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The Megalixir Problem

Image by tatu234 from Pixabay

In the Final Fantasy role playing game series, the Megalixir is usually one of the most powerful healing items in the game, typically restoring all characters’ health and magic points to maximum. The name stems from mega- (one million, but more contemporarily extreme or powerful) and elixir. Most of the time, Megalixirs are also special in that they are not easily obtainable; they are not directly purchaseable for the common in-game currency, and must either be found, obtained (rarely!) from enemy monsters or specifically traded for. When I played the games, these items would often end up sitting around in my inventory unused; I’d maybe use one or two when fighting the final boss or bonus boss, but there would still usually be many left over once the game was 100% completed. There’s always a fear that what’s to come might be even more difficult and necessitate the use of a Megalixir.

On the other hand, when I was in university I used to participate in mobile gaming offers on so-called “offer walls” as a side hustle. Game companies would advertise their mobile game and pay some amount of money to make some defined progress in their game (in exchange for generating usage statistics that would be useful e.g. in the App Store). These played out quite differently; many of these apps were free to play, but had a form of premium currency – players could accumulate some in the game, though they could obtain much more by making in-app purchases. Since the ending point was well known (e.g. complete level 20), I would aggressively burn the premium currency with the goal of reaching the required level with as little slack as possible; most likely if I was playing the game for the long haul I’d not be as relaxed there.

The Megalixir is of course a metaphor for a resource that is high-impact, somewhat scarce and difficult to replace. Whether it is a slug of equities or cash, some nice clothes, a premium bottle of wine or even a stash of airline miles, I find that my first instinct is usually to conserve rather than use them. So the equities are aggressively reinvested, the clothes stay in the cupboard until there’s a really special occasion, the wine remains undrunk and the flights untaken. However, I think a lesson I’ve learned this year especially through the COVID-19 restrictions was that I should be more generous with these resources (and this means being more willing to use them); retaining a Megalixir is a necessary but not a sufficient condition for using it in the future. Of course, one can attempt to make more sophisticated expected-value calculations: that’s the basis for safe withdrawal rates in retirement, for example. These calculations are fairly involve and incur some overhead, though, and thus while it makes sense to plan one’s retirement, it may not be productive agonising over smaller resources. I’m reminded of several instances over the years where I’ve spent hours researching and ruminating over a 10 pound decision. To take the metaphor further, the Megalixirs, while hard to come by, are generally not strictly limited.

The inspiration for this post came up from a video that found its way onto my YouTube playlist this week – that being one for Charlie Puth’s Dangerously. I remember first listening to this song in late 2016 when flying back to Singapore for a Palantir deployment, on Singapore Airlines, and neither having been back nor being able to reasonably return and work from there has been fairly draining. I normally go back every 4 months, while it’s been 11 since the last time I was there. Although I didn’t pay for the flight at the time, it did make me think about travelling, and how I’d be willing to pay up for premium economy (or possibly even business, if it’s cheap) when international travel restrictions loosen.

In general, there has been a shift in my finances and planning in this direction since March or so when COVID-19 began to impact my life in the UK. At least as far as personal finance is concerned, I have quite a strong inherent bias towards delayed gratification, so overcoming that feels strange. It still does feel awkward, unfortunately (and indeed frugality and being critical of one’s purchases is usually a strength).

Some of this is, to go with the Megalixir analogy, actually along the lines of supporting my physical and mental health in a more challenging environment, and I would say my work and personal environment has definitely become more challenging this year (though not purely due to COVID reasons). It’s worth noting here that challenging doesn’t mean bad; for example, I’d say my work environment is several times more challenging than it was in 2016 because the scope of my responsibilities has grown, though this expanded scope also has the potential to be considerably more rewarding. This manifests in terms of better quality food – premium apples and fresh fish, for example. I remember that as a first year in university I facetiously derided the “three for 10 quid” deals on meat and fish in Sainsbury’s, M&S or Waitrose – now I’m actually using these offers quite frequently. The mental aspect of this is probably more damaging to the bottom line, as I do indulge in a little retail therapy.

In spite of this, this year’s saving rate is quite high, because income has held steady while expenditure has not risen dramatically. I’m fortunate that my income flows haven’t really been affected yet, and because of the DPO actually grew. Although I intend to re-work my spending patterns, that’s not readily feasible yet. There’s no point in using a Megalixir when the monsters are very weak; if your HP is say 9900 out of 9999 and a regular potion heals 100 HP, you should just use the regular potion. Most of the stuff I find interesting or purchase as part of said retail therapy turns out to be quite inexpensive, and my tastes in food are simple so there’s really only a limited amount of bottom line damage that can happen here. Hopefully as holidays and international travel start to come back online next year, there’ll actually be opportunities to chase where one of these can be legitimately deployed.

On Speaking English Poorly

I saw an interesting question in my Quora feed:

As a native English speaker, when you hear foreign people speak bad English do you hate it?

Generally speaking, my answer would be no; while I would probably notice it, hate is a strong word. I’d probably react to it more out of confusion or surprise than in an outright negative way; in this context even dislike is stronger than what I would intend. I think it’s implied by the question that the foreign person is from a country where English is not spoken as a first language, and thus their native language is not English.  Learning a second language is difficult. In addition to English, I am also able to speak some Chinese (~CEFR B1) and German (~A2). However, Chinese was generally my worst subject through middle and high school in Singapore and I wouldn’t be confident in delivering an impromptu speech or discussing philosophy in Chinese. So far I’ve found German easier, but speaking is still by far my weakest of the four main language skills there. I almost certainly can’t speak the person’s native language as well as they can. (The case where the person’s native language is English would be a little more frustrating, but even then, hate is a strong word!)

It’s probably more interesting to consider the cases where I would be annoyed and think this is justified. To analyse this, I’d first look at what impact speaking English poorly might have, and then consider the conditions under which these impacts would be unacceptable.

Spoken English is primarily used as a medium of communication, so to some extent I would evaluate the consequences of speaking English badly based on how much this impacts communication.  Firstly, even when communication is achieved without any further questions or disruption, I would find that conversations may still not flow as well as they would with a fluent interlocutor. This is because my brain may need to take extra effort to decode the meaning of what is being said. I do notice small errors such as “There are less apples in my bag” (fewer – apples are generally countable) or “Dancing makes fun” (dancing is fun – this is a word-for-word translation from German, Tanzen macht Spaß). If there is limited grammatical structure to the sentences, even more effort is needed. I can work out what is likely intended when one says “Epistemology means read knowing of part reasoning” (“Epistemology is the branch of philosophy dealing with knowledge”), but I would probably struggle to enjoy a philosophical discussion with such a speaker.

Secondly, even if successful communication is achieved, it may not be as nuanced as intended. If I was to describe a performance that I would rate a 5/10, I could say that. However, a 5/10 could be a 5/10 because it was undistinguished across the board, in which case I might use “unspectacular” or “banal”. On the other hand, a 5/10 with distinct strong and weak components might be described as “a very mixed bag”, or I might feel “ambivalent” about it. I can’t do this as neatly in Chinese (平庸 for 5/10 in general, but distinguishing the types is hard without circumlocution) or German (“nicht so gut”, but I’m not sure how to go about the rest – “nicht so gut überall” vs. “teilweise sehr gut, teilweise furchtbar”?), to the point where I’d not bother communicating these thoughts as precisely, unless how the 5/10 score came about was important to the conversation.

Finally, of course communication breakdowns and errors can happen. These can arise from plain errors in vocabulary (thinking “tomorrow” means “yesterday”, or “sanguine” means “pessimistic”), expressions that are inadvertently inconsistent (“I will eat the fish and drank the white wine with it” – is this meal in the future or past?) or unintended connotations (not realising “imbecile” or “wanker” are general insults and don’t just mean “of low intelligence” and “person who masturbates” respectively).

In general, I think as long as a person is making a genuine good-faith effort and is not in a position where I’d expect them to have fluent English, I would certainly not hate, or for that matter, dislike the situation. It may affect the extent to which I’m able to communicate with them for pleasure, but it doesn’t mean I wouldn’t be pleased to make their acquaintance. It would probably mean I would avoid choosing discussion topics and activities that would require extremely precise, clear communication, but that’s fine – that covers only a minority of my interests. Furthermore, if they’re learning English I would be interested in helping them – maybe they could teach me a bit of their native language too.

I’d see a person as not making a genuine effort at communication if I think they are trying to obstruct it. This is difficult to assess; it’s not clear how to distinguish a lack of ability and a lack of will here. I think this mainly covers deliberate errors when I’m confident the person would know better – this introduces unnecessary overheads or even barriers in communication.

The good-faith part of this would be broken by a breach of trust. For example, the person gets angry or accuses me of speaking English poorly or deliberately obstructing communication. Alternatively, the person blames me for a miscommunication that arises because something was clearly said to mean something following standard English (even if that’s not how the person interpreted it).  Thankfully, I haven’t had many experiences in this vein.

Finally, there are circumstances where I’d expect people to be able to speak English fluently. Typically, this happens as part of a transactional relationship where some communication is required, and English is chosen as the medium of communication (which would be by default in the UK). The level of fluency required here is of course variable, depending on said relationship’s nature. In literary performances I’d expect halting and/or “bad” English to be intentional for some dramatic effect; using English poorly inadvertently here would be highly questionable. Similarly, I’d want my fellow software engineers, lawyers and tax advisors to be fluent because the issues I’d be discussing with them are likely to require this. Statements like “I received a billion US dollars in my account based in Luxembourg in July, converted it to pounds sterling in August and remitted it to the UK two months after that” need to be understood exactly, and any error in the timeline (like thinking a billion is 1012 instead of 109 which is standard in English, the remittance happened in September instead of October, confusing US dollars with Australian or Singaporean dollars, or pounds sterling with Egyptian or Syrian pounds) could have serious consequences. I’d expect some (but less) fluency from say service staff in the UK – enough to understand and complete my requests accurately would suffice. I might be a little annoyed if someone advertises a buy-one-get-one-free offer as an unconditional 50% discount (they’re only equivalent for an even number of items!) or repeatedly brings a fork when I’ve asked for a spoon (once is fine).

On Eating Out to Help Out

The UK Government ran an interesting promotion called “Eat Out to Help Out” for the month of August. Meals eaten in a restaurant, pub or cafe (including soft drinks, but only soft drinks) were eligible for a 50% discount up to a maximum of £10 per person; the customer would pay 50% and the restaurant could claim the remaining 50% of the price back from the Government. The restaurant doesn’t pay anything (apart from the administrative cost of giving the discount and claiming it – there were multiple occasions where I’ve heard staff explain to a diner why the calculation wasn’t as they expected – and possibly some cash-flow/time-value-of-money concerns).

My initial reaction to this was that it was a nice bonus, along the line of “every little helps”. The phrase “bread and circuses” did come to mind, though for me at least this doesn’t really move the needle of my political opinions much. I could see how it could stimulate the economy at least in the short term (granted, with a risk of making the COVID-19 situation worse: the Soho area was extremely packed on the evenings I went there). That said, I also noted that it was a little bit limited in that the rebates are proportional to one’s existing expenditures on food, which may be a (faulty) proxy for existing standard of living. A person who can’t afford to eat out at all can’t benefit, and my individual meals usually cost more like £10 and so I might benefit less than someone whose meals do regularly hit that threshold.

Furthermore, there was no restriction on the number of times the promotion could be used, meaning that one could have an appetiser in one bill, a main course in a second, a dessert in a third and all the soft drinks in a fourth, potentially getting a £40 per head discount (and this could be extended even further if one goes for a tapas-style meal, for instance)! There were also some further loopholes, such as taking away leftovers, and the timing of Sunday dinner (if the bill is issued after midnight, it is Monday and so the discount applies).

Nonetheless, given the scheme was in place it is something I can appreciate, and I figured I might as well enjoy its benefits (while taking appropriate precautions to mitigate risks of actually catching COVID i.e. social distancing, masks, sanitiser, avoiding excessively busy places). I pay a fair chunk of Income Tax plus the odd bit on dividends and capital gains, so in a sense this is reclaiming some of that, especially since I’d expect taxes to rise soon in the light of needing to pay for stimulating policies, this one included. I ate out considerably more than I would have normally – this meant going out for some lunches on days without a lot of meetings, and some dinners at 7.30 or 8 pm after work (instead of eating at 6 pm and then having a short evening session). I figured that the government subsidised my meals to the tune of about £200 over the 13 days of the promotion. However, one could potentially squeeze in 10 restaurant visits per week for the 4 full weeks, and 4 visits on the final Monday, for a total of 44 visits – £440 if one spends £10 at each place, and possibly more because of the aforementioned exploits.

Of course, a consequence of this is that my August food budget went through the roof. My meals at home in terms of raw ingredients probably rarely cost in excess of £3. Although it’s not exactly logical, I often found myself ordering embellishments or more premium options to get the bill close to either side of £20; that is still a cost of £10 to me. My average expenditure on eating out was £190 in the three months to August but £325 in August; on the other hand, for groceries it was £300 in the three months to August and £230 in August. I mitigated some of the costs here by stacking offers – most notably, American Express had a Shop Small promotion that offers £5 off a £10 expense at selected retailers, so £20 becomes £10 from Eat Out to Help Out, and then £5.

In terms of the food itself, it wasn’t as much of an adventure as I originally thought it might be, perhaps because I was tired out from work and was also alone for most of the promotion; I met Tom on the last day for pizza but I think that was it. I didn’t actually end up trying any new places as part of this – it looks to be a string of familiar and comfortable establishments. I ate a lot of Japanese food. I had ramen at three different places (Kanada-Ya, Tonkotsu and the Japan Centre), sushi on a number of occasions (Sagamiya, Sushimania, Dozo and even Wasabi), and other cooked dishes (Eat Tokyo for grilled meats and fish, Coco Ichibanya for Japanese curry, Zen Cafe for melon pans). Four Seasons roast duck and 97 Old Town salted fish and chicken fried rice made the cut too. I also visited pubs a bunch of times (for food; alcoholic drinks weren’t included as part of the promotion). I think my preferences when eating out tend to result in this because a lot of these dishes are difficult for me to make (or make well) at home.

The promotion in terms of the Government subsidising diners has ended, though a number of restaurants and cafes are still continuing with this policy into September, some in a reduced form (e.g. 20 or 25% off instead of 50%, or only including a subset of the days or mealtimes). I haven’t sought out any of these yet, though, and have eaten at home a fair bit more this month – I do like my cooking enough that I would miss it if I don’t get to eat it for some time.

Deficits and Surpluses (UK Sudoku Championship, 2020)

Last weekend I participated in the UK Sudoku Championship 2020. Participants had two hours to score points, by solving as many of 16 puzzles as they could – the top contestant finished everything correctly in 56 minutes 34 seconds, and sixteen people cleared everything within the time limit. It’s pretty impressive how long the tail of the distribution is. I finished thirteen out of sixteen puzzles, scoring 621 of 720 points (puzzles are worth different numbers of points depending on difficulty, and the ones I left behind were generally on the lower-valued side of things), and would probably have taken another 15 minutes to finish out the last three puzzles.

Nonetheless, 621 points was enough to log a 28th place finish. I finished on rank 42 two years ago, and although I missed last year’s I did it offline and ranked somewhere in the 40s as well, so this looks like an improvement. Of course, it could also be the case that the field got weaker.

Most of the puzzles in this contest were fairly common sudoku variants – while I normally scan through the instruction booklet before starting the contest I don’t usually prep explicitly, and here it’s not really needed. The UK championship seems to have had three standard 9×9 Sudokus, a Killer (extra regions are added with target sums; numbers in a region can’t repeat, and must add up to the target), a Diagonal (numbers along the two major diagonals must be unique) and a Thermo (thermometers are added, which introduce inequality constraints: along these numbers must increase). I’d usually go through all of these – based on past data I’m usually reasonably fast at these.

There are also a couple of more peculiar puzzles. There’s “OEBS” (Odd-Even-Big-Small), where a couple of kanji characters (in this case they’re the same as the Chinese words) are placed next to the grid, introducing constraints on the first two numbers seen in that direction. This is usually an 8×8 grid, and is a type I like; it often admits logic around pairs and sets. Then there’s a Surplus and Deficit Sudoku. These puzzles tend to make my head spin a lot, and so I didn’t plan on going for them at all.

  • Surplus Sudoku: Place 1-N in each row and column once each. Some M-cell (M > N) regions are marked – these regions must contain each number at least once. In the contest, N = 7 and M = 8.
  • Deficit Sudoku: Place 1-N in each row and column once each. Some M-cell (M < N) regions are marked – these regions must not contain any duplicates. In the contest, N = 7 and M = 6.

I already find the geometry of an Irregular Sudoku (which has the standard rules, except the 9-cell boxes are weirdly shaped) tough – spatial reasoning is not one of my strong suits. Working with these is even harder, especially because some rudimentary Sudoku techniques become invalid in the regions (but are still valid, and often needed, in the rows and columns!):

  • Naked Single: If all possibilities other than X are present in the same row, column or region as this cell, then X must be the value of this cell. This doesn’t work in Surplus Sudoku regions, because X could be a duplicated digit.
  • Hidden Single: If X can only be placed in one location in this row, column or region, then X must go there. This doesn’t work in Deficit Sudoku regions, because X might simply not appear in the region.

Applying these two techniques is pretty much automatic for me, and these deductions remain valid in almost all Sudoku variants, but are no longer always applicable in these puzzles. The additional checking and context switching slows things down by a lot.

It’s probably a much more involved technical discussion, but to me at least Surplus Sudoku feels like a pretty weird sudoku variant (probably because of having repeated numbers in a region). I don’t find it symmetric: Deficit feels less strange, maybe because “numbers cannot repeat in a specifically marked region” is not too far off from many of the other constraints in common variants (e.g. a Killer with size eight cages – though you’re told which number is missing, or a Thermo with thermometers of length seven or eight – strictly increasing means numbers can’t repeat). I probably don’t have that much experience puzzle-solving, but I can only think of one variant which ends up forcing a lot of equality in sub-regions of the grid: Anti-Diagonal Sudoku (each of the major diagonals has only three distinct digits), though that’s not a super common variant and I find they solve kind of weirdly as well.

Writing those two puzzles off probably turned out to be a mistake, as I had about four minutes at the end with the Surplus, Deficit and Irregular left. I don’t think this was intentional, but the spatial theme here is kind of amusing. I opted to try the 20-point Deficit Sudoku instead of the 30-point Irregular which was unlikely to have been solvable that quickly, but barely got anywhere. Given how close I was to otherwise finishing the set, I might need to be careful about writing off more than one puzzle (and at some point, even about writing off one puzzle).

Nonetheless, I think I performed well, and enjoyed the contest – maybe next time around I’ll actually give these puzzles a shot! (Though probably at the end.)

Different World (Thoughts on COVID-19)

I remember a recent trip where my mum came up to London to visit. It was bright, and we had a relaxed lunch in a Chinese restaurant with a glass ceiling. I thought of how my routine had significantly changed – I would naturally prioritise spending time with her, instead of working an extra hour, studying another hour of German, practicing another set of logic puzzles or just scrolling through eBay.

Similarly, the recent COVID-19 situation has changed things quite a fair bit for me, and I imagine the changes I’m exposed to are still relatively small compared to what many other people face.

The obvious change is working from home, which I’m not particularly a fan of (human contact is important, and pair programming remotely turns out to be really hard – especially for me if I’m doing this more in an instructional capacity, as I like to have newer team members code while I jump in as needed), but I understand it’s necessary given the circumstances. Being able to work from home is certainly a privilege of software engineering, among other jobs. In hindsight, having the ability to work at all is also a privilege, in light of the widespread shop closures that came into effect. The UK government’s proposals should help, but it’s still a 20 percent pay cut (more, if one is earning more than 30,000 a year) – and I don’t know how exactly things are going to work for people on zero-hours contracts or the self-employed.

There is then social distancing: I’d lump the aforementioned shop closures into it, as if it is decided that civil liberties should not be restricted, one can still give strong pushes by decreasing the desirability of engaging in the behaviours that one doesn’t want. I do have a few groups of friends that I meet outside of work, but so far at least the impact hasn’t been too bad – we’ve found alternatives such as doing it digitally which are to me inferior but are still better than nothing. The shop closures are annoying, but in any case I do most of my shopping, apart from groceries and food, online. It hasn’t massively changed a majority of my weekend programmes, because of what activities are involved (usually reading, puzzles, computer games or walks – note that the last of these can still go ahead subject to social distancing). The relative absence of people (which is good!) is highly noticeable.

Travel restrictions are another factor. I had to cancel my Singapore trip over Easter, because the government requires returning Singapore citizens to spend 14 days at home. This is reasonable, but is of course problematic if one’s intended stay is less than 14 days. I didn’t have any other immediate plans to travel, though would likely have drafted some for the bank holidays, perhaps to Germany or Zurich again. In a sense the timing of my Zurich trip was good: I returned just before the number of cases spiked. That’s probably going to be on the backburner – and I wonder if (assuming the airlines survive) prices will be reasonable as we get past the peak.

Grocery shopping has changed as well: a year ago hand soap would have been a mundane entry on a list while I now actively look out for it. I also tried to execute a Sainsbury’s delivery order earlier this week, and saw no availability for three weeks. It’s unclear if the worst has already passed – I’m seeing slightly more stock, maybe because most of the stockpiling has already happened, and supermarkets are actively ramping up the capacity of their food-stocking chains.

Incidentally this makes me think of some variant of the Prisoner’s Dilemma from game theory, where cooperating is not stockpiling and defecting is stockpiling. It’s best if no one stockpiles (since stockpiling has costs: the cost of carry – organising one’s supplies, finding space for them, making sure use-by dates are observed safely – and the opportunity cost of the money going into the supplies, assuming that investment performance outpaces inflation). Normally, this is the case. However, as more people stockpile, seeking to stockpile suddenly becomes rational: not stockpiling may mean that one is unable to obtain important provisions. The game is iterated, as well: one presumably visits supermarkets multiple times, and can see the state of supermarket shelves and derive an estimate of what people are buying.

There is also the recent fall of and heightened volatility on the stock market. I use a broker called De Giro that has a feature to send an email every time a position goes down by 10% (I think additively). I’ve seen a couple of cheerful 40% depreciation emails for some of the riskier assets there. It could be worse: IAG (owning BA, among others) stock is down 72 percent, and Lufthansa is down 65 percent. Interestingly Singapore Airlines is “only” down 40 percent. The numerical magnitude and speed of the damage here, especially if considered in US dollars (which strengthened aggressively) is probably a factor of 10 larger than what I’ve dealt with in the past. In a sense, the speed highlights the perception of loss, because a more drawn-out fall tends to be partially insulated by fresh contributions and pound-cost averaging. However, despite the scale of the numbers involved, they seem a bit further removed (an image of a bonfire of notes equal to the five figure sum would be more concerning).

We’ll see how things go. I guess I was too young to remember and/or make any significant decisions when dealing with SARS in 2002-2003: I was in Primary 6 then, and remember schools being closed briefly. Even when schools reopened there was twice-daily temperature taking and values had to be documented. The numbers of cases were much smaller (I think four-figures globally), though I remember it being somewhat more dangerous (contrast with UK government advice – and this is indeed true – that for most people COVID-19 will be mild).

One risk of FIRE pursuits and/or aggressive savings strategies is for some reason not being able to spend the money one has saved up later on. I’d say the financial strategy I’ve been using from 2016 to early 2020, though not at popular FIRE blogger levels, involves an aggressive savings strategy, and to some extent that risk has manifested. I certainly wouldn’t say it has caused a significant shift in my thinking, but it’s a good reminder that stockpiling of wealth as an end to itself tends to be unhelpful.

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).
    • A 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 D 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.)