Here in Vancouver, the parking bylaw requires one 14m² parking space for every 20m² of supermarket floor space. Adding the necessary lanes to allow cars to get in and out, this law practically guarantees that grocery stores will be more parking lot than store.

In the late ’60s, the government sought to adopt O Canada as the national anthem. The music and original French lyrics had passed into the public domain, but the English version was still under copyright. The government settled the rights for a dollar.

Ironically, the copyright to the English lyrics would have expired anyways by the time the National Anthem Act was finally passed in 1980.

Crayola has a trademark on the smell of its crayons. The odour is described in the Canadian trademarks registry as:

A unique scent of a pungent, aldehydic fragrance combined with the faint scent of a hydrocarbon wax and an earthy clay.

Trademark 1986544

In Canada, a trademark can be anything that is used by by a seller to distinguish their goods or services from those of others, including

a word, a personal name, a design, a letter, a numeral, a colour, a figurative element, a three-dimensional shape, a hologram, a moving image, a mode of packaging goods, a sound, a scent, a taste, a texture and the positioning of a sign.

Trademarks Act

Unfortunately, I have not been able to find records of anyone litigating over a scent trademark.

You can get a really good approximation of a sinusoidal curve from twelve equally-spaced line segments of slope 1/12, 2/12, 3/12, 3/12, 2/12, 1/12, -1/12, -2/12, -3/12, -3/12, -2/12, and -1/12, respectively.

This approximation, known as the rule of twelfths, rounds $\sqrt{3} \approx 5/3$ but otherwise uses exact values along the curve.

I learned about the rule of twelfths from a kayaking instructor and guide, who used it to estimate the tides. In locations and seasons with a semidiurnal tide pattern, the period of the tide is roughly 12 hours, and the rule of twelfths tells you what the water will be doing in each hour.

For example, if you know that the difference between low and high tide is 3 feet, then you can quickly estimate that it the tide will rise by about 3 inches in the first hour, 6 inches in the second, 9 inches in the third and fourth, 6 inches in the fifth hour, and 3 inches in the last hour before high tide.

Recently I’ve been spending my spare time doing two things: solving cryptic crosswords and playing Pokémon Legends: Arceus. The next logical step, then, is to try my hand at compiling my own cryptic crossword themed after the game!

If you haven’t tried one before, a cryptic crossword is a crossword puzzle in which each clue is itself a word puzzle. A cryptic clue is usually misleading if taken at face value, but conceals a definition and a secondary indication (usually wordplay) of the correct answer. For example:

Cryptic clues can be even more complicated and include anagrams, homophones, obscure abbreviations, and all sorts of other devices. Or they can be as simple as a bad pun.

You’ll need to be a Pokémon fan to appreciate all of the surface readings in this puzzle, but don’t worry if you aren’t — none of the clues require any specific knowledge of Pokémon to solve. Once you’re done, you can scroll to the bottom to check the solutions and see how you did. Enjoy!

Confused? Impatient? In the spirit of Fifteensquared, below is the answer key to the puzzle!

As a reminder, a cryptic clue generally consists of a dictionary definition and a secondary indication (usually wordplay) of the correct answer. In the following solutions, I’ve highlighted the dictionary definition hidden in each clue in bold.

I’ll then explain how the rest of the clue leads to the right answer, as well as how each clue relates to the puzzle theme.

Across

Down

Hidden message

Shawn Mendes’ song “Lost in Japan” has had me geographically confused since I first heard it covered by Scary Pockets. If you haven’t listened to the lyrics, the song is about a person who is thinking about their crush and the possibility of taking a last-minute flight to Japan to see them.

The question I can’t get off my mind is: where is the song supposed to be taking place?

The chorus goes:

Do you got plans tonight?
I’m a couple hundred miles from Japan, and I
I was thinking I could fly to your hotel tonight
‘Cause I can’t get you off my mind

from which we can infer that

1. the crush is somewhere in Japan
2. the singer is outside of Japan, and
3. both people are close enough to an international airport for one to entertain the idea of flying to the other’s hotel.

The most likely candidates for airports within “a couple hundred miles from Japan” are in South Korea.

When I first considered the geography of the song, I was satisfied with that answer: couple of hundred miles, South Korea, that sounds about right. But then I noticed the opening lyrics:

All it’d take is one flight
We’d be in the same time zone

That would seem to rule out South Korea, which is in the same time zone as Japan. With the closest locations out of the picture, we have to stretch our interpretation of the song. Here are the possibilities I can see.

The singer could be in Shanghai

Shanghai is 500 miles (in different directions) from Okinawa and Kyūshū Islands. Shanghai Pudong International Airport was the eighth-busiest airport in the world when “Lost in Japan” was released, and has plenty of routes to both Naha and Nagasaki.

This is probably the most plausible answer if we make the thematically appropriate assumption that the lovestruck protagonist is downplaying the distance in order to convince themself that taking a last-minute flight is a good idea.

The singer could be in Vladivostok

Vladivostok is also within 500 miles of New Chitose Airport in Sapporo. This possibility is pretty funny, but unfortunately it’s ruled out by logistics. The only regular flight I could find between the two was only opened by Ural Airlines after “Lost in Japan” was released, and did not last long before world events shut it down permanently.

The singer could be in Taipei

There’s only one location that geographically fits the lyrics exactly: our protagonist is in Taiwan, and their crush is less than 200 miles away in Ishigaki on Japan’s southwesternmost inhabited archipelago.

With a creative enough interpretation, this is also consistent with the second verse:

Do I gotta convince you
That you shouldn’t fall asleep?
It’ll only be a couple hours
And I’m about to leave.

The singer is presumably in a hurry to leave because they have just checked the timetable for the only direct flight from Taipei to Ishigaki, which runs on Wednesdays and Saturdays during the tourist season. The flight is short enough that “a couple hours” is realistic if customs is quick.

The line about falling asleep is, of course, a request for the romantic interest to skip their usual afternoon nap, since China Airlines flight 124 arrives at 11:35am.

The time it takes to properly roast a whole turkey is proportional to its weight to the ⅔ power. My old mathematical modelling textbook specifically recommends 45 minutes per lb^2/3 when cooked at 350℉.

For a spherical turkey of uniform thermal conductivity $\alpha$ and density $\rho$, a precise formula has been derived:

where the oven is set at $T_h$ and the center of the turkey needs to reach a temperature of $T_f$ from $T_0$.

The more general ⅔ power law does not depend on unrealistic assumptions about the turkey’s shape or thermodynamic properties; it can be derived from pure dimensional analysis and applied to turkey-shaped meat-based objects by fitting a curve to specific cook times used by chefs.

Since the ’50s, Alberta has engaged in a deliberate effort to prevent rats from entering the province. Fortunately, rats can’t survive in the wild in Alberta, so they have pest inspectors regularly check every premise within a 29 x 600 km control zone from Montana to Cold Lake. Pet rats are illegal.

The rat-free status of Alberta led to a Wikipedia edit war over whether the province should appear on a map of the brown rat’s habitat. At some point it was decided to remove the map entirely from the English-language entry for Rattus norvegicus, but its presence on other Wikimedia projects means the edit war still rages on to this day.

The official title of Chief Mouser to the Cabinet Office is held by a cat in charge of keeping mice under control at the UK Prime Minister’s office at 10 Downing Street. The current incumbent is Larry.

It was once commonplace to employ cats as mousers, so Larry is not unique; the UK Post Office employed a cat named Tibs the Great for many years and Canada’s Parliament had its own cat colony.

Muji apparently doesn’t have a bug bounty program.

Vessel Finder is a map showing the current location of container ships, cruise ships, fishing boats, and other nautical vessels. It aggregates data from automatic identification systems which all sufficiently large boats are required to be fitted with.

This website was all over the news this week because of the ship stuck in the Suez Canal.

The length distribution of tweets has shifted in response to raised character limits, but it’s still the case that a disproportionate number of tweets use all the characters they’re given.

A sample of tweets gathered in 2019 still exhibit a telltale spike approaching the character limit, but it is smaller than the tweet distribution from a decade earlier. The peak of the curve has also shifted leftwards, to 15 characters, due to a separate change in 2016 that excluded media attachments and certain at-mentions from the character count.

The most interesting feature of the above graph is unfortunately an artifact of the dataset — the massive spike at 105 characters can be blamed on a spambot network broadcasting identical copies of the same tweet when the dataset was collected.

“General particulars” is an excellent phrase that deserves to catch on more widely than its current context of legally-mandated notices on boats.

(Boats are required by international law to have a wheelhouse poster listing their “general particulars”, i.e., a list of statistics, properties, and other bits of information necessary to get a basic view of the vessel.)

The head of a sunflower is actually hundreds of smaller flowers working together to attract pollinators. Each large yellow petal is its own individual flower, and the bits in the middle are tiny five-pointed flowers if you look closely.

Sunflowers share this property with the rest of the family Asteraceae, including daisies, dahlias, dandilions, and dozens of “damned yellow composites”. The heads of these plants can be called inflorescences, or flower clusters.

Pokémon Gold and Silver’s roaming legendary beasts move randomly from route to route instead of sticking to a fixed habitat. By analyzing their behaviour using the math of random walks on graphs, I can finally answer a question that’s bugged me since childhood: what’s the best strategy to find a roaming Pokémon as quickly as possible?

Catching a roaming Pokémon is a graph pursuit game, but in practice the optimal strategy doesn’t involve a chase at all. Raikou and the other roaming Pokémon move every time the player crosses the boundary from one location to another, regardless of how long that takes. So if we repeatedly cross the boundary by taking one step forward and one step back, Raikou will effortlessly speed across the map.

The easiest strategy, then, is to choose a centrally-located location and hop back and forth until Raikou comes to us. The question is what location gives the best results.

Vertices of maximum degree

When left to its own devices, a random walk in a graph G returns to a vertex v every

steps. This suggests that the best place to find Raikou is a vertex of maximum degree on the graph corresponding to the Johto map.

This puts Johto Route 31 as the top candidate, since it’s the only route adjacent to five other routes (Routes 30, 32, 36, 45, and 46) on the roaming Pokémon’s trajectory.

Vertices with minimum average effective resistance

Of course, we don’t intend to leave Raikou to its own devices—we’re going to try to catch it whenever it’s on our route! If it gets away, it will flee to a random location that can be anywhere on the map, regardless of whether it is adjacent or not. This wrinkle means we’re not exactly trying to find the vertex with the fastest return time; we’re really trying to minimize

where $T(u, v)$ is the expected time for a random walk starting at $u$ to first reach our vertex $v$.

How do we compute this value? According to Tetali, we replace all of the edges with 1-ohm resistors and measure the effective resistances $R_{xy}$ between each pair of nodes $x, y$ in the corresponding electrical network. Then

It seems very appropriate to use the math of electrical networks to catch the electric-type Raikou! Unfortunately, there’s no references to effective resistance or Tetali’s formula in its Pokédex entry.

Effective resistance can be computed by hand using Kirchoff’s and Ohm’s Laws, but it’s much easier to plug it into SageMath, which uses a nifty formula based on the Laplacian matrix of the graph.¹

Route 31 comes out on top again by this measure: if Raikou starts from a random location, it will come to this route sooner on average than any other single location.

Vertex pairs with minimum average effective resistance

But this still isn’t the final answer. The above calculations assume we’re hopping between a route (where we can catch Raikou) and a town (where we can’t).1 What if we go to a boundary where either side gives us a chance for an encounter?

There are only four pairs of routes in Johto where this is possible. The expected capture time when straddling one of these special boundaries can be computed using the same kinds of calculations. All four route pairs yield an expected capture time faster than relying on any individual route — enough to dethrone Route 31!

Source code

G = Graph({ 29: [30, 46], 30: [31], 31: [32, 36, 45, 46], 32: [33, 36], 33: [34], 34: [35], 35: [36], 36: [37], 37: [38, 42], 38: [39, 42], 42: [43, 44], 43: [44], 44: [45], 45: [46] })

R_matrix = G.effective_resistance_matrix()

def R(u, v): return R_matrix[G.vertices().index(u)][G.vertices().index(v)]

def hitting*time(u, v): return 1/2 * sum(G.degree(w) \_ (R(u,v) + R(v,w) - R(u,w)) for w in G.vertices())

def avg_hitting_time(v): return mean([hitting_time(u, v) for u in G.vertices()])

{ v: avg_hitting_time(v) for v in G.vertices() }

# account for parity

H = G.tensor_product(graphs.CompleteGraph(2)) R_matrix = H.effective_resistance_matrix()

def R(u, v): return R_matrix[H.vertices().index(u)][H.vertices().index(v)]

def hitting*time(u, v): return 1/2 * sum(H.degree(w) \_ (R(u,v) + R(v,w) - R(u,w)) for w in H.vertices())

def avg_hitting_time(v): return mean([hitting_time((u,0),(v,0)) for u in G.vertices()])

{ v: avg_hitting_time(v) for v in G.vertices() }

# Final

@CachedFunction def H(routes=None): if routes is None: return G.tensor_product(graphs.CompleteGraph(2)) else: x, y = routes graph = H(None).copy() graph.merge_vertices([(x, 0), (y, 1)]) return graph

@CachedFunction def R_matrix(routes): return H(routes).effective_resistance_matrix()

def hitting*time(routes, u, v): H0 = H(routes) R = lambda x,y: R_matrix(routes)[H0.vertices().index(x)][H0.vertices().index(y)] return 1/2 * sum(H0.degree(w) \_ (R(u,v) + R(v,w) - R(u,w)) for w in H0.vertices())

{ (x, y): mean([ hitting_time((x,y), (u,0), (x,0)) for u in G.vertices() ]) for (x, y) in [(30,31), (31,30), (35,36), (36,35), (36,37), (37,36), (45,46), (46,45)] }

Although Raikou will on average arrive at Route 31 faster than any other route, the best place to catch the roaming legendary Pokémon is the boundary between Johto Routes 36 and 37. Hop back and forth between those two routes, and before you know it, you’ll be one step closer to completing your Pokédex!

Not all 26 letters of the alphabet appear on BC license plates. Six are missing — and the reason goes all the way back to 1970, when BC switched from issuing sequential plate numbers to an alphanumeric system.

One [story] is that the stamps used by employees of the MVB for compiling licensing documents in 1960s only had enough space for ten (10) characters.

The other is that when the province upgraded the machinery at the Oakalla Plate Shop in the mid-1950s, it was designed to accommodate a maximum row of ten (10) different dies for each of the six columns that might be used in the license plate’s serial.

Regardless of which, if any of these stories is the correct one, the alphabet was broken into two blocks of ten letters with the first block comprising A, B, C, D, E, F, G, H, J, and K with “I” excluded as it too closely resembled the number one.

Christopher Garrish

The second block of letters used on BC passenger license plates was L, M, N, P, R, S, T, V, W, and X. The letters I and O were obviously too similar to 1 and 0, and Q was apparently also skipped due to its resemblance to zero.¹ I imagine U was excluded instead of X to minimize the number of unusable plate numbers containing rude words. Y and Z missed out just because they’re at the end of the alphabet.

BC has issued several different plate series since the ’70s and the manufacturing process has presumably changed since then, but I believe the letters have remained the same. To this day, I have still never seen an I, O, Q, U, Y, or Z on a BC passenger car.

Keep an eye out for other types of vehicles, though. Motorcycle plates starting with U, Y, and Z were issued in 2012–14, 2017–18, and 2019–20 respectively, and the letters can also be spotted on plates assigned to commercial trucks.