As a traditional Chinese parent, my girlfriend^{1} 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”.]