tens and four’, 'twenty-six from 'two tens and six', etc.  But there remain of course those exceptions to the rule — why is eleven not one-teen, or twelve not two-teen, etc. —  where ‘memorisation [is] the sole educational mechanism’ (Yang, in Berger 2016).  To this end, using the Tolerance Principle, we can find the least amount of words that we need to learn ‘to overcome the [17] exceptions we have to memorise’.

The smallest value of N such that θ_N = 17 is 73:

\[\begin{array}{l}\begin{align}N &= 73\;\\\because17 &= e < {\theta _N} = \frac{{73}}{{\ln \left( {73} \right)}}\\\;where\;{\rm{ }}\frac{{73}}{{\ln \left( {73} \right)}} &= 17.014...\end{align}\end{array}\]

Or in other words, once a child has learned to count to 73, they have learned the rules of the game sufficiently to overcome the cognitive need for memorisation [4].  Once children have passed this 'tipping point', when they come to a number that they are unaware has a word outside of the system they have subconsciously generalised, a thousand for example, they make up a word that fits their generalised system.  Thus, once an English speaking child can count to 73, they can count forever.

Notes & (Select) Links:

[1]  Incidentally, also the product of multiplying 7 and 3.

[2]  Not to be confused with Euler's number, e.

[3]  Yang noted that in Chinese, e = 11, meaning that a Chinese speaking child need only count up to 42 before they are able to continue and thus appreciate the concept of infinity, some one whole year before English speaking children.  

[4]  Welsh may be one of the languages with the smallest threshold at θN = 36.  The numbers 1-10 in Welsh are the only 'exceptions': Un (one), Dau (two), Tri (three), Pedwar (four), Pump (five), Chwech (six), Saith (seven), Wyth (eight), Naw (nine), and Deg (ten).  All numbers beyond this are generalised from them, for example eleven is un deg un (one ten one), twelve is un deg dau (one ten two), seventy three is saith deg tri (seven ten three), etc.

Berger, M.  (2017, October 20). PennCurrent. 'Penn linguist determines tipping point for children learning to count' [Blog post].  Retrieved from: https://penncurrent.upenn.edu/research/penn-linguist-determines-tipping-point-for-children-learning-to-count

Tell Me Something I Don't Know. (2017).  How to Count to Infinity. [podcast] Available at: http://tmsidk.com/2017/10/how-to-count-to-infinity/.

Yang, C.  (2015).  'Negative Knowledge from Positive Evidence', Language, Vol. 91, No.4, pp938-953.  Retrieved from: https://www.linguisticsociety.org/sites/default/files/07_91.4Yang.pdf

Yang, C.  (2005).  'On Productivity', in Linguistic Variation Yearbook 5, John Benjamins Publishing Company.  Retrieved from: http://www.ling.upenn.edu/~ycharles/papers/l.pdf

On Santamaths (An xmaths conversation with a 5 year old)

OK, let's get straight to the point: it is a fallacy, or to put it another way, a gross oversimplification to think that Santa will deem you, in binary fashion, to have been either naughty or nice, discretely, categorically, one way or the other, and that this will in turn delineate whether you receive presents or not.  This is — plainly, to put it politely, and as I will show in what follows — utter horse nonsense.  There are gradations of naughtiness or niceness: one person can be naughtier than another, or nicer, and it is evident to boot that naughtiness is a function of niceness, and vice-versa.  To put it another way, an increase in naughtiness (say η) and a decrease in niceness (say η) are biconditional:

\[\rlap{-} \eta \;\; \Leftrightarrow \;\;\eta \]

To elaborate: Sometimes you will be naughty and sometimes you will be nice. And of course the more you are naughty the less time you have to be nice — and vice-versa: the more you are nice the less time you have to be naughty.  So, and in other words, the naughtier you are the less likely it is that you will be nice, and the nicer you are the less likely it is that you will be naughty.  Yes?  The truth is that two naughty people will be naughty to different degrees: one will be more or less naughty than the other.  (The same is obviously true for two nice people.)  And of course, one naughty act might not be as naughty as another, so there are gradations of naughtiness and, indeed, gradations of niceness.  It is nice to say please and thank you, for example, but it is nicer to slot a pound into a Noddy car ride in the shopping centre and leave it to be discovered by an unsuspecting — but soon to be overjoyed — toddler.  Similarly, it is naughty to take your sister's cuddly Pooh bear when she doesn't want you to, but it is naughtier to hide Pooh and tell your sister that he’s run away because he doesn’t love her her anymore.

Santa is not stupid; he knows all of this.  And he uses Santamaths to deal with the inherent variability in η and hence η.  Put simply, he uses relative frequencies to assign a weighted 'niceness index' (known as the η_i — pronounced /niː/) to each person (𝓍), and the presents each person receives are thus dependent on such.  In short:

\[\begin{array}{l}\forall x,\;0 \le {\eta _i} = \frac{\eta }{{\eta  + \rlap{-} \eta }} \le 1\\\\and\;as\;{\eta _i} \to 1,\;\forall x\left( {\rlap{-} \eta  \Rightarrow \eta } \right),\\and\;P\left( {x\;gets\;what\;s/he\;wants} \right) \to 1\\\\Similarly,\;as\;{\eta _i} \to 0,\;\forall x\left( {\eta  \Rightarrow \rlap{-} \eta } \right),\\and\;P\left( {x\;gets\;what\;s/he\;wants} \right) \to 0\end{array}\]
Santa then uses probability theory and combinatorics to calculate what and how many presents you get.  As such, this year, as you know, I have been keeping a tally of all the times you have done something naughty (η) and all of the times you have done something nice (η).  You get one scratch for a naughty act and one for a nice act, an act of kindness if you will, and dependent of course on the act.  Remember that time you fooled your sister —  your type 1 hypersensitive to tomatoes sister — into thinking that ketchup you had mixed in her ice-cream was actually raspberry juice, well that got you ten scratches in the naughty column, for example, whilst tidying up your Lego without being asked that one time you did it in the whole year got you five scratches in the nice column.  

Here's a little section of the tallies I've been keeping for this year:

Overall you had 740 naughty scratches this year and 623 nice scratches, giving η_i = 0.46:

\[\begin{array}{l}\begin{align}\frac{{623}}{{740 + 623}} &= \frac{{623}}{{1363}}\\\\ &= 0.46\end{align}\end{array}\]
Now, this doesn't mean that you have a 46% chance of getting your presents this year.  Oh no.  You wrote to Santa to ask for the following five presents: 

• An umbrella in the shape of an Octopus, 
• A Lego Batman car [I presume you mean the Batmobile], 
• A green and orange ukulele, 
• A dinosaur that can really move and roar and scare people like a monkey on a horse, 
• A helicopter (remote-controlled will do if Santa can't source a real one for you). 

Hence the following Santamaths model applies:

\[\begin{array}{l}{\left[ {{\eta _i} + \left( {1 - {\eta _i}} \right)} \right]^5} = {}^5{C_5}{\left( {{\eta _i}} \right)^5}{\left( {1 - {\eta _i}} \right)^0} + \\\quad \quad \quad \quad \quad \quad \quad \quad \quad {}^5{C_4}{\left( {{\eta _i}} \right)^4}{\left( {1 - {\eta _i}} \right)^1} + \\\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad {}^5{C_3}{\left( {{\eta _i}} \right)^3}{\left( {1 - {\eta _i}} \right)^2} + \\\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad {}^5{C_2}{\left( {{\eta _i}} \right)^2}{\left( {1 - {\eta _i}} \right)^3} + \\\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad {}^5{C_1}{\left( {{\eta _i}} \right)^1}{\left( {1 - {\eta _i}} \right)^4} + \\\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad {}^5{C_0}{\left( {{\eta _i}} \right)^0}{\left( {1 - {\eta _i}} \right)^5}\end{array}\]
and inputting the respective values for η_i and 1 – η_i...

\[\begin{array}{l}{\left[ {{\eta _i} + \left( {1 - {\eta _i}} \right)} \right]^5} = 1{\left( {0.46} \right)^5}{\left( {0.54} \right)^0} + \\\quad \quad \quad \quad \quad \quad \quad \quad \quad 5{\left( {0.46} \right)^4}{\left( {0.54} \right)^1} + \\\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad 10{\left( {0.46} \right)^3}{\left( {0.54} \right)^2} + \\\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad 10{\left( {0.46} \right)^2}{\left( {0.54} \right)^3} + \\\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad 5{\left( {0.46} \right)^1}{\left( {0.54} \right)^4} + \\\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad 1{\left( {0.46} \right)^0}{\left( {0.54} \right)^5}\\\quad \quad \quad \quad \quad \quad \quad  = 0.0205... + \\\quad \quad \quad \quad \quad \quad \quad \quad \quad 0.1208... + \\\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad 0.2838... + \\\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad 0.3332... + \\\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad 0.1955... + \\\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad 0.0459...\end{array}\]
So as you can see, there is therefore only a 2.06% chance that you will receive all the presents you have asked Santa for this year — indeed, it is more than twice as likely that you will receive nothing.  The most likely thing to happen is that you will receive 2 of the 5 presents you have asked Santa for, but there is more than a 50% chance that you will receive 2 or 3 of the presents you have asked for. 

I am sharing this analysis in the detail I am with you so as to prepare you for any possible disappointment on Christmas morning.  I must of course emphasise that this is a purely theoretical model and thus that the experimental error that is life may get in the way, and result in you having all of the presents you have asked for, or, indeed, even more.  (Your sister, incidentally, has asked for 4 presents.  I wonder what proportion of her behaviour would need to have been deemed nice in order for her to have a 90% or more chance of receiving at least 3 of them?)

Night-night.

Problem... The Sophie Germain Identity

#AlgebraicManipulation,  #ExpandingBrackets, #Proof

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• For a blog post on a solution, see this post.

On The Sophie Germain Identity

Marie-Sophie Germain (1776-1831) was, in short, an extraordinary mathematician.  Noted for her work across a range of disciplines, from number theory to physics and astronomy, she is widely recognised as one of the first women to make significant and original contributions to mathematical research.

Germain's story is, in more ways than one, a remarkable one, and it is not hyperbole to describe her achievements, whether framed within the context of time and place or not, as inspirational. Forbidden to attend University and thus unable to make a career from her mathematics, Germain, with the financial and moral support of her father, famously taught herself.  She was inspired by Jean-Étienne Montucla's Histoire des Mathématiques, read Newton and Euler, and, under the nom de plume of a Monsieur LeBlanc, corresponded with preeminent mathematicians of the time, most notably Adrien-Marie Legendre, Joseph-Louis Lagrange and Carl Friedrich Gauss.  It was through her persistence, resilience, ingenuity, inquisitiveness and most importantly the sheer, outrageous brilliance of her enduring work, that she overcame — to an arguable extent — the prejudices of 19th-century French society that threatened to impede her way in mathematics.

Germain was inspired by Ernst Chladni's beautiful, awe-inspiring 'figures' to formulate a mathematical theory of elastic surfaces that described Chladni's experimental observations, eventually publishing Récherches sur la théorie des surfaces élastiques in 1821 (see Gardner's and Kowk's fascinating paper), and in doing so winning the prix extraordinaire from the esteemed Académie des sciences.  She was also one of a select number of mathematicians to have worked on Fermat's Last theorem, including 'Euler, Legendre, Gauss, Abel, Dirichlet, Kummer and Cauchy' (Dickson, 1917), resulting in arguably her most notable achievement, what is now known as the Sophie Germain Theorem (for more detail see here, this typically insightful account from Simon Singh, and this from Robin Whitty of theoremoftheday.org, who notes that Germain provided 'the first real breakthrough on Fermat's Last Theorem since Fermat’s death in 1665').

Whilst this post is not about Germain's life or accomplishments per se, it is an advocation — it is about sharing Germain's story with our students, but in a way moreover that achieves more than mere reach, or appreciation, or admiration.  This post is about sharing Germain's story with our students in a way that supports their learning and enhances their mathematical experience.  It is about using narrative to personify, if you like, Germain's mathematics — in an effort to make it more accessible, to make the seemingly difficult be more 'learn-able', to undermine in turn the pervasive (and often persuasive) fear of 'hard maths'.  For all of our students, boy or girl, for the sake of their own mathematical maturation and in the perspectivist sense of the messages we inadvertently transmit as teachers, it is the way in which we tell Germain's story that matters more than the story itself [1].

Germain, along with Emmy Noether, Hypatia of Alexandria, Sofia Kovaleskaya and Ada Lovelace, are often the women schools refer — or should that be, if you will forgive me a degree of provocativeness here, defer — to when celebrating the women of mathematics, usually on a perfunctory once or twice a year, such as on the albeit superb Ada Lovelace Day[2].  But of course every day is (or should be) Ada Lovelace Day, and of course the stories of these amazing mathematicians should be told, celebrated, and learned from.  However, to encourage the greatest effect (i.e. deep and lasting learning for our students), we must take care to share these stories at appropriate — i.e. not overly contrived — points in the curriculum, and that when doing so, in the face of such overwhelmingly compelling socio-cultural narratives, we do not lose or dilute the underlying story of their astounding mathematics and mathematical achievements. 

As I have elaborated upon in more detail in an earlier post, we must do more as teachers than just not deny (inadvertently or otherwise) the girls we teach access to the positive gender role models they need (see, for example, Dennehy and Dasgupta, 2017; Lockwood, 2006; Allen and Eby, 2004).  And this is about more than just telling their stories, it is also about considering what it is that makes them positive role models, considering what it was that drove their intellectual achievements.  To this end, we should want all of our students, boy or girl, to be inspired by Germain's mathematics, to be inspired by Germain because of her mathematics and the manner in which she did it, not just by the albeit inspirational fact that her mathematics was done in the face of almost insurmountable odds. 

I am suggesting, therefore, that we look for genuine ways of exposing our students to the accessible aspects of Germain's mathematics (and the mathematics of her ilk) as early as possible in their mathematical careers — wherever that is possible, wherever that is doable.  It is a deliberate effort, in other words, to expose students to what they will perceive to be ‘hard maths’ earlier than is perhaps typical, when they may not ordinarily be considered 'ready', to increase therefore the opportunities for ‘inspiration’ and support our students' mathematical maturation, to sail against the prevailing cultural and pedagogical winds and make students begin to see themselves as mathematicians in the making.  Exploring Sophie Germain primes when working on properties of number, for example, or, indeed, when working on algebraic manipulation, this:

Prove the Sophie Germain Identity
\[{a^4} + 4{b^4} \equiv \left( {{a^2} + 2{b^2} + 2ab} \right)\left( {{a^2} + 2{b^2} - 2ab} \right)\]

The outline presented as follows is a ‘thinking through’ of how this problem could be approached in a way to make it accessible to secondary students yet to have formally learned how to factorise expressions.  The problem is intended, thus, to be shared and explored with students before they typically learn the relatively more advanced techniques that would normally (and more efficiently) be used, i.e. completing the square and factorising by finding the difference between two squares.  It is intended in other words for use and exploration with secondary students who have some knowledge of simplifying and manipulating algebraic expressions by collecting like terms, multiplying a single term over a bracket, expanding products of two binomials, and the multiplication index law.

The idea behind this exercise is to consider with students how they could prove the identity, using their existing knowledge, without the direct verification of the identity that they are likely to suggest — without, in other words, ‘going that way’, multiplying out the right-hand side and simplifying (which should still of course be applauded and reinforced as a sound proof).  This then situates students at the limits of their existing knowledge and frames their approaches within a more ‘problem solving’ context.  Students’ understanding of the techniques they have already learned — or are in the process of learning — should deepen as a result of working with this problem.

The accompanying text is intended purely to help teachers consider how to approach the problem with students, structurally [3] and pedagogically, and consider what are the questions that can/should be asked of their students, should they not yet be able to ask them for themselves. (This from Brilliant.org could be used to illustrate how the identity is often used in modern mathematics, and  thus perhaps lead onto other areas of learning.)

(A .pdf version of this can be downloaded here.) [4]

Notes & (Select) Links:

Problem... The Most Fantastic Magic Square Ever Discovered

#MagicSquares, #AlphmagicSquares, #LeeSallows 

• Download pdf format.
• Download png format.
• Click here for Lee Sallows work on this and other squares.
• Watch this from Matt Parker (@standupmaths)

Problem... The Repeating Number

#FundamentalTheoremOfArithmetic, #PrimeFactors, #MathsMagic 

• Download pdf format.
• Download png format.

Problem... Planar Network

#Factors, #PlanarGraphs, #Topology 

• Download pdf format.
• Download png format.
• Adapted from Matt Parker's 'Things to Make and Do in the Fourth Dimension'

Problem... Vampire Numbers

#NumberProperties #Factors #VampireNumbers

• Download pdf format.
• Download png format.
• For Cliff Pickover's original post defining Vampire Numbers.
• For results, as given in the OEIS.
• Vampire Numbers on Wolfram Mathworld.
• Watch this video from @Numberphile:

Problem... Cake

#Combinations, #Rotation, #Computational Thinking 

• Download pdf format.
• Download png format.

The problem was adapted from the 'Ice Cream Cake' problem in Peter Winkler's 'Mathematical Mind Benders' (p111 with solution on pp115-118), and via Stan Wagon's take on it in Gil Kalai's blog.  It's likely origin was as problem 31.2.8.3 in Olympiad 31 (1968) Moscow Mathematical Olympiads (p90).

SPOILER/SOLUTION:

The values of \(f\left( \theta  \right)\) for \(\left\{ {\theta |\theta  \in \mathbb{N},1 \le \theta  \le 360} \right\}\) can be generated thus:

\[\begin{array}{l}Let\;k = \left\lceil {\frac{{360}}{\theta }} \right\rceil \\\\f\left( \theta  \right) = 2k\;for\;\left\{ {k|k = \frac{{360}}{\theta },1 \le \theta  \le 360} \right\}\\\\f\left( \theta  \right) = 2k\left( {k - 1} \right){\rm{ for }}\left\{ {k|k \ne \frac{{360}}{\theta },1 \le \theta  \le 360} \right\}\end{array}\]
Giving \(f\left( {181^\circ } \right) = 4\) and \(f\left( \theta  \right)\) for all \(\left\{ {\theta |\theta  \in \mathbb{N},1 \le \theta  \le 360} \right\}\):

θ	f(θ)
1	720
2	360
3	240
4	180
5	144
6	120
7	5304
8	90
9	80
10	72
11	2112
12	60
13	1512
14	1300
15	48
16	1012
17	924
18	40
19	684
20	36
21	612
22	544
23	480
24	30
25	420
26-27	364
28-29	312
30	24
31-32	264
33-35	220
36	20
37-39	180
40	18
41-44	144
45	16
46-51	112
52-59	84
60	12
61-71	60
72	10
73-89	40
90	8
91-119	24
120	6
121-179	12
180-359	4
360	2