## The Game of 24

As a traditional Chinese parent, my girlfriend1 Ying is concerned with our daughters’ arithmetic skills.2 To this end, Ying has played a game with them from an early age and still plays it with them: the game is called 24. The rules of 24 are simple: deal four cards, and then use all four cards and any basic arithmetic operations (and brackets) to make a total of 24. Given the four cards above, for example, we could get to 24 by

### (3 x 6) + 10 – 4

For clarity, suits do not matter, only the four basic operations are permitted, each card must be used exactly once, aces count as 1, and jacks, queens and kings count as 11, 12 and 13, respectively (or can simply not be used for younger players).3 Of course some card combinations are very easy and can be solved in multiple ways, but others are much more difficult. Some combinations are impossible. A great challenge, courtesy of Tony Gardiner, appears below.

Although it’s a natural and excellent game, I had never paid 24 much thought. That may have been partly because I had a snowflake’s chance in hell of ever beating Ying. A game with her almost invariably consists of four cards being dealt, a nanosecond later Ying calling out “Got it!”, and then four more cards being dealt, and so on. It wears thin.

I became more interested in 24 when I read about the game in Alexandre Borovik and Tony Gardiner’s Essence of Mathematics.4 I then became much more interested in 24 and its origins after Ying introduced the game to our daughters’ primary school’s Maths Club. The game has really taken off, with both strong and less strong students eagerly taking part.

The attractiveness and value of the game is self-evident but also, in an email reply to me, Tony Gardiner wrote an excellent, considered reflection on the game. I’m hoping Tony will post some version of that reflection as a comment or will permit me to post it as an addendum. (20/09/23 Tony’s thoughts are now added as an Addendum, below.)

The history of the game is less clear but also very interesting, at least to me. Ying first played the game with her best friend in the late 80s, when they were upper primary students. Her friend in turn was introduced to the game by their uncles and grandfather in the early 80s. From Ying’s hunting, it seems possible that the game was first popularised in a 1979 Chinese book, Interesting Mathematics.

Compiled by a team led by two mathematics teachers, Interesting Mathematics is an excellent collection of puzzles and games and challenges. It was a very successful book, selling over a million copies. The game of 24 appears as the sixth puzzle:

And finally, to end, scroll the document below for Tony Gardiner’s challenge, a sequence of 24 puzzles, seven in total. Please avoid providing solutions in the comments, so as not to spoil the challenge for others. You may however, indicate how long it took you and/or how many of the seven puzzles you were able to solve. Good luck!

1. Ying demands that I refer to her as my partner. I steadfastly refuse.

2. I, of course, couldn’t care less.

3. A 500 deck works perfectly, but in practice the court cards do not seem to be a problem.

4. Ying claims that I scoffed at 24 until reading about the game in Gardiner and Borovik’s book. I deny the accusation.

## UPDATE (19/09/23)

Courtesy of commenter Centurion, here is a translation puzzles 5 and 6 above, from Interesting Mathematics.

5. Arithmetic expressions involving four “4”s

Four “4”s are provided to you. Can you use arithmetic operations to make them expressions, such that the results of calculation are 0, 1, 2, …, 9, 10, respectively? There is more than one solution. Compete with others. See who would have more expressions? Who could get it done fast?

6. Who calculates faster?

One judge will be selected for this game. Each time the judge will take four natural numbers from 1-13 (poker cards can be used, numbers can be repeated). All game players will do four operations (+, – , ×, ÷) such that the result will be 24. The fastest player will score 1 each time. In the end, the one who accumulate the highest score wins. For each time, after a certain period of time, if no one can figure out a suitable calculation, the judge can revoke these four numbers and take another four numbers.

For example, if the judge takes 3, 5, 7 and 11, a calculation can be done as follows:

(7 – 3) × (11 – 5) = 4×6 = 24.

Sometimes there are multiple ways to calculate. For instance, 3, 9, 9, 11 can be calculated as follows:

(3 + 9) × (11 – 9) = 12 × 2 = 24;

alternatively these numbers can be manipulated as follows:

9 ÷ 3 × 11 – 9 = 33 – 9 = 24

There are instances where the numbers can not computed to 24, such as 1, 1, 2 and 2.

Also there are very challenging instances that needs meticulous observations.
For example, 10, 10, 4 and 4; 1, 5, 5 and 5.

Okay! Now you understand the way to calculate and compete. You can try the following groups of four numbers:

1,2,4,10
3,6,9,12
10,10,4,4
1,5,5,5
3,3,7,7
2,4,10,10
1,3,4,6

Additionally, you can also adjust the rules of competition above a little bit. For example, you may use a larger range of numbers; the final result can be varied to 12, 36, 1 and even 0; more operations may be allowed – other than four operations, squares, square roots and even logarithms; more numbers may be taken each time, etc.

## UPDATE (20/09/23)

OK, commenters can now feel free to discuss their solutions. And, below are Tony Gardiner’s ponderings on the game.

## ADDENDUM (20/09/23): Tony Gardiner’s Reflections

Perhaps it was GH Hardy who said of Ramanujan:

“It was as if every positive integer was one of his personal friends.”

Effective school systems achieve something more modest for every young person – first taking time to explore the many facets of each number 1-10, and later conveying the idea that larger numbers can be even more multi-faceted.

Once all four operations (and brackets) are available one can begin to notice unexpected differences between integers: some numbers become familiar from their (possibly repeated) occurrence in the multiplication tables; others hide out of sight (such as primes > 11); and some *feel* like primes even when they are not (such as 91).  Moreover, some integers stand out because they arise in lots of ways (like 20, 24 and 36).  (This idea lies at the root of that remarkable book by David Wells: “The Penguin dictionary of curious and interesting numbers”. This should be on every maths teacher’s bookshelf, and is more-or-less what it says on the cover: a dictionary, starting with “-1 and i”, then “0”, and so on.)

Most of us can never emulate Ramanujan. But we can absorb indirectly the intriguing idea that the basic facts of arithmetic provide each integer with *multiple personalities*, and that some of these personalities lie hidden in plain sight – yet can be revealed as a result of (sometimes sustained) searching. In other words, that elementary facts provide us with tools for discovering more elusive results.

In some sense one might prefer such lessons to be absorbed implicitly, without the intervention of artificially designed *didactics*. So we should encourage children and families to enjoy the 24 Game *as a game*, where speed is at a premium (as a measure of the robustness with which addition facts and multiplication tables have been internalised in a useable way).

However, the important aspects of elementary mathematics have never been learned on the street corner. If such lessons are to be learned (and are worth learning), then someone has to design sequences of individual lessons and tasks that cumulatively convey the intended message. It is in this spirit that I would encourage all maths teachers of those aged 9-13 (and even older students and adults) to introduce and use the 24 Game (for small integers) as part of the ongoing challenge to achieve a more profound mastery and enjoyment of elementary arithmetic.

This is the intended spirit of the following task:

For each set of four inputs, use the four integers (once each), three operations (and brackets) to “make 24”.

(a) 3, 3, 2, 2

(b) 3, 3, 3, 3

(c) 3, 3, 4, 4

(d) 3, 3, 5, 5

(e) 3, 3, 6, 6

(f) 3, 3, 7, 7

(g) 3, 3, 8, 8

[Unlike Marty, I would always put the two 3s at the front to underline the fact that they are the invariant part of the input.] [Marty: I actually thought about this, and then screwed it up.]

Commentary

The kind of robust arithmetic that is needed later on (e.g. when one comes to work with and to simplify, fractions) presupposes fluency with addition facts and multiplication tables. But this is not sufficient (even if one also gives students occasional hard challenges). Something else is needed – and is generally missing.

Addition facts and multiplication tables are the easy part. Students first need to be at home with numbers, and to calculate directly: that is to carry out prescribed operations, following instructions, to obtain answers to given calculations (or “sums”).

But they also need lots of experience with inverse tasks: how to obtain a given answer, using only given inputs and rules.

The power of elementary mathematics depends largely on our ability to handle such inverse problems. And inverse problems are always much more demanding than direct calculations.

(i) Addition is important; but its power and application derives in large part from the fact that fluency in addition allows us to handle the more awkward task of subtraction.

(ii) Multiplication is important; but its historical and educational power derives largely from the way it allows us to handle the more elusive art of division (and extracting square roots).

(iii) The idea that multiplying numerator and denominator by the same factor produces an equivalent fraction is important; but it is much more important to be able to do the inverse, and to recognise that 18/36 and 12/18 are familiar fractions in disguise.

(iv) Using the distributive law to multiply out brackets in algebra is a key stepping stone; but what really matters is to master the inverse – namely to factorise.

(v) Plotting simple graphs needs to be learned; but what matters more is to be able to distinguish different kinds of function.

(vi) Differentiating a given function has its uses, but eventually comes down to remembering a few facts (as with multiplication tables); however, serious problems that need calculus depend on the inverse operation of integration – an art which depends on a prior mastery of differentiation, but which is significantly harder.

This is a general phenomenon. LEGO bricks are brilliantly designed, but they are not that interesting in themselves. Very young children simply handle them (or chew them), or count them: it is not immediately obvious that they are for *making more complicated things*. But in the end, what matters is what you can do with them.

This is absorbed first (and often exclusively) by the direct experience of following instructions to assemble preordained models; and learning to do this reliably is already a valuable and rewarding skill. However, the greater challenge is the inverse engineering art of envisaging a desired model or component, and then working out how to select pieces, and how to arrange them, to achieve the required effect.

In mathematics most operations come in such direct-inverse pairs; and it is always the inverse form that is the most important and that is the hardest to master (but which is generally given the least class-time). The 24 Game conveys this message without thumping a didactical tub (as I am doing here).

24 is neither too small nor too large.

More importantly, 24 has many faces (additive; multiplicative; mixtures), so there are lots of ways of making 24.

And once the message is silently internalised for 24, it may then become clear (without actually doing it) that the same may be true for other numbers.

The game also reinforces (without saying) the fact that juggling small numbers in this way (not just to get answers as in “doing sums”, but to select from what one knows so as to “make 24”) is a satisfying mental activity for everyone. The competitive element adds to this – especially since inverse problems often favour a different kind of student.

Inverse problems are what maths is really about, what makes it hard, and what makes it useful/powerful/interesting: solving equations – not just calculating answers; and solving problems – not just calculating.

The sequence of seven “3, 3, n, n” examples also illustrates the distinction between (i) tasks that turn out to be relatively straightforward, and (ii) tasks which may prove to be surprisingly elusive, but where persistence can lead to eventual enlightenment (which is about as close as one can come at lower secondary level to the experience of the unexpected power of elementary mathematics).

[If anyone is sufficiently masochistic to want to explore this theme further, they might download my free book, Teaching mathematics at secondary level, and explore the 76 references to “inverse”.]

## ACER’s Guide to Gender Correctness

ACER, which began life ninety years ago in Camberwell as a tiny educational research institute, is now a worldwide, um, thing. Courtesy of ACER’s UK branch, we have a very informative guide, titled,

The assessment community has promoted gender stereotyping for decades. How can we stop?

The guide, written by a single ACER “Research Fellow”, is labelled as a comment piece. As such, the guide presumably does not rise to the level of ACER policy. Nonetheless, it’s there on ACER’s website and it seems fair for ACER to take the credit.

## Giant Reporting Blues

Here’s a quickie, courtesy of Mr. Big.

SEN, a sports radio station, has an article up today, on the performance of lower ranked teams in the AFL semi-finals. The article first notes that, over the last ten years, the results aren’t all that bad:

Over the last 10 years, the lower ranked team holds a[n] 8-12 win-loss record which does give both the Giants and Blues somewhat of a fighting chance.

But, recent news is not so good: Continue reading “Giant Reporting Blues”

## NSW’s Great Backward Leap Forward

A couple days ago the Sydney Morning Herald had a report and an editorial, on changes to New South Wales’s mathematics curriculum. The SMH combo is weirdly contradictory. It is difficult to make any proper sense of the story, and it appears the SMH writers tried but didn’t quite succeed. Continue reading “NSW’s Great Backward Leap Forward”

## Kicking Around the Birthday Paradox

Ed Barbeau, who is possibly a little exhausted from the doomsaying on this blog, has pointed out to us a recent BBC article on the birthday paradox. By mathematician Kit Yates, the article is framed around matching soccer players’ birthdays at the recent Women’s World Cup. There’s nothing grandly original about Yates’s article but it’s very nicely done, and it may form the basis of a good classroom discussion and/or assignment.

Regular doomsday programming will continue tomorrow. Continue reading “Kicking Around the Birthday Paradox”

## ACER, and Masters’ Level Nonsense

We’ve never written about the Australian Council for Educational Research, except in passing. Unlike VCAA and ACARA, for example, ACER has not appeared to be omnipresently idiotic. For the most part ACER just professionally goes about its business, quietly screwing things up.

## Quick Notes on the Herald Sun’s Exam Errors Article

There is report today in The Herald Sun (Murdoch, paywalled), titled,

Mistake-riddled VCE maths exams robbing students

Regular readers will know pretty much the lay of the land. However, there may be some non-regular readers in the next few days. So, a few clarifying remarks are probably worthwhile. (This is quick: I’ll adjust as I can through the day.)

First of all, without reflecting at all on the accuracy or the merits of the report, I want to make clear that I had no role in the creation of the report.

Secondly, at one point the report makes quick reference to this blog:

A Bad Mathematics blog run by a professional mathematician with a PhD in maths has identified more than 90 serious problems with specialist maths exams and 77 in maths methods, including sample exams and Northern Hemisphere exams going back to 2006.

More specifically, this appears to refer to the Specialist and Methods (and there’s also Further) error list posts (and the subsequent links included there). The report refers to “serious errors”. Without rejecting that language, the language I use on these posts is of “major” and “minor” errors:

To be as clear as possible, by “error”, we mean a definite mistake, something more directly wrong than pointlessness or poor wording or stupid modelling. The mistake can be intrinsic to the question, or in the solution as indicated in the examination report; examples of the latter could include an insufficient or incomplete solution, or a solution that goes beyond the curriculum. Minor errors are still errors and will be listed.

With each error, we shall also indicate whether the error is (in our opinion) major or minor, and we’ll indicate whether the examination report acknowledges the error, updating as appropriate. Of course there will be judgment calls, and we’re the boss. But, we’ll happily argue the tosses in the comments.

In recording and characterising such errors, I have made no attempt to determine or guess the effect of such errors on students’ scores. That seems to me to be a very difficult thing to do, for anyone.

Thirdly the report refers specifically to three questions in error on the 2022 Specialist Exam 2. That exam is discussed generally here. (The other 2022 exams are discussed here and here and here and here and here.) The specific questions are discussed here and here and here. These three questions (and others on the 2022 exams) appear to me to be unquestionably in error.

Fourthly, and finally for now, for me the prevalence of errors on the VCE exams is simply the tip of the iceberg. The many posts on this blog concerning VCE and VCAA indicate my more general concerns with VCE mathematics. (My broader maths ed concerns are probably best captured by this post.)

That’s it for now. I’ll update this post if something occurs to me, or if someone suggests in the comments that I somehow should.

## New Cur 30: The Complete Pain Words

### WRITING STYLE

I am not a good writer. Primarily, I use the monkey-typewriter method: if you rewrite a sentence sufficiently many times then you’ll eventually wind up with something at least serviceable. Then, if you rewrite a paragraph sufficiently many times … And so on. It is not a very efficient method.

Even if not efficient, however, the method works well enough for me. Of course I’m not creating great literature, but I don’t think it’s too boastful to claim that I get my ideas across clearly enough and engagingly enough.

Less monkeyishly, my writing style is fuelled mainly by an undergraduate humour and a stubbornness to rewrite and rewrite and rewrite, steered by the lessons and the spirit of the famous style guides, such as Strunk & White and Gowers. These guides are expressly not about creating great art, but rather about lowering your eyes to a much more achievable goal: recognising your intended readers and getting your ideas across to those readers in a clear, uncluttered manner. The powerful message of these guides is that anyone can write well enough, if they simply recognise the proper goal and work sufficiently hard to achieve it.

In the editions of the The Elements of Style to which he contributed, E. B. White quotes and then comments upon a key paragraph from William Strunk‘s original:

“Vigorous writing is concise. A sentence should contain no unnecessary words, a paragraph no unnecessary sentences, for the same reason that a drawing should have no unnecessary lines and a machine no unnecessary parts. This requires not that the writer make all his sentences short, or that he avoid all detail and treat his subjects only in outline, but that he make every word tell.”

There you have a short, valuable essay on the nature and beauty of brevity — fifty-nine words that could change the world.

Sir Ernest GowersThe Complete Plain Words has an even pithier summary, in an epigraph, a quotation from historian G. M. Young, to the Prologue:

“The final cause of speech is to get an idea as exactly as possible out of one mind into another. Its formal cause therefore is such choice and disposition of words as will achieve this end most economically.”

In his revision to The Complete Plain Words, Sir Bruce Fraser expressed it about as economically as possible:

Be simple. Be short. Be human.

The Complete Plain Words is even more specifically aimed than The Elements of Style, having famously originated as a booklet for UK civil servants. Which brings us, finally, to the Australian Mathematics Curriculum.

### THE AUSTRALIAN CURRICULUM

I have already whacked some of the larger aspects of the curriculum writing, as well as some of the structural aspects. Here I want to consider a smaller aspect: the phrasing of the curriculum content descriptors. These descriptors comprise the core of the curriculum, the part that the teachers have no choice but to read and to decipher. It is thus in this part, more than any other, that the writing must be clear and clean. Of course, I have already indicated dozens of poorly written descriptors, in particular in the Awfullnesses post. But the focus was almost always on the poor content; here, I will focus on the poor language.

I will start at the very beginning, as Mary Poppins sang, with Foundation Number. The first descriptor is,

name, represent and order numbers including zero to at least 20, using physical and virtual materials and numerals (AC9MFN01)

Is this OK? Well, adequate, maybe. It’s clear enough that the kids are supposed to learn the numbers from zero 0 to 20. But why not just,

Learn the numbers from 0 to 20.

OK, for ideological reasons ACARA wants to be more detailed and explicit in the descriptors. But “physical and virtual materials and numerals” is very clunky conjuncting, and the three listed activities don’t match the three listed aids: the kids will be “using” none of the aids in naming numbers. Here’s a suggested alternative:

Learn the numbers from 0 to 20, including their names, order, physical representations and representations with numerals.

Is this not clearly better?

On to the next descriptor:

recognise and name the number of objects within a collection up to 5 using subitising (AC9MFN02)

The preposition “within” is a poor choice, since it distracts from considering the collection as a whole, which is the entire point. Then, “collection of up to 5” hangs there, and it should be “five” not 5. Finally, does one really require the five dollar word “subitising”? Here’s a suggested alternative:

Identify the number of objects in a group of up to five objects.

On it goes. Here are the remaining four descriptors, and suggested alternatives:

quantify and compare collections to at least 20 using counting and explain or demonstrate reasoning (AC9MFN03)

Identify and compare the number of objects in groups of up to twenty objects.

partition and combine collections up to 10 using part-part-whole relationships and subitising to recognise and name the parts (AC9MFN04)

Combine and partition groups of up to ten objects, identifying and naming the sizes of the parts and the whole.

represent practical situations involving addition, subtraction and quantification with physical and virtual materials and use counting or subitising strategies (AC9MFN05)

Represent and analyse problems with physical materials, using counting, addition and subtraction.

represent practical situations involving equal sharing and grouping with physical and virtual materials and use counting or subitising strategies (AC9MFN06)

Represent problems of equal sharing, using number recognition and counting.

I don’t for a minute claim that these rewrites are perfect. But they seem unarguably a damn sight better, and they were the product of about three minutes’ contemplation. And, sure, maybe I skimped a little, with a little detail here or there left out. But it was ACARA’s idiotic idea to shove everything but the kitchen sink into the content descriptors. If that then means the content descriptors are unwritable, it just implies that ACARA screwed up earlier on. Which of course they did. And also later on.

Even accepting the needlessly bloated content, the descriptors are a muddy mess. What are “practical situations”? Is the intention that the kids use counting or (subitising strategies), or that they use (counting strategies) or (subitising strategies)? What even are “subitising strategies”? Isn’t subitising that you either get the number or you don’t? So what could be a strategy for that? To peek when you think the things are not looking?

Every single descriptor is like this. Every descriptor is muddied by vague and inaccurate words, by clumsy grammar, and by imprecise direction and goals. The kids must forever “communicate” and consider “questions” of unstated character, describe “features” and “investigate” god knows what, and “reason” in god knows what manner. The meaningless and excruciating “situations” occurs 136 times.

With absolute honesty, I do not believe the Curriculum contains a single content descriptor that could not be improved with two minutes’ thought from a half-competent mathematical editor. For most descriptors, a monkey with a typewriter would have an even money shot. It is astonishing, and it is a disgrace.

The Curriculum writing is a disaster. There is nowhere a single hint of even an attempt to be simple or short or remotely human. For the writing alone, and for so many other reasons, the Australian curriculum should be discarded.