# 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”.]

## 47 Replies to “The Game of 24”

1. Matt says:

Stumped on Puzzle E and G, although I suspect when I wake up tomorrow morning I’ll solve them in a flash.

This was a nice read, thanks for sharing!

1. marty says:

Thanks, Matt. Your stumpedness is interesting.

1. Back in Black says:

Letters and Numbers (which put Lily Serna on the map) uses this idea to entertaining effect. And there’s a 30 second time limit. Whoever gets closest to the target number wins.

I think I read somewhere that out of all possible four card combinations (picture cards excluded), 25% of them cannot make 24, that is, there’s no solution. I haven’t seen any way of proving in general that there’s no solution except by brute force using a computer (and maybe the ‘wisdom of the crowd’). I wonder if there’s a simple general condition that must be met. I also wonder why 24. Why not 20, say? (Is the number of factors a reason?)

1. Back in Black says:

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

I had noticed the “two 3s [being an] invariant part of the input” and had pondered (once grappling with puzzle G) whether there was perhaps a simple relationship between the 3’s and the n in 3, 3, n, n for which no solution was possible. A relationship that perhaps the first six puzzles were nudging towards.

And then Marty’s comment “Your [(Matt’s)] stumpedness is interesting” made me think further about this and that maybe there was no solution. It wasn’t a natural comment …

2. Red Five says:

I found E to be one of the easier ones. G is bothering me a bit, I think there must be a simpler solution than mine but I cannot find it.

In keeping with the rules I will not post a solution though, but look forward to (eventually) seeing if anyone found a simpler solution.

Edit: by “simpler” I mean easier to calculate quickly. My solution does not use many operations, but took a minute to verify was actually equal to 24.

1. Back in Black says:

RF, we might have found the same solution. It took me a while to find mine. I was ready to give up and decide there was no solution. But I felt that no solution would be unfair (how could you prove it?). So I repeated my efforts and with some rational thought, succeeded.
As I say to my students, it helps to keep in mind that a question (usually) has a solution. However, in the case of G, I think some students will be more advantaged than others (in fact, it may be fair to say that for some students there is no solution).

I have heard of more ‘advanced’ versions of the game where operations such as factorial and exponentiation are allowed. Combinations such as Ace, Ace, Ace, Ace then become possible.

1. Red Five says:

I’ll send my solution to Marty who can forward it to you if he agrees that it is a valid solution.

1. Back in Black says:

No need, RF. It looks like you can post it tomorrow and then we’ll all know. I suspect there’s only one solution.

2. Dr Mike says:

When I was a kid my grandfather played with me Game of 100. Instead of cards, we used numbers on the public transport tickets. Each number was 6 digits from 0 to 9. One has to get to 100 using all digits and only 4 operators; plus, minus, multiplication, and division. First one to get to 100 is a winner. Each number can be used only once operators can be used more than once.
A very simple example is 250250 one can get to 100 by 2x5x2x5-0-0=100

1. marty says:

That’s great. Of course “use these numbers to get to this total” puzzles are common, and always good value. Puzzle in 5 in Interesting Mathematics, above, is another example. But I really like the idea of using the (semi)random numbers on a ticket.

1. Dr. Mike says:

These days I am playing the game of 100 with my son. We generate the six figures number randomly and then compete who solves first.

3. Centurion says:

I really like this blog post.
It is pleasing that you share the history of “the Game of 24”, and the book excerpts are lovely readings.

I never came across with the 1979 book when I was in China. Good to know such a book! They way the writers phrased the instructions makes me feel the targeted audience is primarily 6-8 years old but the language is indeed engaging.

Happy to translate the relevant part provided in this post later.

1. marty says:

Thanks very much, Centurion. A PDF of the 1984 edition of the book can be purchased cheaply here, although I don’t know how easy is it is to do. (Ying had a friend in China use that link to get it for her.)

The book does seem to have lots of good stuff, and I plan to post more on it. Ying was very excited to find the book and plans to print it out and bind it to use with our kids.

As for translating the text above, sure, thanks. If you do I’ll post it as an update. I had Ying translate-read it to me, but figured the game was clear enough, and I had already made Ying work hard enough.

1. Centurion says:

Marty,

Here is my translation. Attempted it word-by-word. Hopefully nothing is lost.

V. 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?

VI. 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.

1. marty says:

Thanks! Close enough. I’ll add as an update soon.

2. marty says:

Done.

3. banacek spaces says:

“There are instances where the numbers can not computed to 24, such as 1, 1, 2 and 2”
22+1+1 ?

1. marty says:

No. The numbers are the numbers.

4. Gaia says:

I am definitely going to play this with my son, and introduce it to my intervention kids. This looks amazing.

1. marty says:

Thanks, Gaia. It is interesting to see how more and less skilled kids approach the puzzles. Unsurprisingly, younger and weaker kids cross their fingers and hope addition and subtraction will suffice. Then, some will begin to throw in a simple multiplication. Division is harder to employ, of course. And, as Tony’s challenge demonstrates, the puzzles can be remarkably difficult.

5. Red Five says:

Puzzle F took me the longest, but I felt a bit silly afterwards… should have been easier than G.

So can confirm all of the puzzles A to G have at least one solution.

If it matters, E took two attempts, F took four attempts and G took three attempts.

Was not timing the length of any attempt, but had to go away and come back between attempts in some cases.

Absolutely loved the challenge though.

1. marty says:

Thanks, RF. Maybe I’ll give it one more evening for people to try the challenge without having to avert their eyes, and people can post tomorrow.

1. Red Five says:

No worries. I’ll wait for the green-light before posting any solutions.

It may be an odd thing to consider, but I was thinking about Mathologer’s cubic formula video recently and how real solutions come about after the algebra wanders around a bit in complex space.

I like the way integer solutions can come about in certain problems after wandering around a bit in rational space (if there is such a thing). Makes the problem quite deep in many ways and now I’m wondering how it can be tweaked to push a few high-achieving middle-years students into some seriously deep thinking…

1. Back in Black says:

I suppose one could say that some solutions come more naturally than others.

1. marty says:

I guess I indicated I was ending the embargo, but neither of you guys is very good at being quiet.

1. Back in Black says:

I think understanding our cryptic comments might be harder than solving the puzzle.

1. marty says:

They weren’t that cryptic.

1. Back in Black says:

I was tempted to say that some solutions come naturally than others. not cryptic.

2. Banacek Spaces says:

C’mmon, lets stretch the rules a bit.

Happened in the past, but can happen in 2023. Using these 4 digits, in this exact order but otherwise in any way you like, create expressions for all whole numbers from 0 to 100. Prizes for first three in class/school…

2. Dr Mike says:

G took me the longest. I started to think that there was no solution. However, yes, there is one.

A, B, C, D, and E went very fast. I had to spend some time with F but also went fine.

1. Red Five says:

The beauty of problem G is that you can easily get 24 with only two of the cards – which makes it really frustrating…

I now wonder if similar issues arise if you use, say 4,4,6,6 or 2,2,12,12 and am hoping that there may be some pattern in the solutions…

Great post Marty – this is a gift that keeps on giving!

3. Johnald says:

Puzzles F and G gave me a really hard time, too. I gotta echo that for younger students, the solution might not even exist…

1. marty says:

Of course. No one was suggesting that Tony’s puzzle was for young ‘uns, or that young ‘uns should generally be expected to hunt for such solutions.

6. marty says:

OK, people are now free to discuss their solutions and/or frustrations with Tony’s challenge. (For the competitive out there, Ying took about twenty minutes to do the seven problems.)

I’ve also added Tony Gardiner’s very interesting thoughts on the game, as an Addendum to the post.

1. Back in Black says:

I’m guessing she took about 2 minutes for the first six …

1. marty says:

20 seconds.

1. Back in Black says:

I was assuming she’d try to make you feel good.

7. Red Five says:

Here are my solutions for the last two puzzles:

With apologies for not keeping quiet…

1. Back in Black says:

The first is what I got too. I realised that if I could make 1/3 from 8, 3, 3 then . It was simple after that.

(Once I realised the murderer was left-handed, red headed, club footed and was lactose intolerant, it was simple after that to eliminate all but one suspect).

The trick was rational thinking rather than natural thinking. Younger students who haven’t met fractions (rational numbers) might feel cheated, then (hopefully) interested.

1. Solange Amato says:

Some people enjoy puzzles. I am more interested in students’ understanding of why 8 ÷ 1/3 = 24. I remember a maths educator in a conference saying that there was no possible explanations for division of fractions using pies, cakes, pizzas, etc. What do you think?

1. Joe says:

It doesn’t seem too hard to me:

Group x pizzas into y [of a group|groups] of pizzas. How many pizzas is in the group?

E.g. for the above: Group 8 pizzas into 1/3 of a group of pizzas. How many pizzas is in the group?

However, I would never introduce division by fractions this way.

1. Back in Black says:

There are two 1/2’s in 1.

There are three 1/3’s in 1 …

2. marty says:

I think “understanding” is oversold; kids have more interest in puzzles and challenges and they get more out of them. And, I think pies are way oversold. By the time you’re teaching division of fractions, I think the pies should have disappeared.

8. Tony Gardiner says:

Solange Amato reminds us how much lies behind the puzzle, and how few experts understand fraction arithmetic, or why it matters.

I apologise for perverting a beautiful game that should be played and enjoyed with a *didactic sequence*. But this has a place – especially in the classroom. So here is a bit more didactics.

Getting the right answer to 8 ÷ (1/3), and understanding why it makes sense, is not so hard provided one has been consistently encouraged to think of 24 ÷ 8 as “How many 8s in 24?”.
However, dividing by fractions, this will always cause trouble *if integer division with integer answers has not been handled in a way that extends naturally to fractions*.

Forget the pies, and stick to quantities.
The problem is that division is not symmetric – so if we use a model, there are always going to be three different kinds of quantity here.

Suppose that the 24 in “24 ÷ 8” is a “number of counters”.
Then we have two possibilities:
(i) First, the “8” may be a completely different kind of quantity, such as the “number of *piles*” required. In this case, the answer “3” is the “number of counters per pile”.
(ii) Second, the “8” may be a rather subtle kind of quantity – namely, the “number of counters per pile”. In that case, the answer “3” gives us the “number of piles” required.

So when teaching division with integers, with integer answers, we have to make sure (over and over again) that the calculation is understood in both senses:
(i) “If we divide 24 into piles of 8, how many piles?”
and
(ii) “If we divide 24 into 8 piles, how many in each pile?”
These are psychologically quite different, but give the same answer, as the 3×8 array shows
. . . . . . . .
. . . . . . . .
. . . . . . . .
“If we have 3 per pile, the there are 8 piles”; and “if we have 3 piles, then there are 8 per pile”.

The “three different kinds of quantity” in the 3×8 array are: the number of dots in the picture; the number columns (or the number of dots per row); and the number of rows (or the number of dots per column).

The first kind of question (i) should be repeatedly paraphrased as “How many 8s in 24?”

The second kind of question (ii) should be repeatedly paraphrased as “If I put 24 into 8 piles, how many are there in each pile?”

Then (i) gives an easy meaning for 8 ÷ (1/3): “How many lots of “1/3″ do you need to make 8?” (3 of them make 1, so we need 3×8 altogether).

(ii) needs to be chewed over, and is probably best delayed until students are comfortable with getting the right answer. But it still makes sense as something like: “If 8 is (1/3) of a pile, how many are there in one whole pile?”

1. Solange Amato says:

Hello Tony. I remember meeting you in Mexico and talking about strange “recent” ideas concerning teaching maths. Now they became old ideas!

Yes. That is the explanation. Of course you know how to explain 8 ÷ (1/3)!

It is not difficult to explain using pizzas. I just use different words with the same meanings you wrote: “8 pizzas divided into portions of a 1/3 of a pizza. It gives me 24 portions”.

Division seems to have 3 different meanings. I only found the third meaning of division in a book by Haylock and Cockburn.

9. Red Five says:

Perhaps I am wearing my teacher hat a bit too tightly here, but I think there is quite a bit more to the “division by a fraction” debate than seems to be the assumption.

It is not at all difficult to show that because and if you have a standard definition of equal fractions, then the two expressions must be equal.

However… fractions within fractions are potentially quite problematic. Consider for example and I can imagine the difference between these two expressions taking a while to explain to some students in a way that they would fully comprehend.

As someone who has worked with fractions a lot, I have no issue manipulating expressions such as but this doesn’t mean everyone will immediately understand why this is (or even should be) equal to 24.

The concept of a “fraction” may seem rather instinctive, but when you really think about it… I’m not sure the pizza analogy really helps that much after a while.

1. marty says:

I think the heart of any understanding is Tony Gardiner’s “how many”.