Patterns are the root of math, all else is arithmetic. Unfortunately, most of my early education consisted solely of arithmetic and calculation – that strict application of rules and transformations, imagination not required. That’s not math! And you don’t realize if no one tells you. There’s a certain conspiracy of math professors and teachers of all stripes to re-form elementary math education into a different mold that I hope to give you a taste of here.


Can you cover a chessboard with a set of blocks shaped like a horse move from chess, or equivalently, a four tile L-block from Tetris (if you prefer to think of it that way)?


The answer is yes. Think for a bit and you’ll see that it’s so. Can you cover any similar rectangle of tiles? To answer this, we consider all possible types of these rectangles by splitting them into three categories.

Case 1: Number of board tiles is not a multiple of four

In the world where your rectangle doesn’t have some multiple of four number of tiles, it is impossible to cover it with Ls. Since each L contains four tiles within it, any set of non-overlapping Ls is going to use a multiple of four tiles (e.g. 2 Ls have 8 tiles, 3 Ls have 12 tiles, and so on).

Among the rectangles ruled out by this case are those with two sides of odd length.

Case 2: Number of tiles is a multiple of eight

We’ll split this case into three types.

case 2a: the one by eight rectangle


Not happening. You can’t even fit one L on something that narrow.

case 2b: one side is a multiple of four and the other is a multiple of two

We can lay two Ls together to get a two by four shape.


By laying many of these on the board with the side of length four oriented with the side of the board that is the multiple of four, the board will be covered without holes. With this trick, we can easily cover the chessboard of the original question.

case 2c: one side is a multiple of eight and the other is odd (but greater than one)

This time we’ll make a more complicated shape; I’ll have to use less pretty letters.


Lay enough of these shapes against the side of the board that is a multiple of eight to cover the entire length. The area of the board that remains uncovered will have one side that is a multiple of eight while the other side will have a length that is an odd less three. An odd less three is even and eight is a multiple of four so the still uncovered board matches the boards that we just tiled in (2b). We’ll use the same trick with the two Ls rectangle shape to cover it.

As a side note, there is a philosophy of mathematics that demands proofs of this type. Sometimes, a mathematician demonstrates that a solution exists by proving that it is impossible for a solution not to exist. To a constructivist, this is a cheat and a shortcut and, as such, unacceptable. Quora has some good examples of these unacceptable proofs.

Case 3: Number of tiles is a multiple of four but not of eight

This is the tricky one. It also turns out to be impossible but it’s non-trivial to sort out why. Bear with me as I sketch the reason.

Each L has four tiles so covering the entire rectangle requires an odd number of Ls. Now break each L into two parts, a horizontal set of two tiles and a vertical set of two tiles. It’ll be impossible to cover the whole with an odd number of horizontals and an odd number of verticals, even ignoring the requirement of connecting them into Ls.

Place all your horizontal pieces. In each row, an even number of tiles has now been taken up. If the width of the board is odd, an odd number of verticals is needed per every two rows and since there are a multiple of four rows, we’ll need an even number of verticals to cover the empty space entire (which makes it impossible). If the width of the board is even, then you need an even number of verticals per row so an even number of verticals entirely (again, impossible).

Continue Questioning, Friends

Hopefully, you’re disappointed that we have an answer so quickly. No reason to stop, the natural question is how we can extend. Well, what about hexagonal tiles? There are L-esque patterns that can be laid there. How about non-rectangular boards of square tiles? The following L shaped board can be covered (do you see?).


What about boards with holes? There are endless variations. Now you have an outlet the next time you’re really bored and staring at a tiled floor. See if you can make some rules and work out a pattern. Then change the rules to make it harder.


Crying is the most manly of expression. It is a pure demonstration of the depth of your passion and emotion. Not crying means you don’t have the emotional depth to be hurt. Or that you’ve fully embraced stoicism.

Gender roles are not inherently bad. They’re clusters of traits and clusters are not inherently bad. I happen to object to drawing firm boundaries on loose clusters when enforcing such rules results in (1) individual suffering and (2) the loss of a beautiful diversity of possibilities. Often we find the sublime in random experimentation so I will not support a brutal pruning before the realization of that promise. Evaluate gender traits on their own merits and decide what works for you.

Wang Wei

During my favorite dynasty, Chinese philosophy passed through a period called the Hundred Schools of Thought. What a fantastic time to be a scholar! Granted life for most people was fairly turbulent as the period was marked with weak political control, chaos, and war. Perhaps this is a necessary soil for producing such a rich diversity of thought. Each School of the period was attempting to construct a distinct system. They influenced each other though a cross-pollination of ideas and they influenced each other through conflict and argument. Experimentation was the order of the day. The Schools that survived from the period were strong enough that they still exist twenty-two hundred years later. It’s tragic that the same richness of thought didn’t survive through more peaceful times. I feel that organizations can be poisoned by a surfeit of agreement as surely as they can fall from a lack of internal unity.

Moving into my second favorite dynasty, we find a fantastic poet, Wang Wei. A half-remembered translation relayed in a conversation of his poem “A Song at Weicheng” first brought him to my attention.

“Wait, friend, and share another drink.
Tomorrow you’ll be past the mountains
and there will be no more
another drink, friend.”

It turns out that the original doesn’t literally translate to this in English. A professional translation is below.

“A morning-rain has settled the dust in Weicheng;
Willows are green again in the tavern dooryard….
Wait till we empty one more cup –
West of Yang Gate there’ll be no old friends.”

Translation is a tricky activity. Depending on how tight the rules you impose, many works are simply untranslatable because there’s no way to convey that same experience and meaning that a native speaker would hear. Word choice, rhythm, and references are some of the obvious difficulties to translate. So by necessity, I’ve been fairly comfortable with fairly loose translations – including reinterpretation. Which is mostly to say that I like the original version that I remembered, as wrong as it is to the purist. I would rather live in a  world where one hundred translations can exist and the audience can choose among them than live in a world where the Immortals decide the canonical translation.

If you’re interested in more Wang Wei, I’d recommend “Walking In Mountains In The Rain” (translated by David Young) to give a taste of his style.

Seeing Then Believing

A good diagram is a precious thing. Words are amazing workhorses but reading never stops being that acquired skill a semi-arcane technology. There’s translation effort there. Pictures leap into your head as if they were born there and we have effortless recall. Data visualization is a powerful way of aggregating a information and I have great respect for those who can do it well.

The canonical example for most school kids is the inscribed squares that show the proof of the Pythagorean theorem. Geometry and topology are full of these moments to the point that other branches of mathematics deride them for practicing “proof by picture”. But most people will probably find them impenetrable so let’s look at some of Seth Kadish’s work.

I loved wandering the streets in London. My home town is small and most of the few streets fall into a grid. The jogs in the road are where it dodges around some bit of forest or hill. Very few alleys, very few surprises. London was nothing but narrow winding roads and surprises. There were streets running parallel to each other separate by a single row of shops but you wouldn’t know if you had not glanced down that alley two intersections back. Four right turns may not leave you facing your original direction. I’ll never complete The Knowledge but if I lived there, the urge to map the place would be irresistible.

Consider our expectations of an ancient country’s boundaries. At its smallest level, it would arise naturally from divisions between neighboring tribes and villages. Since these presume a single central point, they would appear as a rough circle with jagged edges for natural features that make easier delimitation marks. Aggregating several of these together will deform the circle we’d expect to still be able to discern some small number of fixed central points. In fact, the cantons of Switzerland form more-or-less these shapes. Thus we expect that for self-determined countries, the graphs in the visualization above would form roughly the same shape as the continent for which they’re drawn. The long outliers are indicators that decisions about a border was not made at the local level. Assuming a continuous habitation of the entire landmass, this implies that people were not given a choice about which country they would join and so were never given the chance to agree to that countries version of a social contract.