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Chips on the First Floor

Every so often, I spend some time filling out a crossword puzzle. There are quick crosswords where only definitions of terms are given, and cryptic crosswords which include both a definition and some wordplay. Especially for quick crosswords, these clues can be ambiguous – for instance, Fruit (5) could be APPLE, MELON, LEMON, GUAVA or GRAPE; OLIVE if one wants to be technical, and YIELD if one wishes to be indirect.

To resolve this ambiguity, about half of the letters in a quick crossword are checked. This means that their cells are at the intersection of two words, and the corresponding letters must match.

With a recent puzzle I was attempting, I had a clue with a definition for ‘show impatience (5, 2, 3, 3)’. I didn’t get this immediately, but with a few crossing letters in the middle I quickly wrote down CHOMP AT THE BIT. This was fine until I had a down clue with definition ‘problem (7)’ which was D_L_M_O. This should clearly be DILEMMA. It was a cryptic crossword, so I was able to check CHAMP AT THE BIT with the wordplay part, and it made sense. (The clue was “show impatience in talk about politician with silly hat, I bet” – which is CHAT around (an MP and then an anagram of HAT I BET).) The “original” expression is actually CHAMP, though I’ve only heard of the CHOMP version before.

I sometimes have difficulty with crosswords in the UK (and sometimes with crosswords from the US as well) owing to regional variations in English. Singaporean English follows the UK in terms of spelling. However, in terms of definitions, things vary. For example:

  • Common with UK usage:
    • Tuition refers to additional small-group classes (like in the UK), not the fees one might pay at university (US).
    • biscuit is a baked good that’s usually sweet (like in the UK) and probably shouldn’t be eaten with gravy; an American biscuit is a bit more scone-like.
  • Common with US usage:
    • Chips are thin fried slices of potato (same as US). The word refers to fried strips of potato in the UK (which themselves are fries in both Singapore and the US); the thin slices are called crisps in the UK.
    • The first floor of a building is the ground floor (same as US); in the UK that’s the first floor above ground (which is the second floor in Singapore and the US).

Without venturing into Singlish (which incorporates terms from Chinese, Hokkien, Malay and several other languages), there are also terms that aren’t in common with either American or British English. Some of these pertain to local entities. Economy rice is a type of food served in food courts, and the MRT is Singapore’s subway network – though I’ve heard several uses of it as a generic term, much like Xerox for copying.

Others seem a little more random. Sports shoes refer to trainers specifically, and don’t refer to water shoes or hiking boots which are used for sport. The five Cs refer to cash, cars, credit cards, country club memberships and condominiums – five things starting with the letter C that materialistic Singaporeans often chase.

I’ve been resident in the UK for around six years now. This is obviously fewer than the number I’ve spent in Singapore (about 21), though the years in the UK are more recent. I’ve gotten used to the British expressions, especially for life in the UK (I generally like chunky chips more than crisps, and correctly distinguishing the first and ground floors is important for getting around). I don’t think I’ve had too many issues with remembering the correct versions of terms to use when in Singapore or in the US – having had to deal with these inconsistencies has helped here.

A Quiet Winter Night

I made my way down the staircase at the end of the bridge. It was late on Thursday, somewhere between 10 and 11 pm. The snow was still falling (though quite lightly). What struck me the most, however, was the silence. For a few brief moments, it seemed like it could have been some kind of post-apocalyptic world, or the Rapture; I was alone, at least as far as I could tell.

The UK has witnessed an unexpected patch of cold weather over the past week or so. I have pretty good cold resistance and thus don’t normally detect weather changes that much, though this was of course painfully obvious. It’s possible that more snow fell in London in the last week than in the past five years – at least for the times I’ve been around.

I enjoyed the snow on the first day, though that’s about as long its welcome lasted. For some reason, I generally associate snow with bitterness and harshness, rather than the alleged fascination that people in the UK might tend to have. The reason cited in the article is that it tends to remind people of youth or Christmas, which is fair – I certainly don’t associate the former with snow, and although there is much media about snowy Christmases, I don’t think I have experienced one. I think more Dead of Winter (a board game about surviving winter with zombies) than Jingle Bells. It was interesting to experience a different kind of weather, nonetheless, but the practical pains of dealing with it came to the forefront very quickly.

For me at least, the most pertinent issue was travel disruptions. My commute is typically a 20 minute walk. though because of the snow and ice I had to be substantially more careful with my footing. I easily took 30-plus minutes to traverse the same route. Thankfully I don’t have to drive or take the train in to work, but it certainly affected my colleagues too (and it did mean the office was quieter, which in turn affected me). Expectedly, quite a few flights were cancelled as well; I wasn’t travelling, but this would have certainly been an annoyance if I was.

This wasn’t a factor the last time round, but snow can also cause problems in supply chains. I don’t think I was affected this time, but there was a similar incident in New York early last year; I was waiting on a package to be delivered. Thankfully it was delivered on the morning of the day I was flying back to London, though I got somewhat anxious about whether it would arrive on time. In general of course this could be a much larger logistics problem.

Low temperatures were another factor, though much less significant for me. To paraphrase Elsa, at least most of the time the cold never bothered me anyway – though there were two instances where I decided it made sense to switch on the heating. Apart from that, I’m not sure I did much to deal with the temperatures on their own. I did dress marginally more warmly than normal (which in say low single digit weather is a light jacket), but that was about it.

It’s also suggested that there are correlations between cold weather and various types of illnesses, such as on this NHS page. Of course, I recognise that the impacts on people who spend more time exposed to the cold (e.g. people who work outdoors, rough sleepers) would be substantially greater.

The recent weather also seems to have had a sobering effect on my thoughts. Some of this might be driven by confounding factors (shorter days, more darkness, fewer social gatherings etc). This isn’t bad per se, though I find myself easily engrossed in thoughts that may be counterproductive at times. I also read an article in the Guardian questioning why the UK was unprepared for the snow; while I’m not sure I agree with the central thesis (I don’t have the data, but this may be viewed as an actuarial decision; spending $2 to prevent a 1/10th risk of losing $10 may not be worth it), but there was a point on extreme weather making societal inequalities starkly obvious which I can follow.

The weather is forecasted to return to more normal levels in the week ahead. If that holds, I’ll appreciate the easier travel and that more people are in the office. I’ll count myself fortunate that it hasn’t impacted my routines and plans that much, at least for now.

Running the Gauntlet (Hotseat: OCR Mathematics C1-C4)

Background

The GCE A Level is a school-leaving qualification that students in the UK take at the end of high school. Students usually take exams for 3-4 subjects. The exams are graded on a scale from A* to U (though not with all characters in between); typically an A* is awarded to the roughly top 8-9 percent of students.

This is a rather different type of challenge – previous installments of this series have featured especially difficult exams (or rather, competitions; only the MAT is realistically speaking an exam there). I’ve usually struggled to finish in the time limit (I didn’t finish the AIME and barely finished the MAT; I had some spare time on the BMO R1, but still not that much). I could of course do this in the same way as the other tests, though the score distribution would likely be close to the ceiling, with random variation simply down to careless mistakes.

Interestingly, the UK has multiple exam boards, so for this discussion we’ll be looking at OCR, which here stands not for Optical Character Recognition, but for Oxford, Cambridge and RSA (the Royal Society of Arts). The A level Maths curriculum is split into five strands: core (C), further pure (FP), mechanics (M), statistics (S) and decision (D); each strand features between two and four modules, which generally are part of a linear dependency chain – apart from FP, where FP3 is not dependent on FP2 (though it still is dependent on FP1). For the Mathematics A level, students need to take four modules from the core strand, and two additional “applied” modules; Further Mathematics involves two of the FP strand modules plus any four additional modules (but these cannot overlap with the mathematics A level ones). Thus, a student pursuing a Further Mathematics A level will take 12 distinct modules, including C1 – C4 and at least two FP modules, for example C1-4, FP{1,3}, S1-4, D1 and M1.

(In high school I took the IB diploma programme instead, which did have Further Mathematics (FM), though I didn’t take it as I picked Computer Science instead. That was before Computer Science became a group 4 subject; even then, I think I would still have wanted to do Physics, and thus would not have taken FM in any case.)

Setup

I attempted the June 2015 series of exams (C1 – C4). Each of these papers is set for 90 minutes, and is a problem set that features between about seven and ten multi-part questions. The overall maximum mark is 72 (a bit of a strange number; perhaps to give 1 minute and 15 seconds per mark?). To make things a little more interesting, we define a performance metric

P = \dfrac{M^2}{T}

where M is the proportion of marks scored, and T is the proportion of time used. For example, scoring 100 percent in half of the time allowed results in a metric of 2; scoring 50 percent of the marks using up all of the time yields a metric of 0.25. The penalty is deliberately harsher than proportional, to limit the benefit of gaming the system (i.e. finding the easiest marks and only attempting those questions).

Most of the errors were results of arithmetical or algebraic slips (there weren’t any questions which I didn’t know how to answer, though I did make a rather egregious error on C3, and stumbled a little on C4 with trying to do a complex substitution for an integral, rather than preprocessing the term). There are a few things I noted:

  • The scores for the AS-level modules (C1, C2) were considerably higher than that for the A-level modules (C3, C4). This is fairly expected, given that students only taking AS Mathematics would still need to do C1 and C2. Furthermore, from reading examiners’ reports the expectation in these exams is that students should have enough time to answer all of the questions.
  • The score for C1 was much higher than that for C2. I think there are two reasons for this – firstly, C1 is meant to be an introductory module; and secondly, no calculators are allowed in C1, meaning that examiners have to allocate time for students to perform calculations (which as far as I’m aware is something I’m relatively quick at).
  • The score for C4 was actually slightly higher than that for C3 (contrary to a possibly expected consistent decrease). While there is meant to be a linear progression, I certainly found the C3 paper notably tougher than that for C4 as well. That said, this may come from a perspective of someone aiming to secure all marks as opposed to the quantity required for a pass or an A.

We also see the penalty effect of the metric kicking in; it might be down to mental anchoring, but observe that perfect performances on C1 and C2 in the same amount of time would have yielded performance numbers just above 5 and 3, respectively.

Selected Problems in Depth

C3, Question 9

Given f(\theta) = \sin(\theta + 30^{\circ}) + \cos(\theta + 60^{\circ}), show that f(\theta) = \cos(\theta) and that f(4\theta) + 4f(2\theta) \equiv 8\cos^4\theta - 3. Then determine the greatest and least values of \frac{1}{f(4\theta) + 4f(2\theta) + 7} as \theta varies, and solve the equation, for 0^{\circ} \leq \alpha \leq 60^{\circ},

\sin(12\alpha + 30^{\circ}) + \cos(12\alpha + 60^{\circ}) + 4\sin(6\alpha + 30^{\circ}) + 4\cos(6\alpha + 30^{\circ}) = 1

This might have appeared a little intimidating, though it isn’t too bad if worked through carefully. The first expression is derived fairly quickly by using the addition formulas for sine and cosine. I then wasted a bit of time on the second part by trying to be cheeky and applying De Moivre’s theorem (so, for instance, \cos(4\theta) is the real part of e^{i(4\theta)} which is the binomial expansion of (\cos \theta + i \sin \theta)^4), subsequently using \sin^2 x = 1 - \cos^2 x where needed. This of course worked, but yielded a rather unpleasant algebra bash that could have been avoided by simply applying the double angle formulas multiple times.

The “range” part involved substitution and then reasoning on the range of \cos^4\theta (to be between 0 and 1). The final equation looked like a mouthful; using the result we had at the beginning yields

f (12 \alpha) + 4 f (6 \alpha) = 1

and then using a substitution like \beta = 3 \alpha, we can reduce the equation to 8 \cos^4 \beta - 3 = 1. We then get \cos \beta = \pm \left( \frac{1}{2} \right)^{(1/4)} and we can finish by dividing the values of \beta by 3 to recover \alpha.

C4, Question 6

Using the quotient rule, show that the derivative of \frac{\cos x}{\sin x} is \frac{-1}{\sin^2x}. Then show that

\displaystyle \int_{\frac{1}{6}\pi}^{\frac{1}{4}\pi} \dfrac{\sqrt{1 + \cos 2x}}{\sin x \sin 2x} = \dfrac{1}{2}\left(\sqrt{6} - \sqrt{2}\right)

The first part is easy (you’re given the answer, and even told how to do it). The second was more interesting; my first instinct was to attempt to substitute t = \sqrt{1 + \cos 2x} which removed the square root, but it was extremely difficult to rewrite the resulting expression in terms of t as opposed to x. I then noticed that there was a nice way to eliminate the square root with \cos 2x = 2 \cos^2 x - 1. The integrand then simplifies down into a constant multiple of \frac{-1}{\sin^2x}; using the first result and simplifying the resultant expression should yield the result. That said, I wasted a fair bit of time here with the initial substitution attempt.

Meta-Analysis

To some extent this is difficult, because students don’t generally do A levels in this way (for very good reasons), and I’m sure that there must be students out there who could similarly blast through the modules in less than half the time given or better (but there is no data about this). Nonetheless, the A level boards usually publish Examiners’ Reports, which can be fairly interesting to read through though generally lacking in data. The C3 report was fairly rich in detail, though; and the 68/72 score was actually not too great (notice that “8% of candidates scored 70 or higher”). Indeed the aforementioned question 9 caused difficulties, though the preceding question 8 on logarithms was hardest in terms of having the lowest proportion of candidates recording full marks.

Navigating Tube Fares

A bit of an additional post for the week, as I’ve had a little bit more spare time! This post is a more fully-fleshed out response to a question my friend Andrea had, about the value of an annual travelcard.

I’ve started doing my preliminary accounts for 2016, and one of the things I examined was my transport expenditure. I typically try to use what’s known as zero-based budgeting (that is, each category and the value assigned to it is justified from fresh assumptions, rather than say raising the previous year’s data by RPI and calling it a day). Of course there’s some flexibility (I’m not going to pass up a social gathering just because of finances, unless it’s insanely expensive – which is unlikely given the background of my friends, or at least the activities we take part in together).

There’s a column of 86.50s, corresponding to a string of monthly zone 1-2 Travelcards purchased on student discount. We then have a crash to two low months as I was in the US and Singapore respectively, a figure just over 100 for November, and December looks to be closing around 50; I didn’t purchase any Travelcards after August. At the time, I made these decisions because I was unsure if going for the annual Travelcard was a reasonable idea, especially given that I would frequently not be in London owing to international travels, both for work and for personal affairs. The total cost for the category for the year was 894.68; this is lower than normal because I didn’t purchase any flights this year. I’ve been a bit cautious having been deployed internationally on quite a few occasions; I didn’t realise that you can refund the remaining value of a Travelcard!

This would have been 924 if I bought an annual zone 1-2 Travelcard (sadly, I’d now need 1,320 as I’m no longer a student); that said, with one I might have travelled more as well. Also, I was out for two months and started occasionally walking to the office in December. You can get refunds on the remaining value of a Travelcard – that said, I’m not sure repeatedly canceling and then repurchasing annual Travelcards is permissible, and it seems like it would certainly be inconvenient. Loss shouldn’t be too major of a concern, as Oyster cards can be registered to an online account which one can use to transfer a season pass away from a lost card. (I’ve done this before, though with a monthly pass.)

I think a question would then be as follows: exactly how frequently (in terms of number of days) do I need to use the Tube to make pay-as-you-go (PAYG)/monthly/annual Travelcards the best choice? We can examine that under a few assumptions:

  • The traveller is an adult.
  • All journeys are within Zone 1.
  • PAYG is implemented through contactless, so weekly caps apply.
  • The year begins on a Monday (this matters for weekly capping computations).
  • 16/7 trips per day (that’s reasonably realistic for me).
  • (Somewhat cheeky) If one travels for N days one travels for the first N days of the year.
  • Journeys on day are made between 0430 of D and 0430 of day D + 1.
  • The “greedy monthly flexible” (GMF) strategy works as follows:
    • It buys monthly travelcards as long as there are full months remaining.
    • For the partial month (if one exists), it uses the cheaper of:
      • a monthly travelcard
      • PAYG (with weekly capping)

Obviously GMF dominates a pure PAYG strategy, because for full months a monthly travelcard always beats PAYG (consider February), and for partial months GMF considers PAYG, so it does at least as well as PAYG. If I’m not wrong GMF is optimal under these contrived conditions: it intuitively seems difficult to recover from burning through February, the shortest month, without buying the monthly travelcard as you’d need four weekly ones. However, in the general case GMF is certainly not optimal (consider the period February 28 – March 31; you can buy the Travelcard on February 28, which expires March 27, and then pay for four days of fares, or pay February 28 and buy the Travelcard on March 1; the optimal strategy saves three days of fares).

The fare if one has to travel for N days is reflected in the graph below; and unsurprisingly the flexible methods are superior for small N but inferior for large N. Our model has a break-even point at about 314-315 days.

The final decision, unsurprisingly, boils down to the level of certainty you can have about your travels. If you don’t expect to be spending more than around 50 days outside of the UK, the annual travelcard seems like an idea worthy of consideration especially if you know when said days lie. That said, we have made two key assumptions, one of which favours the monthly strategy and one of which favours the annual one:

  • An upfront lump-sum payment is needed if you’re using the annual scheme. Our analysis did not account for the time value of money (you would need to discount the monthly payments to today to get a fairer comparison of the two).
  • However with the monthly strategy we’ve assumed that plans are known well in advance (at least a month) and implementation is done perfectly. In practice, there are likely to be some minor errors or plans not aligning neatly on month boundaries that will result in slightly higher fares.

I personally don’t expect to travel more than that, but I won’t be getting an annual card next year, for other reasons. (In particular, that “16/7 trips per day” assumption is unlikely to be valid, but that’s a subject for another post.)