paving the way

When I was growing up, the pavements in Penrith’s main streets were constructed using thin rectangular slabs of the local stone—Penrith sandstone—known as ‘flagstones’. Sadly, these were all ripped up in the 1960s and replaced by either concrete or tarmac. However, in the last couple of decades, amorphous concrete has been replaced by parallel courses of precast concrete blocks, which are certainly more interesting than featureless concrete.

I don’t think I’d have noticed any of this had my attention not previously been attracted by the brick pavements in Fanling and the incredible range of patterns to be seen there. There are no deliberate colour variations in the pavements of Penrith, although the basic hue of the blocks does vary from location to location. However, when walking over them, I couldn’t help but wonder what criteria were used by the paviour to determine which block to lay next.

This is a typical example of block paving:

The blocks have been arranged in parallel courses running transverse to the direction of travel of most of the foot traffic hereabouts. This is the standard arrangement. The first thing I noticed was that there appears to be an absolute prohibition against having the corners of four blocks meet at a single point, what I will refer to subsequently as a ‘crossroads’.

The next thing I noticed was that the blocks are not of arbitrary sizes. In the example above, there are three distinct and consistent sizes, examples of which I’ve labelled L, M and S. Closer inspection shows that the mathematical relationship between the lengths of the blocks is S+M=L (I’ve defined length in terms of the orientation of the course, which in the case of the S block means that it’s wider than it is long). The other point to note is the frequency: it makes sense to use an L block wherever possible, and, conversely, S blocks are relatively uncommon.

However, in the next example, M blocks have been used most frequently—the total is significantly more than the combined totals of the frequencies of the two other blocks. And there is no obvious mathematical relationship between the lengths of the blocks although it may be that 2S=L:

I had thought that more than two contiguous blocks was a rare occurrence—check out the first photo, where even two blocks of the same size touching is a relatively infrequent occurrence. However, there are examples of six in a row in this photo, and in the area where the photo was taken, I’ve noted several such sequences that are significantly longer! In this location, the courses are up to 6–7 metres long, and there are benches to sit on, so I can imagine a game for bored children: who can find the longest sequence of blocks that are all the same size?

It seems to me that the easiest way to decide which size of block to lay next is to make alternate courses identical, as in this example of a narrow pavement:

The next photo shows something similar, although in this case, the relevant section is part of a larger area of paving with courses running in different directions. The thing to note here is that the mathematical relationship between the block sizes is 3M=L+2S. Also, while there is only one way to arrange three contiguous blocks that are the same size, an L and two S blocks can be arranged in three different ways, none of which result in a crossroads. However, none of the alternatives has been used here.

Note too the single narrow course near the bottom of the photo. This feature can be seen in a number of locations, although where it does occur, the number of standard-width courses between each narrow course is arbitrary.

The next photo shows an added complication. With a narrow section of parallel courses like this, it’s probably going to be necessary to cut blocks to fit. The blocks that have been cut are marked:

I had begun to think that every block-paved area consisted of three sizes of block, but I eventually discovered a location where four sizes have been used:

In this photo, there are alternating wider and narrower courses. Four blocks have been used for the wider course, indicated by the numbers 1–4 in ascending size order. Only three blocks (1–3) have been used for the narrower course. Notice that while the smallest and largest blocks in each course are the same length, the intermediate lengths are different.

Unbelievably, I found a location where only two sizes have been used, although as usual I’m defining size as the dimension parallel to the orientation of the course. This is another example of alternating narrower/wider courses, but the lengths are the same in both courses:

I mentioned above the requirement that there should be no ‘crossroads’ in the gaps between blocks, but of course I did find an example:

You might think that I’m being picky here, because there is a slight offset, but I regard this as a mistake anyway, because the next two junctions to the right have an only slightly larger offset.

However, I do not consider the next example a mistake. I think it was deliberate:

Look again! The crossroads here is right in the middle of the pavement and would thus be noticeable by anyone looking down as they passed. It stands out. At least, it does for me. And it could have been avoided merely by exchanging the two blocks marked X!

And there’s a lot more to block paving than you think.

update
Sod’s Law in action! When I posted this analysis yesterday, I did so because I thought that I’d covered all the significant features of block paving. However, this morning I was walking through a yard that I rarely pass through, because there is a parallel yard a few metres to the north with an excellent example of English bond brickwork. I took the following photograph:

This is the only example I’ve found of a repeating sequence of blocks—in this case SSSSL—in each course. And there are only two block sizes. There is also a narrow course following each run of five wide courses.

I took the photo because the repeating sequence of four S blocks seemed obvious, but when I examined the photo, I noticed that there is a run of five S blocks in the foreground. And it doesn’t seem necessary.

above water gardens

Over the Easter weekend, it has been necessary for me to attend the Jockey Club clinic in Sheung Shui (‘above (the) water’) for treatment because the Fanling clinic has been closed. Naturally, we don’t simply just travel there and back. we’re on unfamiliar turf, so we want to take a look around.

On the first day, a short distance from the clinic, in the angle between Jockey Club Road and Lung Sum Avenue, we discovered the prosaically named Sheung Shui Garden No. 1. It was still early in the morning, and a mass dance-a-thon was taking place under a huge permanent canopy:

Some of the dancers appeared to be following a set series of moves, while others just did whatever they felt like. This kind of thing is a common sight in Hong Kong’s parks.

Once we’d walked past the dancers, it didn’t take me long to notice the way this park was paved:

 The pink pavers are granite, while the bluish grey ones are probably a type of slate. The other points of interest in this photo are the gazebo—there are several in this park—and the man asleep on the right. He is resting on a small table between two chairs, and the tabletop has a Chinese chess board engraved on it, which reminds me to advise that you should never play Chinese chess in public, because you will be surrounded by kibbitzers suggesting your next move. And almost all these suggestions will be incorrect!

This is a closer look at the pattern in the foreground of the previous photo:

These patterns occur at regular intervals throughout the park. The whitish central points of the star are another type of granite, while the dark rock appears to be dolerite, which is the shallow intrusive equivalent of the volcanic rock basalt.

This photo shows the reason for our initial foray into the park:

In case you can’t distinguish the male/female icons, just remember that it’s ‘pink for girls, blue for boys’!

On the second day, Paula planned to travel into Kowloon to see her father, so I thought I’d look for an alternative route home. Having noted that the park I’ve just described is ‘No. 1’, I assumed that there must be others. It didn’t take long for me to discover No. 3:

This park is completely out of sight from any road, and it is surrounded by commercial buildings:

The covered walkway in the previous photo has some interesting paving:

There are 1×1, 1×2 and 2×2 blocks, but there would need to be an extra 1×1 block, or one fewer 1×1 block and one more 1×2 block, for this arrangement to cover a larger rectangle completely.

The colonnade in the photo above is the central focus of this garden, although there is a second colonnade to one side:

The pattern that I illustrated above in connection with No. 1 is repeated here, but with a subtle difference:

Notice the eight red diamonds surrounding the central pattern. This photo also shows an exit from / entrance to the park. They are all like this, presumably to deter cyclists!

Sheung Shui Garden No. 2 is located across a road from the exit shown in the previous photograph. Unlike Nos. 1 and 3, it is a narrow strip juxtaposed between roads and a multi-storey car park, but it still has some interesting features. For example, this is the repeating pattern here:

This narrow garden bends through 90 degrees halfway along its length, and there are two gazebos, one on each leg:

I sat on the seat seen in the second photo for quite a long time, because I wanted to take the next photo with nobody in it:

From a distance, the circular colonnade on the corner appears to be utterly nondescript, but it is the location for by far the most impressive paving in any of these gardens:

There are in fact twelve concentric rings of granite blocks, and if you stop to think about it, there must be an equal number of blocks in each ring. The blocks must be wider the further you are from the centre for this to work, and because there are no obvious gaps between the blocks, whoever laid these blocks must have performed some mathematical calculations beforehand. Unfortunately, the central pattern has been damaged, possibly by someone dropping a heavy weight on it.

Although I cannot say definitively that there are only three gardens in this collection, I am assuming that this is the case. However, I will check out the area again, just in case I’ve missed something.

a chinese board game

I grew up playing board games, starting with dice-based games such as ludo and snakes and ladders, and later, Monopoly. However, once I’d discovered strategy games such as chess and draughts (checkers), I stopped playing games where chance is a significant factor in the outcome. Even a game like backgammon, which does require a certain level of skill, is unsatisfactory because the outcome is ultimately determined by the roll of the dice.

By the time I came to Hong Kong in 1974, I’d become familiar with Chinese games such as wei chi and Chinese chess, but soon after my arrival a Chinese colleague showed me an apparently simple board game that I’d not encountered previously. It isn’t listed in Board & Table Games from Many Civilizations, a book that I’d bought several years earlier, and I’ve been unable to find any references to it on the internet, although I did find an illustration of the board with the pieces in their starting positions. However, this game was attributed to Korea, and the object of the game was different. Mind you, this kind of thing is not that unusual; after all, draughts is played on a chess board.

I mention this because I was browsing through some of my old notebooks a few days ago when I came across a detailed analysis of the game that I’d been shown in 1974, written at the time. The following photograph shows the first page of that analysis:

Like wei chi and Chinese chess, play takes place on the intersection of the lines rather than on the squares. The starting position is shown in the following diagram:

The rules are simple:

• Black moves first.
• Players take turns to move any one piece to the next intersection along any of the lines that emanate from that piece’s current location. Obviously, diagonal moves are not possible.
• A player loses if they have no legal moves available (‘blockade’).
• A player loses if they have only one piece left.

The last rule implies that there is a method for capturing one’s opponent’s pieces, and this procedure is explained with the aid of the following diagram:

In this scenario, if it is black to move, they can capture the white piece on C1 by moving B2–C2 or the white piece on D2 by moving C3–C2. However, if it is white to move, they can capture the black piece on B2 by moving C1–C2 or the black piece on C3 by moving D2–C2. Notice that in each case, capture is effected by lining up two of your pieces with one of your opponents. The fourth intersection in the line must be empty, and that blank point must be on an outside line. A player can move a single piece into alignment with two of their opponent’s without penalty.

Referring to the above diagram, there are four possible first moves for black (A2–B2, A3–B3, B1–B2, B4–B3). The first two are topologically identical, as are the last two, and playing either of the first two would result in the immediate loss of a piece (white plays C1–C2 in the first case, C4–C3 in the second). Effectively, therefore, black has only one opening move. However, the game quickly becomes more complex. I don’t plan to attempt a detailed analysis, although I believe that black should win with correct play by virtue of having first move.

If you’re interested in trying this game, just mark out the grid on an A4 sheet of paper—use a ruler if you must, although it shouldn’t be necessary. Almost anything of uniform appearance will suffice for use as pieces—we used bottle tops in the old days. And if anyone can tell me what this game is called, I’d appreciate their letting me know. My Chinese colleague called it simply ‘chess’, but I soon discovered that many Chinese call all board games ‘chess’.

nuclear chicken

If you’ve been watching BBC News, then you will know that the year of the monkey came to an end at midnight on Friday, to be superseded by the year of the rooster. Except that this is ‘fake news’ (or an ‘alternative fact’ if you prefer). As happened two years ago with the year of the sheep (or year of the goat—the Chinese language does not distinguish), which was frequently announced at the time as ‘the year of the ram’, the commentators have fallen into the trap of seeing the animals of the Chinese zodiac in gendered terms, which is incorrect.

You may be wondering about the title of this post. Chinese astrologers will tell you that following the generally lighthearted mischief that characterizes years of the monkey, the world is heading into darker times. Although most people will be aware that there are twelve animals in the Chinese zodiac, not so many will know that the astrological cycle actually takes sixty years to complete, because overlapping the cycle of the animals is a cycle of the five elements. And this year is a fire year. There will be turbulence. There will be turmoil. And there will be trouble.

I’m not basing this assessment on astrology, which is complete hogwash, but on the new geopolitical climate following accession to the throne of Donald J. Trump. And nothing is more pressing in that sphere than the possible use of nuclear weapons. It is not a coincidence that the Bulletin of Atomic Scientists advanced its Doomsday Clock by two minutes last Thursday, and it is now at the latest it has been set since the end of the Cold War (three minutes to midnight). Trump has already stated that he will cancel the nuclear agreement with Iran signed by his predecessor, which can only destabilize even further the most volatile region of the world by encouraging Iran to return to its nuclear program (and Israel to attempt to stop it). And Stuxnet probably won’t work next time.

Even worse, he has already tweeted, in response to news that North Korea is close to having a missile capable of delivering a nuclear warhead to cities on the west coast of the USA, that “It won’t happen!” Which means that he is, in effect, playing chicken with a man who is even more deranged than he is. And the only workable strategy in a game of chicken is to know, in advance, at what point you will always pull out. I don’t think that Trump understands this principle, which is a serious worry given that the only certain way to stop the fat man is a pre-emptive nuclear strike.

Anyway, whatever the future, the new Chinese year always comes in with a bang in my village:

There are several different types of lion dance, and we were treated to one I’d never seen before today. In this first photograph, the lion is in the process of leaping onto the first platform, a feat that requires considerable leg strength from both performers. My friend Tom Li, who was the front end of a lion in his youth, used to tell me that I’d have made a good rear end, but this looks too much like hard work to me.

In the next photograph, the lion has just landed on the double bench between the two platforms:

Performing on the double bench requires some intricate footwork. This is our local lion dance troupe, and I notice that they have a new bass drum this year:

At the end of its performance, the lion is fed lettuce (Cantonese: sang choi); sang choi sounds like a phrase meaning ‘grow prosperity’, which is why the spectators try to catch pieces of the lettuce when the lion spits it out again:

A second lion now appeared on the scene, and it looked as if it was challenging the first lion. I didn’t understand the movements in this performance, but here are four photos that give some idea of what was going on:

Kung hei fat choi.

hughie’s game

When I was a pupil at my local grammar school between 1957 and 1964, I did my best to dodge playing rugby, for reasons that I described in All Must Have Prizes. I was largely successful, and in four of those years I didn’t even set foot on a rugby pitch. However, I do recall one occasion, when I was in the sixth form, when my doctor’s note kept me off the rugby pitch, but it didn’t stop the games master, Brian McVey, sending me to walk around the senior cross-country course in the company of a friend, Hughie Taylor, who also had a doctor’s note.

The cross-country course was 7–8km long, but within 500 metres it passed under the main railway line, which was on an embankment, so even if Mr McVey had been keeping an eye on us with binoculars, he wouldn’t have been able to see what we were up to once we’d passed beyond this point. Naturally, we cut across the fields on the far side of the line to rejoin the course, thus cutting out 6–7km of needless walking.

We then had to pass some time idling about to avoid raising suspicions by returning to school too early, so we played a little game. After more than fifty years, I cannot recall the fine details of the game, but I do remember the general principles. One of us would start by saying “McVey is a …”. The other would respond by saying “McVey is a … and a …”. We probably used fairly offensive terms to describe our nemesis, but what precisely these were I no longer have any idea. However, we also used quite a few made-up words, and I can actually remember some of these, so I will use them to illustrate how the game was played:

Hughie: McVey is a peroot.
Me: McVey is a peroot and a prannock.
Hughie: McVey is a peroot and a prannock and a maroot.
Me: McVey is a peroot and a prannock and a maroot and a ….

And so on. The loser of the game was the first person to misremember the sequence as it grew longer, although who actually won this particular game I cannot now recall. In fact, we probably played the game several times anyway.

If you’ve read Memory Games and Memory Games #2, you will know that my previous suggested tests of memory are solo games, a kind of mental solitaire, so I thought that a competitive game was needed to provide some balance, and Hughie’s game could well provide some amusement in a social situation. It was originally a two-player game, and this is probably the optimum number, but there is no reason why more players couldn’t be involved, especially if the game is fuelled by alcohol. Smoking cannabis before a game probably isn’t a good idea.

Instead of using nouns to describe the object of derision, in this updated version of the game, I propose to use adjectives. Having watched from afar, with increasing dismay, the inexorable rise of an utter mountebank towards the US presidency, I have absolutely no hesitation in using this charlatan as an example of how the game might be played by four people:

1: Donald Trump is vain.
2: Donald Trump is vain and arrogant.
3: Donald Trump is vain and arrogant and bigoted.
4: Donald Trump is vain and arrogant and bigoted and rude.
1: Donald Trump is vain and arrogant and bigoted and rude and pompous.
2: Donald Trump is vain and arrogant and bigoted and rude and pompous and bombastic.
3: Donald Trump is vain and arrogant and bigoted and rude and pompous and bombastic and obnoxious.
4: Donald Trump is vain and arrogant and bigoted and rude and pompous and bombastic and obnoxious and crass.
1: Donald Trump is vain and arrogant and bigoted and rude and pompous and bombastic and obnoxious and crass and ignorant.
2: Donald Trump is vain and arrogant and bigoted and rude and pompous and bombastic and obnoxious and crass and ignorant and boorish.

I could add a lot more words to this sequence, but this should be sufficient to illustrate how a game might pan out. If you try this game, you could incorporate an additional rule that bans the use of inaccurate descriptors, so that someone who said “Donald Trump is compassionate” can be challenged by his opponents. If the challenge is ruled to be valid, then that player is eliminated. In a multi-player game, the last to be eliminated becomes the winner.

I still run into Hughie (not literally) when I’m in the UK, because he’s often out walking his dog when I cycle through the village where he lives. I always stop for a chat, but I don’t think we’ve ever reminisced about that day we were both able to skive off playing rugby but were ‘punished’ for doing so. I shall have to remind him the next time I see him.

memory games #2

In Memory Games, posted earlier this month, I suggested a way to alleviate the tedium of commuting to work in trains so crowded that it is impossible to use a smartphone. Here’s another. This time, the objective is to produce the longest possible ‘story’ using only three-letter words. As the name implies, you have to commit the expanding story to memory.

Unlike in the earlier game, you are allowed to use words more than once. A dictionary is likely to be of little use, even if you could access one. This is what I came up with before I decided to stop and write it down:

His wig did not fit, was far too big and now hid one eye, but the old man did see his son hit the fat dog and run off via the red tin hut and the big oak. Why did the boy hit the dog? For fun, you may say, but yes his act was bad.

Yet the wee boy was not the one who hit the cat. The rat was the one who hit the cat, who ate the hot eel pie. The dog did not eat the pie, and the rat got the bag out for the cat.

memory games

I’m fortunate that I’ve only ever had to commute to work for two short periods: from Highbury in north London, where I lived at the time, between 1979 and 1981; and from Sai Kung in the eastern New Territories to Aberdeen on the south side of Hong Kong Island, between 1985 and 1987. Nowadays, most people fill these utterly empty minutes by fiddling with a tablet or smartphone, but in those days, there were no Angry Birds, no Candy Crush, to while away the time, and although I no longer work, let alone commute, I still employ the same strategy to pass the otherwise empty minutes whenever I travel on public transport.

The idea is to construct the longest possible sentence using only words that begin with the same letter. No word may be used more than once, a dictionary or thesaurus may not be consulted, and the sentence cannot be written down (hence ‘memory games’) as it grows longer. I can no longer remember the exact sentence I came up with at the time, although I do recall that it involved ‘alarmingly aggressive alligator armies’, but I would say that the longest possible sentence uses words that begin with the letter A—there are several conjunctions and prepositions beginning with this letter—so for the purposes of this post I’m going to use the letter O.

The sentence will grow in stages, but the obvious place to start is with the main verb and its subject. The existence of the conjunction or means that the subject can be a list, and the verb should be transitive to allow for an object. This is my proposed starting point:

opticians, optimists or onlookers offer opinions…

Obviously, each of the four nouns needs an adjective to complement it:

obese opticians, obsessive optimists or ordinary onlookers offer orthodox opinions…

This can be expanded by adding adverbs and a preposition:

only outrageously obese opticians, obstinately obsessive optimists or otherwise ordinary onlookers offer orthodox opinions on…

I won’t go through all the iterations, but this is what I eventually came up with:

Originally, only outrageously obese opticians, obstinately obsessive optimists or otherwise ordinary onlookers offered orthodox opinions on officially organized operations, occasionally opposing oppressive orders of obnoxious oafs.

Remember, you shouldn’t be writing anything down or consulting a dictionary if you try this game. To do either of these is to render the exercise pointless. You may think that it’s already pointless, but when you’re crammed so closely to other people that it’s impossible to do anything other than think, what are you going to do?