“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.

Brick pavements and tatami mats are traditionally laid out so that no four meet at a single point to form a ┼ shape. Only a few ┼-free patterns can be made using $1\times 1$ and $1\times 2$ tiles, but the addition of $2\times 2$ tiles provides a lot more creative flexibility.

When I discussed tatami tilings with my relative Oliver Linton, he suggested applying similar rules to other brick sizes to make beautiful tiling patterns. The tatami condition alone does not provide enough of a constraint to mathematically analyze tilings with arbitrary shapes and sizes, but it is a good starting point when looking for interesting patterns.

With the addition of $2\times 2$ square tiles, it’s possible to construct rectangular blocks that fit together to tesselate the plane while preserving the four-corner rule.

This opens the door to self-similar tilings, which I’m very interested in! The goal is to use $1 \times 1$, $1 \times 2$, and $2 \times 2$ tiles to construct $n\times n$, $n \times 2n$, and $2n \times 2n$ blocks which can be put together in the exact same way to make increasingly intricate $n^k \times n^k$ tilings that maintain the tatami condition.

The simplest non-trivial example I could find involves a set of $5\times 5$, $10 \times 10$, and two $5\times 10$ rectangular tilings.

Starting with any of these four layouts, we can replace each of the $1\times 1$, $2\times 2$, and $1\times 2$ bricks with a corresponding $5\times 5$, $10\times 10$, or $5\times 10$ rectangular tiling in the correct orientation. (This will produce a few four-corner intersections, but we can fix these by merging adjacent pairs of $1\times 2$ bricks.)

Repeatedly performing this operation gives an infinite sequence of tilings, but can we say they converge to anything? A tiling $T$ can be identified with its outline $\partial T$ (i.e. the set of points on boundaries between two or more tiles). Note that if a point is in $\partial T_i$, then it will be in every subsequent $\partial T_j$ unless it is one of the few bricks merged in $\partial T_{i+1}$. So we might sensibly define the limiting object of the tiling sequence $T_i$ as the union

This self-similar dense path-connected set satisfies the topological equivalent of the “four corners rule” — a pretty interesting list of mathematical properties!

The same strategy could be applied to other sets of $(2n+1)\times (2n+1)$, $(4n+2)\times (4n+2)$, and $(2n+1)\times (4n+2)$ tiles with similar boundaries. What’s the prettiest brickwork fractal you can find?

Although I’ve left academia, Bojan Mohar and I have published a new paper in the proceedings of SODA exploring the “perimeter” measure that plays a key role in my doctoral research. It is mostly based on Chapter 4 of my PhD thesis.

Mount Baker was named by George Vancouver after his third lieutenant, who was the first on his ship to see it.

Although the name is better than a few others in the Pacific Northwest, being the first person to see a giant mountain isn’t a particularly notable claim to fame. Especially when you have to ignore not only tens of thousands of people who lived there but also the Spaniards who had gotten there the year prior.

The mountain appears as la gran montaña del Carmelo on a map drawn by Gonzalo Lopez de Haro, first pilot on Manuel Quimper’s six-week expedition to the Juan de Fuca strait. The Spanish name is apparently a reference to a religious order whose white cloaks resembled the snow-capped peak. Who knows, if the Nootka Crisis had been resolved differently, it’s entirely possible that Europeans would have ended up calling it Monte Carmelo.

Instead, the name Mount Baker stuck after Vancouver described it in his published memoir.

About this time a very high conspicuous craggy mountain … presented itself, towering above the clouds: as low down as they allowed it to be visible it was covered with snow; and south of it, was a long ridge of very rugged snowy mountains, much less elevated, which seemed to stretch to a considerable distance … the high distant land formed, as already observed, like detached islands, amongst which the lofty mountain, discovered in the afternoon by the third lieutenant, and in compliment to him called by me Mount Baker, rose a very conspicuous object … apparently at a very remote distance.

George Vancouver

Vancouver’s diary mentions encounters with different indigenous groups of the area, some friendly and some indifferent,¹ but he never stuck around in the same place long enough to learn their names or pick up their languages.² If he had, he might have recognized Mount Baker by the name kwelshan, the term used by the Lhaq’temish (Lummi) people around Bellingham and the San Juan Islands, or swáʔləx̣, reportedly used by the nəxʷsƛ̕áy̕əm̕ (Klallam) people on the Olympic Peninsula.

The mountain itself is surrounded by the traditional lands of the Nooksack and Upper Skagit peoples. The Nooksack use kwelshán for the high open slopes of the mountain and kweq’ smánit for the glacier-covered summit.

😖 + 👻 = 🤧

I have succesfully defended my PhD thesis! It’s “packed” with results on graph immersions with parity restrictions, and “covers” odd edge-connectivity, totally odd clique immersions, and a new submodular measure that’s intimately connected with both.

I am grateful to NSERC for funding my degree with a Alexander Graham Bell Canada Graduate Scholarship, and to my supervisor Bojan Mohar.

US CONSTITUTION: No person except a natural born citizen shall be eligible for president.

MACDUFF: 😢

Urban planning oddities in Pokémon:

• Kanto Route 17: because a steep hill is the perfect place for a Cycling Road.
• Cave lighting and sliding block puzzles are an essential part of a transportation network.
• Hospital-adjacent land is considered prime real estate.
• Thirsty guards are a major source of traffic bottlenecks.
• A nonprofit society needing funding convinced Lavender Town council to rezone their memorial tower for radio use.

“May you live in interesting times” is typically claimed to be a Chinese expression, but it actually originated with the British. Joseph Chamberlain — Neville’s dad — used the phrase “interesting times” frequently in speeches:

I think that you will all agree that we are living in most interesting times. I never remember myself a time in which our history was so full, in which day by day brought us new objects of interest, and, let me say also, new objects for anxiety.

Joseph Chamberlain

Joseph’s other son Austen was the first to claim it originated as a Chinese saying. Quote Investigator theorizes that Austen, in conversation with his diplomat colleagues, learned about a Chinese proverb that expresses apprehension about living in what his father would call “interesting times” and assumed that was the source of Joseph’s phrase. But the wording of the real proverb is entirely different:

寧為太平犬，莫作亂離人

Better to be a dog in days of peace, than a human in times of chaos.

Feng Menglong