Hannah’s sweets and Archie’s socks

This question, from a GCSE maths exam, has been causing a stir. There’s been quite a bit about it in the media, although you’ll struggle to find anyone who can be arsed to print part (b). Why is that?

There are n sweets in a bag.

6 of the sweets are orange.

The rest of the sweets are yellow. Hannah takes at random a sweet from the bag.

She eats the sweet. Hannah then takes at random another sweet from the bag.

She eats the sweet. The probability that Hannah eats two orange sweets is 1/3. (a) Show that n2 – n – 90 = 0 (b) Solve n2 – n – 90 = 0 to find the value of n

What makes this question so hard? Might we ask it differently? What’s going on? At undergraduate level (first years, mostly) I’ve been setting and marking exams: my lot are not much older than the pupils puzzling with Hannah. It’s good to try to think about these things. (It’s just possible my niece sat that exam, which is an extra cue to be less reflexive and more reflective.)

Correspondingly, I have a variety of crackpot theories, but before I waste your time with them, let me trouble you with some more respectable theory from David Perkins, specifically his paper Beyond Understanding.

It’s not just a matter of what you know. What does what you know enable you to do? Perkins characterises an escalating scale of what I might call knowledge weaponisation (which remind me of the old joke about solving, stating or colouring in the Navier-Stokes equations):

possessive knowledge can be recited and applied to routine calculations performative knowledge can be called upon more flexibly to solve problems proactive knowledge can be deployed outside its domain of obvious applicability

The “Hannah’s sweets” puzzle clearly demands more than possessive knowledge of basic probability, algebraic manipulations of fractions, and solving quadratics. Given part (a), part (b) should be a routine problem. It’s part (a) that sticks out as a bit peculiar, pulling an equation like a rabbit from a hat then asking the students to find a hatful of rabbit shit. It doesn’t demand anything they don’t know, but it isn’t a routine calculation.

If I take my specs off, metaphorically I mean, the detail becomes indistinct but the general story arc remains perceptible. The question goes like this

There are some unknown quantities, described in words. If they are not represented by named variables, then introduce named variables for them. There are some mathematical facts about those quantities trapped in a slab of prose. Scan the prose for quantities you can represent as formulae. Extract constraints on those formulae. Deduce the values of the unknown quantities. Solve the constraints by whatever means is appropriate.

That’s like lots of questions. Once upon a time, there might have been a question like this.

Archie own six black socks, some number of pink socks and no other socks. He is too lazy to pair them up before he puts them in his sock drawer all in a jumble. He always gets dressed in the dark, picking two socks at random. The day after laundry day, when all the socks are available, he typically wears a pair of black socks one third of the time. How many pink socks does Archie have?

My question, with its stereotyped male protagonist and bias against pink, probably needs a bit of rethinking before we can issue it to the youth of today. But it starts by introducing a mystery quantity, gives us some chat which constrains that quantity, then finishes by asking us to find that quantity. That is, it’s coded as an instance of that standard problem type. Moreover, it doesn’t name a variable, let alone confront us with a quadratic formula, and it doesn’t specifically invoke the concept of probabilty. Socks motivate the interest in “two the same” better than sweets, but they might predispose us to guess that they are found in even numbers. The latter is a good red herring, and besides, Archie has probably lost the odd sock here and there. But I digress. The issue I’d like to open is the relative difficulty, from a human perspective, of “Hannah’s sweets” and “Archie’s socks”.

You see, my crackpot theory is that “Hannah’s sweets” is knocked off an older question uncannily like “Archie’s socks”, but when revising it, the examiners named the number of sweets and added the intermediate goal to establish the quadratic constraint in an attempt to make the problem require less initiative. If so, I think they also made it more intimidating. However, they also made the last phase of the problem, part (b), completely routine, in the hope that people without the initiative for part (a) would still collect some marks. Whether students notice that they can do part (b) without a clue for part (a) is another matter. Many students stop doing a question at the first part which causes trouble and read only on demand, which means that in “problem” questions, they miss the clues in the later question parts for what might constitute useful information in the prose.

The media have, by and large, not printed part (b) of the question. They make it look like the question is “there are some sweets; prove an equation”, rather than “there are some sweets; find out how many”. See, for example, Alex Bellos’s piece in the Grauniad. Why is this? Crackpot theory time again.

There’s one photo of the question paper which seems to show up a lot. The Guardian credits its source as Twitter. It cuts off after part (a). The story is clearly less time-consuming to deliver just as Twitter churnalism: bothering to find out what the whole question might have been is extra work and who likes doing that? (I got part (b) from the BBC, which shows what you can achieve with a mandatory licence fee.) The impact is to make a dishonest protest about the extent to which the problem is undermotivated. That shows good politician skills on the part of the original poster, and is a good way to increase the sensation-value of the story. Many secondhand reports compound the problem by publishing not this image but its transcription in plain text, dropping the (a) label to make it look like the question stops at the quadratic equation. They also write n^2 rather than n2, necessitating a comment on notation, making it look as if the original question introduced weird notation on the fly, when in fact the extra bend is introduced in quotation.

Especially in its curtailed form, “Hannah’s sweets” looks superficially weirder than “Archie’s socks” because some al-gebra terrorism has been added and the purpose has been taken away.

What both questions have in common is that they are in code. They must be decoded before the algebra can begin. “Hannah’s sweets” uses weaker encryption than “Archie’s socks”. I’m afraid that’s because I wrote “Archie’s socks”: here are some more excerpts from my archive. I quote them selectively, not to give you whole questions, just a sense of my style of distraction.

Disco Mary is rummaging through a collection of old electronic spare parts. She finds a multicolour lamp which has three Boolean inputs, labelled red, green and blue. … The green input signal is connected to the output of a T flip-flop. The blue input signal is connected to the output of another T flip-flop. … Mary’s favourite song is “What A Blue Sunset” by Ray Dayglo and the Thunderclaps, so she decides to wire the lamp and flip-flops to make a repeating sequence of colours, changing with each clock cycle: cyan, blue, yellow, red, then back to cyan and round again.

Madame Arlene teaches the Viennese Waltz. When she is teaching beginners, she finds that she has to shout “1, 2, 3, 1, 2, 3,. . . ” repeatedly for ever, to keep her pupils in time. When she teaches advanced classes, she doesn’t need to shout, because they listen to the music. … Madame Arlene builds a shouting machine and wires it into her music player: it gets a 2-bit unsigned binary number as its input and a clock signal generated by the player in time with the music. At each tick, the shouting machine checks its input: if it gets 0, it shouts nothing; otherwise it shouts the number it gets. … The challenge is to generate the input signal for the shouting machine, so that both counting and silent behaviours can happen.

A control panel has four switches on it, named S, T, U and V, respectively. Each switch sends a 0 signal when its handle points downward and a 1 signal when its handle points upward. This is the current setting: [S down, T up, U down, V up] The control panel is wired both to the lock of a safe and to a burglar alarm. The setting on the control panel represents a number in 4-bit two’s complement binary notation. … To open the safe, you need to construct a circuit which connects the switches S, T, U, V to the release control R which will set R = 1 if and only if the correct combination is entered. An informant has discovered that the correct combination is -6. … [alarm circuit diagram] … You can flick only one switch at a time. You need to flick switches in sequence to change from the current setting to the setting with the correct combination. You must not set off the alarm.

Professor Garble is a researcher in multicore programming techniques, attempting to explain a recent trend in processor performance. ‘Moore’s Law is finished! That’s why processor clock speed has levelled off. And that’s why processors have exponentially increasing numbers of cores.’ Tick the box or boxes for whichever of the following is true. … [] He is correct in none of the above ways.

That’s just the sort of thing that occurs to me in the bath.

All of these questions require you to extract a model of what is going on from some chat. That’s the skill I am trying to test. I make the chat blatantly spurious partly to be clear that it is a “decode the puzzle” question, but mostly because I am habitually facetious. It occurs to me that maybe exams should be no place for facetiousness from them or from me: why should I have a laugh when they’re not having quite so much fun? This question style is at least routinely visited upon them in the course of the class: if you have paid attention to past papers, it is exactly what you expect. Still, I can see how it could be exclusionary, like an in-joke that you detect but don’t get. I think perhaps that I should do less of that stuff in exams and more in class, where there’s less pressure and we can afford a bit of a laugh while learning to decode problems.

But I really do digress. The point about encoded problem questions is that you need to recognize when you are being told Something Important, and what that something is. In that respect, it’s a lot like doing a cryptic crossword: crossword clues use a stylised language that takes time and practice to acquire. I was taught to do crosswords by my father’s colleagues, who always appointed me the writer-inner for the lunchtime crossword and were happy to indulge my queries: what signalled an anagram, an inclusion, a pun, and so on. In the same way, I instinctively decode the declaration “The probability that Hannah eats two orange sweets is 1/3.” as the instruction “Write a formula for the probability that Hannah eats two orange sweets and set it equal to 1/3.”. It’s familiarity with this sort of code which pushes puzzles like “Hannah’s sweets” back down Perkins’s scale of Ps. And that’s teachable. When I see part (a), I’m a bit spooked, and I think “Why are they asking me to deduce this equation? I already have an equation? Why are they not just asking me what n is? Ah, that’s part (b). Oh well, I expect I should be able to deduce the part (a) equation from mine by doing a wee bit of algebra.”. I’m already on course, and part (a) threatens to throw me off it: that’s why I think “Archie’s socks” is easier. I’m reminded of Whitehead’s remark:

It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.

I agree, sending for the thought-cavalry is a desperate measure, but it would be a shame if an examination intended to assess knowledge and reward the performative (or even proactive) were entirely devoid of decisive moments. For “Hannah’s sweets”, the thought-cavalry can be avoided if you recognize the way in which you are being instructed. Moreover, thought-cavalry tactics, when needed, are greatly assisted by the key extra exam-puzzle knowledge that the question contains a sufficiency of clues: we don’t get that luxury in real problem-solving. I often tell students that my role is to be simultaneously Blofeld and Q: the fact that they are in a James Bond movie means that there is necessarily a strategy to escape Blofeld’s menaces with Q’s gadgets. I do not expect them to die. In fact, I’m trying to arrange their survival. In that sense, “Hannah’s sweets” already comes with the expectation that whatever information is packed in the opening prose must be sufficient to ensure the equation demanded of us: the game is to unpack it.

We could present “Hannah’s sweets” in a more decoded form. Here’s a kind of stream-of-consciousness translation.

Hannah has a number of sweets. It doesn’t matter that they are sweets or that Hannah is called Hannah. What matters is that there are things and we are going to find out how many: call that n. 6 of the sweets are orange and the rest are yellow. There are two different sorts of thing: “orange” and “yellow” are arbitrary labels whose only role is to be distinct. You are told that there are 6 orange sweets, but not how many yellow sweets (perhaps call that y, so n = 6 + y). Hannah selects two sweets at random without replacement. It doesn’t matter whether she eats them or throws them at pigeons. It does matter that the two selections are random, and that the second selection is made from one fewer than the first. Their randomness tells you that you can base probability on proportion and that selections are independent, so you can compute the probability of a particular pair selection by multiplying the probabilities of the separate selections. You are told the probability of a particular outcome: she selects two orange sweets with probability 1/3. The probability of getting two oranges clearly depends on n: write down a formula for that probability and set it equal to 1/3. Rearrange that equation (by clearing fractions) to obtain the quadratic equation n2 – n – 90 = 0, then factorize the equation to obtain two candidate solutions, only one of which makes sense.

I think it’s reasonable to teach that decoding skill and to expect school pupils to acquire it. The persistent complaint that they haven’t seen anything like it on a past paper seems wide of the mark, and more worryingly as a perceived entitlement to be tested only on possessive knowledge. We should be clear in our rejection of that entitlement. But we should also acknowlege that the boundaries between Perkins’s kinds of knowledge is fluid, and that our responsibility as teachers is to rearrange their positions relative to the student by acting on both, in the direction marked “progress” by Whitehead.