Browse Month

# Odd Sums Sudoku

The puzzle above is from the WPF Sudoku GP, Round 4 2016 #13. To solve it, each row, column and marked 3×3 box must contain the digits 1 to 9 once each, and the solution must have, in the marked rows, maximal odd substrings summing to the given totals in the correct order.

I’ve always enjoyed a good puzzle, though I’ve never considered myself especially good at them. That said, I’ve done my fair share of sudoku, including several books where puzzles were labelled as very difficult; yet, fairly often identifying triples, or the odd X-wing, XY-wing or unique rectangle would be sufficient to blow through the puzzles. At least, I thought that clearing some “very difficult” puzzles in 8 or 10 minutes was blowing through them; I found the World Puzzle Federation’s Sudoku GP, where some of these had benchmark times along the lines of 3-4 minutes!

Furthermore, in addition to your standard “classic” puzzles which feature the standard rules (i.e. each row, column and marked 3×3 box must contain the digits from 1 to 9 once each), there are a couple of interesting variants that require understanding new rulesets and figuring out how to deal with them. After a while, as mentioned the classic puzzles largely tend to be solved with standard techniques, and the challenge there is dispatching them with extreme speed; the variants, on the other hand, tend to maintain some degree of freshness (at least as far as I’m concerned).

The above puzzle was an example of this; several of the rows are marked with additional clues, and for these rows, the largest odd substrings in the row must sum to the values indicated by the clues. For example, if a row was 178356924, the clue associated with it would be 8, 8, 9. Note that this is also generated by several other valid sequences, such as 532174689 or 174283569.

For the above puzzle, the 9, 5, 3, 8 clue is a good starting point, and indicates that we have 9, 5, 3 and (17 or 71) as substrings of the solution to that row in some order; given the position of 6, we can conclude that the 9 and the 3 must occur before the 6 (though we don’t know). We thus have 9 in the bottom left hand corner, an even number (either 2 or 4), and then a 3. Now, reading the first column clue 1, 12, 3, 9; we have 1, (57 or 75), 3 and 9 as substrings. The 9 is at the end, and 3 cannot occur before the 4 as there’s no space to fit it. Thus 3 occurs after the 4, but it has to be separated from the 9 by an even number. More simply, the second column clue 10, 9, 6 indicates that our 5 has to be preceded by an even number and then a 1. There’s only one spot left for a 7 in this lower box, so we can insert it.

We thus reach this position:

Moving on, we can consider the middle column. 3 can only be satisfied by a singular 3, so we know that the 3 in the bottom row cannot be in the middle column. It thus must be in the sixth column, with 1 and 7 taking up the eighth and ninth columns (in some order).

Now consider the rightmost column. It is 3, 10, 12; 3 can only be satisfied by a singular 3, so the odd substrings must be 3, (19 or 91) and (57 or 75). Given the position of the 2 (fourth row), it must be the case that the column has 1 and 9 in the fifth and sixth rows, an even number in the seventh, and 5 and 7 in the eighth and ninth. We thus clearly have 7 in the corner, and 5 above it.

Finally, we can actually decide on the ordering of 1 and 9 as well; consider the hint for the middle row, which ends with 8; a 9 would automatically exceed this. Thus the middle row must contain the 1 and the sixth row the 9. The hint gives us for free that we have a 7 to the left of the 1.

There are more interesting deductions we can make as well. Consider the seventh column. The clue is 16, 9 and this can be satisfied in two ways: {1357} in some order and then a singleton 9, or {79} and {135} in some order. However, we can prove that the latter construction is impossible; the middle row is an even number, and also we have already used the 1 and the 5 in the bottom right hand box so we can’t place those numbers there. Thus, the first four numbers must be {1357} in some order and there is a 9 somewhere in the bottom right box.

The clue in the first row requires a 5 at the end. Given the 10, 10, 5 clue and the 4 already placed in the sixth column, the only odd substring in the last box must be the 5. We thus must have the 5 in the first row, and now only the 3 can fit in the fourth row. We also gather that the last two columns of the first row must be even, and we have enough clues to place these (using standard Sudoku rules).

Finally, we can place the 4 in the middle right box (now that the 3 is there, there’s only one space left where it can fit). Similarly, we can place the 5 (knowing that it can’t be in the seventh column any longer).

From this point on, solving the rest of the puzzle should be reasonably straightforward. The next step would probably involve placing 1, 9, 6, 7 and 3 in the first row (in that order – using the column hints and the knowledge that we have two tens)!

For these contests, I’d say the challenge lies largely in figuring out how the ruleset can be used to help solve the puzzle (and, well, speed; you’re thrown 13-15 puzzles to solve in the space of 90 minutes and I can only usually manage about seven or eight, and for that matter the ones I do finish are the ones on the easier end of the scale!).

I travelled to Denmark for a recruiting event last week. The nature of the travel (2-hour flights plus an hour or so of ground transportation on each end) was not something I was used to; admittedly most of the flights I can remember taking are pretty long-haul ones (London-Singapore, London-San Francisco are by far the most common; London-New York exists too, though is less common). I had a great time, in any case – although being awake for about 24 consecutive hours especially when slightly ill was not too enjoyable, it was nice to meet up with students and flex some speed coding and debugging muscles. I think I still coded well even when really tired, though it did lead to some confusing moments as I tried to understand what exactly “PAY THE PRICE” meant in terms of a discount (it means no discount, but given that there was already “HALF PRICE” and a sliding reduction scheme, this did confuse me quite a bit).

This is just the second trip so far this year, though I anticipate there’ll be quite a few more. The two trips I’ve made so far were for business reasons; I’ve been hammering away at a major project at work that should be drawing to a close soon so there should be scope for personal time off. There are also likely to be more work trips in the pipeline – within the scope of my major project, for company events at large as well as for recruiting purposes. Of course, there would be even more trips if I was a Forward Deployed Engineer as opposed to a dev.

I remember reading an article in the BA inflight magazine en route to Billund, and noted that in some survey, 30 percent of people would accept a lower paying job if it meant that they could travel more for work. I think I subsequently found the survey they were referencing. I certainly would be in the 70 percent of people that would not, though it’s hard to draw an accurate conclusion given the way the question is framed (what’s the baseline here? I’d think people who already travel 50 percent compared to people who don’t travel at all might respond differently to being asked if they would do this to travel “more”). The survey was also commissioned by Booking.com, which might have a vested interest in portraying people as more willing to travel (so that companies would engage travel provider services more).

Of course, in business terms there are the obvious benefits of having in-person interaction between team members and/or getting the people who know what they’re doing on the ground, to see what’s going on. They can be really useful to get things done fast; communicating over instant messaging or even via a video conference doesn’t tend to be as effective.

I would say the upside of travel as far as an individual is concerned includes the opportunity to experience new places and see new sights, though I’m not sure how much of that you could achieve in a business trip. I find that many of the past trips I’ve had had been high-octan fire-filled affairs (though there have been exceptions – and not the programming kind). Other benefits involve meeting up with existing friends and acquaintances in the area (again, the tight schedules of many of my trips in the past precludes this). The article does make reasonable points concerning extending one’s stay, though – I actually did that two years ago when I stayed on in California for an additional week after one such trip – and fluidity of plans leading to last-minute bookings.

One of the things that makes me wary of travelling too much is actually long, unpredictable delays through security and customs – this might partially be a result of the routes I typically fly on (which go through major hubs with lots of traffic). I do have strategies to mitigate this (typically, I book an aisle seat near the front, and walk at a very brisk pace once disembarking from the plane; I also tend to choose flights that arrive at relatively unpopular times). This wasn’t a problem at LCY nor in Billund Airport in Denmark; queues were very short and handled very quickly.

To some extent this can be mitigated by Registered Traveller in the UK, the ability to use the electronic terminals in the US and the biometric gates in Singapore; it’s especially useful for me since I don’t typically travel with checked baggage, making the whole process of leaving the airport and getting where I need to go much faster. I did some computations to check whether the UK Registered Traveller scheme was worthwhile, and in the end decided it was reasonable. I used an assumed number of 12 trips per year with an estimated time savings of 25 minutes per trip (a crude estimate, but there are 20 minutes between the UK Border Force’s benchmarks for non-EU and EU passport queues; going by benchmarks ePassports are probably even faster hence the + 5) to figure out that the net cost was 14 pounds per hour saved post-tax (or 24 pre-tax). It could be argued that those specific hours might have unusually high value (e.g. stress at the end of a long flight), perhaps boosting the case for it. I’ll probably travel even more with Registered Traveller in place, actually.

I guess the other thing that has recently dampened my appetite for travel (at least on the personal front) would be the depreciation in the pound; relative to many people I know my proportion of overseas expenses is actually already fairly high (because of investments; 90% of my equities and 50% of my bonds are based overseas – and I hold much larger quantities of equities than bonds). To some extent I overreacted to Brexit (my savings, even in US dollar terms, have increased from the “Remain” case model I had) and there’s certainly still scope for travel around the EU where the pound has fallen more like 10 as opposed to 16 percent, though the length and luxury of such trips is likely to be reduced.

I’ll be travelling again soon, though it’ll be another of those long flights instead. Nonetheless, at some point in the future I definitely want to revisit Denmark. This trip was focused rather closely on executing the recruiting events in question (well, there isn’t really much you can do with one day anyway; I was in Denmark for less than 24 hours) – while I did get to see a little of the city, there was certainly much more to do.

# Revisiting Mathematics Competitions (Hotseat: AIME II, 2015)

I remember I had a pretty good time doing the Haskell January Test earlier this year, and thought it might be both enjoyable and useful to revisit some of these past mental exercises. Hotseat is intended to be a series that discusses my thoughts, ideas and reflections as I take or re-take examinations or competitions; many of these will end up being related to mathematics or computer science in some way, though I’ll try to also include a few that go against that grain.

#### Background

I didn’t actually participate in the American Invitational Mathematics Examination (AIME) as I don’t think my high school offered it (the only maths competitions I remember taking myself are the Singapore Math Olympiad and the Australian Mathematics Competition). This is typically the “second round” of the main mathematical competition in the US (the first being the AMC, and the third being the USAMO), and features 15 problems, all of which have integer answers from 0 to 999. The point of this is to facilitate machine scoring. While the questions might indeed have integer answers, they can be very difficult especially towards the end.

2015 II was supposed to be an easy paper, though I didn’t find it to be very much easier than what I recall these to be. You can find the paper here.

I managed to fight my way through 12 out of 15 questions, leaving out two hard problems towards the end and a not-so-difficult one in the middle (mainly because I struggle somewhat with geometry in three dimensions). I think I’m not used to doing these as quickly as I would have in the past as well; problem 14, for instance, would probably have taken under 15 minutes in the past while it now took 32. 8 minutes on problem 1 is rather bad, as well.

The values below indicate the time spent on each problem; here green means the problem was answered correctly while grey means no answer was given. (In practice, in a contest you would take the 1-in-1000 shot and guess, since there are no penalties for guessing!). I’ve also outlined the subject areas – you can see that I tend to be stronger with combinatorics and probability, and struggle somewhat with geometry!

I think I remember mentioning in a post a very long time ago that when I look at data, I tend to initially look for things that seem unusual. Bearing in mind that the difficulty of these papers tends to rise, we could say the 32 minutes spent on question 14 or 15 on question 7 is perhaps not too far from expectations (the target “pace” here is 12 minutes per question, if you’re trying to finish). Bearing that in mind, we can look at a couple of outliers:

• Problems 1, 6 and 11 took unduly large amounts of time.
• Problems 2, 3, 8 and 12 were done well above pace.

Some of these were actually pretty interesting, which is one reason I tend to enjoy these competitions as well.

#### Selected Problems in Depth

Problem 1: Let $N$ be the least positive integer that is both 22 percent less than one integer and 16 percent greater than another integer. Find the remainder when $N$ is divided by 1000.

For this question you were actually supposed to find $N$; the “divided by 1000” shenanigans was so that the answer would be an integer from 0 to 999. The challenge here was really formulating a requisite equation from the problem statement, which would be

$\dfrac{39}{50} k_1 = N = \dfrac{29}{25} k_2$

and then reasoning that $N$ must be divisible by both 29 and 39, such that $k_1$ and $k_2$ were both integers. Since these are relatively prime $N = 29 \times 39 = 1131$ and the answer was 131. I guess I hadn’t really warmed up yet, and wanted to be very careful to set up the expression correctly, so this took rather long.

6 was a bash with the Vieta formulas, though my brain temporarily switched off when I somehow thought a cubic equation should have four roots. 11, on the other hand, was a rather messy geometry question which I initially found tough to visualise. The solution on Art of Problem Solving involves constructing perpendiculars from the centre of the circle to the two lines, but I didn’t think of that (and for that matter wouldn’t have known that was a property of a circumcentre), and instead had a rather painful trigonometry bash to find the answer.

On the brighter side, 2 and 3 were rather simple questions (an elementary probability question, and a rather quick modulo-arithmetic one) and I cleared them off very quickly. I dealt with 8 using some rather quick-and-dirty casework that yielded the desired result, though it wasn’t too interesting. I’d say 12 was a bit more of a fun one.

Problem 12There are $2^{10} = 1024$ possible 10-letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than 3 adjacent letters that are identical.

I thought in terms of recurrences and dynamic programming, and realised that valid strings (in the long case) must end in BA, BAA, BAAA or their flipped versions (alternatively, with a blank space in front). To end a string with BAA, it must previously have ended with BA, and so on for BAAA. Thus, the following equations drop out:

$BA(n) = AB(n - 1) + ABB(n - 1) + ABBB(n - 1)$

$BAA(n) = BA(n - 1); BAAA(n) = BAA(n - 1)$

Since the AB and BA cases are symmetric, we can simplify the first equation above to

$BA(n) = BA(n - 1) + BAA(n - 1) + BAAA(n - 1)$

And we have of course the base cases; $BA(1) = 1, BAA(1) = 0, BAAA(1) = 0$. Computing $2 \left( BA(10) + BAA(10) + BAAA(10) \right)$ is then not difficult, and yields $548$.

In mathematical terms I would say problem 14 required some insight, though for seasoned competitors it would probably still not have been too difficult.

Problem 14Let $x$ and $y$ be real numbers satisfying $x^4y^5 + y^4x^5 = 810$ and $x^3y^6 + y^3x^6 = 945$. Evaluate $2x^3 + (xy)^3 + 2y^3$.

I initially tried squaring the resultant expression, and combining the two given results in various ways. This went on for about 20 minutes with little meaningful progress. I then took a step back, attempted and finished up with problem 11, and came back to this. I suddenly had the idea of combining the two givens to force a cubic factorisation:

$x^3y^6 + 3x^4y^5 + 3x^5y^4 + x^6y^3 = 810 \times 3 + 945$

And that simplifies down to

$x^3y^3 (x+y)^3 = 3375 \leadsto xy(x+y) = 15$

We then proceed by factoring the first result:

$x^3y^3 (xy(x+y)) = 810 \leadsto x^3y^3 = 54$

We can get $17.5$ for $x^3 + y^3$ by factoring the second given to $x^3y^3(x^3 + y^3)$. The final result is then just $35 + 54 = 89$.

I subsequently figured out solutions for 13 (my brief intuition about complex numbers was kind of on the right track, but I didn’t have time) and then 9 (not good with 3D geometry), and had a look at the rather painful solution for 15.

#### Meta-Analysis

It’s worth taking a look at how this performance was relative to the actual score distribution, which is available on the AMC statistics page. A score of 12 would be in the 96.09th percentile, which is interesting; I did the 2015 AIME I a few weeks ago and scored 11, but on that contest that was the 97.49th percentile!

There’s also a report on item difficulty (i.e. the proportion of correct responses to each problem). I’ve graphed this below (items I answered correctly are in green, while those I did not are in red):

It’s interesting that problem 12, a problem I found to be one of the easier ones (assuming you sort by success and then time, I found it to be the 8th hardest) turned out to be the second “hardest” problem on the paper! I’d attribute some of this to a programming background; if I took this contest when I was in high school I think I would probably struggle with 12.

13 and 15 were hard as expected; the next hardest problem in general after that was 10 (just a notch above 12 in difficulty for me) though again it relies on tricky combinatorics. We then have 6, 11 and 8 before 14; I found 6 and 11 relatively harder as well, as mentioned above, though 8 was fairly trivial for me. It turns out that my geometry weakness is fairly real, as 9 was completed by more than half of the participants! We can also observe problem 1 being challenging relative to its location in the paper.

#### Conclusion

This was a good mental workout, and it felt nice to be able to hack away at the various problems as well. I think a 12 is pretty solid (the 96th percentile is that of AIME qualifiers, which itself is something like the top 5 percent of high school students taking the AMC, so that would be the 99.8th percentile – and you could argue that considering only students taking the AMC already introduces a selection bias towards those with greater aptitude). I’d probably have been able to solve 9 and maybe 15 if I was doing this back in high school, though I’m not sure I would have managed 12.

I can certainly feel a loss of speed, though I think my problem-solving ability is for the most part intact, which is good.

# On Hints in Puzzles

I played a game of Paperback with a couple of friends from Imperial yesterday. The game involves players making words from letter cards in their hands, and using them to buy additional, more powerful letter cards and/or cards worth victory points. It’s pretty much a deckbuilder crossed with a word game; while anagramming skills are indeed important, they aren’t always necessary (for instance, I managed to form the word WORK to buy the second highest tier victory point card) and certainly aren’t sufficient (keeping one’s deck efficient is certainly important).

In any case, the game reached a point where one of my classmates seemed to be agonising over what was presumably a tricky hand. The game does have a number of cards that use digraphs such as ES, TH or ED, which can make searching for words a little trickier. The common vowel (a vowel that players are all allowed to use in their words in addition to the cards in their hand) was an O, and the hand in question was [IT], [R], [B], [U] and two wilds. I immediately saw a solution when he sought help (BURrITO or BURrITOs; none of the cards had abilities dependent on word length) and let him know that a ‘perfect’ solution for his hand existed, though we decided to let him figure it out for himself. This started to drag on, though, and giving him the answer directly seemed excessive; I thus decided to drop hints that would hopefully lead him to the solution.

Technically, the search space that he would have had at this point would have been $5! + 6! \times 26 + {7 \choose 2} \times 5! \times 26^2$ which is about 1.7 million possibilities, though of course one does not simply iterate through them; combinations like OqRzBITU would probably not even be considered directly, especially since the distribution of letters in English certainly isn’t memoryless (the Q being a good example; it is rarely followed by any character other than U). I gave a few hints, in this order:

1. The word is a common word that a person in middle school would likely know.
2. The word can be made with just one of the blanks.
3. The blank you use matches one of the letters already on the table.
4. You can eat it.

We could argue that the point of a hint could be to reduce the search space being considered by disqualifying some solutions, and/or serve as a way to encourage the algorithm to explore cases more likely to be correct first. Interestingly, it seems that clue 1 was neither of the above, though. It didn’t really cut down the search space much, and as far as I know it’s not easy to use this information to organise one’s search. Its point was mainly to reassure my classmate that the solution wasn’t too esoteric and he would know it was a valid solution upon considering it. I didn’t think I had played any particularly uncommon words earlier in the game (the rarest might have just been TENSOR or something), though I think I might have played Anagrams with him before, where I know I’ve called out some more… interesting words.

The next two clues do actually substantially reduce the search space in a sense though; the second clue slashes it to just $6! \times 26 = 18720$ and the third further cuts it down to $6! \times 6 = 4320$. I would think that after the third hint, the search should be pretty fast; I would probably try doubling up on the R and T first as they’re common.

Nonetheless, hint number 4 seemed to be the hint he responded best to, and it falls into the second category (encouraging the search to explore cases more likely to be correct). I’m not sure how humans solve anagrams, especially when blanks are involved; I personally seem to look at letters, intuitively construct a word that looks like it incorporates the key letters present, and then verify that the word is constructible (in a sense, it is a generate-and-test approach). Given the point values of the tiles, I saw including the B, U and IT cards as being the most important; thinking of words that use these patterns resulted in BITUMEN and BURRITO fairly quickly, the latter of which was feasible.

I noticed that I tend to bias towards using the constraint letters near the beginning of the word though; this might be a cognitive issue with trying to satisfy the constraints quickly, in an unusual form of the human brain often being weak at long-term planning. This wasn’t really the case with the burrito example, but another friend had a hand of ten cards (I think) at some point with an [M], [IS], [T], [R], [D] and a couple of blanks. With the common O and knowing that the M, IS and D were higher-value, I saw MISDOeR and struggled to incorporate the T, finding it difficult to back out of starting the word with a MIS- prefix. It turns out it might have been okay in that I could have found MISsORTeD, there were easier solutions such as aMORTISeD or MODeRnIST.

On hindsight, though, my hinting might not have been ideal. This is because BURrITO and its plural form weren’t the only solutions; there was OBITUaRy, which some people might find easier to see (though I didn’t actually see this during the game). I think clue 1 would probably already be invalid; I knew the word ‘obituary’ in primary school, since it was a section in the newspapers that I would browse through in the mornings, but I’m not sure it would be fair to expect middle school students in general to be familiar with the term. Clue 2 would certainly be incorrect.

# The Boy Who Would’ve Reached for the Stars (Q1 Review)

The past few weeks have been incredibly intense; I had been root-causing and then fixing a fairly complicated issue on AtlasDB that involved unearthing rather messy relationships between various components. I thus took three days off this week to cool off a little, and also to collect some of my thoughts on the quarter gone by.

Looking at my public GitHub statistics, I merged twenty-eight pull requests over the last three months, ranging in size from one-liners to an almost 3000-line monster that should probably never have been. That’s about two per week, which is a reasonably healthy rate, especially bearing in mind that much of the last three weeks was spent debugging the big issue. (I acknowledge this metric is pretty coarse, but it does at least reflect positively in terms of the pace at which development work is getting reviewed and merged to develop.)

It seems I’ve done quite a fair bit of pull-request reviewing as well; recently this has to some extent shifted towards benchmarking and performance, which is something I find to be personally interesting. Concepts that a few months ago were merely abstract ideas I was vaguely aware of, such as volatile variables, blackholes and false sharing now actually do mean a fair bit more to me. I’d say there’s progress here.

Other goals in terms of computing such as learning more about distributed systems, contributing to interviewing and sharpening my knowledge of Java have certainly been trundling along acceptably well. My work with MCMAS development also continues, and on a bright note my submission (together with Prof. Alessio Lomuscio) of a paper on symbolic model checking CTL*K got accepted at AAMAS 2017. I’m also in the process of setting up a more modern version control infrastructure for MCMAS, and merging my own CTL*K-related changes from the project down into MCMAS’s master branch. This adds an end-to-end test framework and also fixes some old bugs in MCMAS, including deadlocks in counterexample generation.

Financially, the market continued its march upwards and my portfolio gained 4.8 percent (inclusive of reinvested dividends). Interestingly, I thought it had largely been flat across this January-March period. I was of course aware of the Trump reflation trade, but thought it was already largely priced in by the end of last year. Looking at things within my control, I continued my monthly regular investment scheme, and successfully completed an admittedly ill-conceived challenge in February to limit discretionary expenditures for the month to a hundred and fifty pounds (clocking in at 148.16). I decided to replace my laptop in March, which was a significant budget bump, but not unreasonable given my previous machine had an SSD failure and had served me well for about two years.

Amidst all this, it can be easy at times for me to simply lose myself in the music rhythm of getting things done, to paraphrase a certain rapper, without considering why said things are being done. I had to make a trip to New York to do some important work, and on the flight back to London I noticed X Factor winner James Arthur’s new album, “Back from the Edge” on the inflight entertainment system. I liked “Impossible” and “Say You Won’t Let Go”, and I was aware he had a great belting voice, so I decided to give it a spin.

The title track, among other things, flags a desire for the speaker to return to his beginnings. It’s delivered with a lot of vocal power (as I would expect), which works for me at least (viewed in the sense of announcing or declaring one’s return):

Back from the edge, back from the dead
Back from the tears that were too easily shed
Back to the start, back to my heart
Back to the boy — who would’ve reached for the stars

Thinking back, things have changed quite a fair bit for me over the last few years as I moved from Singapore to London. I’d say the most obvious changes were in terms of academic/professional expectations (they’ve always been fairly high, but skyrocketed after placing 1st in first year), work ethic (I do remember being pretty lazy) and personal finance (I’m now much more frugal and also invest in stocks and funds). In many cases, I’m relatively satisfied that these changes have taken place; I wouldn’t say that after these four to five years and reflecting upon them I’m now seeking to come “back from the edge” or “back from the dead”. However, I’m sure there are some less obvious changes that are certainly less positive, such as relatively less interest in and less time spent on learning outside of engineering and finance; these are often exacerbated by high expectations leading to more time/effort spent on ensuring said high expectations are met. There definitely exist things which I used to like about my past self that aren’t as present in my present self.

I’ll be finalising Q2 targets soon; apart from the usual engineering/finance/friendships (EFF) targets which are certainly very important, I will include one or two other things as well. I’ve generally found identifying what I want to accomplish beyond EFF which are often my primary foci difficult, and other things tend to be very quickly deprioritised if specific objectives for them aren’t established.