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Designer Diary: Of Ice and Isomorphism, or The Story of Expedition: Northwest Passage

Yves Tourigny
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In 1845, Sir John Franklin led a British expedition which undertook to find the Northwest Passage. He had two great state-of-the-art ships — the Terror and the Erebus — 128 men, and provisions for several years. They sailed into Lancaster Sound, then promptly disappeared forever. Understandably worried, numerous British and American expeditions were sent to look for survivors — eventually simply to figure out what happened — and to find the Northwest Passage.


This diary isn't about them. If you're curious about what happened to the Franklin Expedition, this sums it up nicely (**spoiler alert**):

"...and then I said: Send more expeditions!"

Let me instead introduce you to a friend of mine: the isomorphism. It is no exaggeration to say that without his constant companionship, I would not be designing games. He was instrumental to the process of designing Expedition: Northwest Passage, for example, so I'll tell you some pertinent and interesting things about him, and then about the game.

Isomorphism is derived from iso, meaning "same", and morphism, meaning "shape". Two objects are mathematically isomorphic if there is a one-to-one mapping of their elements, or if they are indistinguishable when considering certain features. This is an incredibly powerful idea from mathematics that can be broadly applied to many different aspects of game design. Before I tell you any more about it, we'll have to take a little detour into combinatorics first. It won't take long.

Say you are designing a tile-laying game, which is a commendable idea. For instance, you want tiles of a certain shape with certain features on them. In the case of Expedition: Northwest Passage, I wanted tiles that were rectangular (2:1), and I wanted each of them to have six different areas which could be either water or land. In order to refrain from taxing your imagination too much, I have generously provided a diagram showing how the tiles tile:


How many tiles would be necessary if I wanted to exhaust all possibilities? It seems like a straightforward question: Each of the six areas can be either water or land, therefore for each of the areas there are two possibilities. The Rule of Product tells us tha...

Oh dear. You don't know the Rule of Product? No, I won't tell a soul. It is a basic principle of enumeration. Enumeration can refer to counting the exact number of elements in a set, and to creating an ordered list of those elements. It should be clear by now that this is the task that I faced. The Rule of Product simply states that if there are ''x'' ways of doing something (no, it doesn't matter what) and ''y'' ways of doing another thing, then there are a total of ''x'' times ''y'' ways of doing both things.


To put it in terms you can relate to, if you have two flavors of ice cream, there are two ways of choosing the first scoop and two ways of choosing the second scoop. That makes for four different cones. Yes, sometimes the order DOES matter. No, I don't have any ice cream.

So, to get back to it, the Rule of Product tells us that if there are two ways of filling the first area, two ways of filling the second, and so on until the sixth, there are 2x2x2x2x2x2 (2^6) ways of assigning land and water to the areas on a tile, that is 64 ways. Pretty straightforward.

Ah, but this tells us only about the content of the areas. It tells us nothing about how these areas are connected. Yes, it does matter! In Expedition: Northwest Passage, you see, you are leading an expedition into the maze of the Arctic Archipelago. You are looking for a navigable passage for your ship, so the connection between the areas matters very much. For instance, you would much rather encounter the tile on the extreme right, assuming that the dark areas are water.

Matagot decided to go a different route with the art

This complicates things — unless we simply rule that all water areas will always be connected. Yes, that's much simpler. Now each of the 64 permutations of water and land correspond to one and only one tile. One-to-one mapping? Yes, now we're approaching the isomorphism, and we're moving on to the second type of enumeration: creating an ordered list.


Each tile can be described using a string of zeroes and ones in which "0" is land and "1" is water. If you are familiar at all with binary notation, you know that any number can also be written as a series of zeroes and ones. Each position in the number represents a power of two, just like each position in a number written in our everyday decimal notation represents a power of ten, e.g., "10" in decimal notation is a shorthand for one group of 10 and zero groups of 1, whereas "10" is binary for one group of 2 and zero groups of 1 (2 + 0 = 2). There is a one-to-one mapping between the tiles and the numbers 0-63 (which is 000000 to 111111 in binary).

Hand. Drawn. I am available for commissions.

So what? Quality control. When making the tiles and laying them out for printing, they can be ordered numerically. This makes spotting errors and omissions simpler. Not simple, mind you, but simpler. Is that it? Not at all. If you license your game to a publisher, you may be asked, periodically, to check the work of the artist for errors.


Go ahead, I'll wait. Not easy, is it? You could simply give them a quick look over and write back saying, "Looks good!" but that would be risky. Further complicating things is the fact that these tiles are double-sided, and some of them are identical under rotation or reflection. This means that while tile 1 of the middle row and tile 14 of the top row seem like duplicates, they are in fact part of a set of four tiles which, when rotated or reflected, are identical. They are literally isomorphic, that is, the same shape. They are 101000 ("40", top row tile 12) 010001 ("17", top row tile 14), 010001 ("17", middle row tile 1), 000101 ("5", middle row tile 15). The second "17" is actually "10" (001010) in disguise.


Each of the tiles is isomorphic to one or more other tiles — with very few exceptions, which are identical to themselves under rotation and reflection. I'll let you discover those for yourself. Without a good way to enumerate the tiles and a clear understanding of their features, you might find mistakes where there are none or fail to see the mistakes that are there.

The process of checking and double-checking the tiles occurred on several different occasions, not least of which was when the final proofs were being sent to be printed.

The theme of a game is also an isomorphism, broadly speaking. (Recursively, that would mean that there is an isomorphism between themes and isomorphisms. If that doesn't delight you, then you and I are very different.) A good theme makes an abstract system comprehensible in terms of real-world phenomena.

In Expedition: Northwest Passage, for example, you explore the Arctic Ocean. In reality, you are doing no such thing: You are actually moving a wooden token in the shape of a boat on cardboard tiles with pictures of an ocean on them. If you consider the appropriate features, however, you can see interesting similarities. The ships move on water, and the sleds move on ice, obviously enough. The sled is stored on the ship until it is needed, at which point it is deployed. Nothing groundbreaking here.


Consider the ice. One frustrating and deadly feature of Arctic exploration was that, before global warming seriously curtailed it, the ocean surface was frozen for most of the year. This left a very short season during which navigation was sometimes possible. In the game, the Sun token moves once per round in a graceful arc along the edges of the board, from East to West. As it moves North along the arc, the ice moves along with it, leaving a greater portion of the board ice free. At its zenith, the entire board is navigable. When the Sun starts moving — excuse me, when the Sun token starts moving South along the arc, the ice returns. Ships caught North of the solar disc are trapped in the ice, leaving them stranded until the Sun returns. The crew may still explore using the ship's sled, but progress is slower and more treacherous.

Now consider the crew. The crew as a whole performed the work necessary to navigate the maze of ice and land, to survey their surroundings, and to investigate the disappearance of the Franklin Expedition. Rather than give you a large crew to manage, feed, and discipline, you need only worry about seven crew members on your expedition.

"Hold on, do you think these skeleton people know where Franklin went off to?"

Each round, you use your crew to perform actions. Many of the actions require a single crewman. The players take turns executing these actions until they choose (or are forced) to pass. When all players pass, the round ends.

You may decide that time is of the essence and may want to perform more than one action on your micro-turn. Your crew is willing to oblige, but each action after the first requires one extra crewman to perform. There are limits to how much any man can endure, after all (or woman, but women were spared the horrors of Arctic exploration and had to endure instead the horrors of Victorian life).

"The last journal entry reads 'too many... actions... in one turn...'"

I will grant you that playing Expedition: Northwest Passage is not the SAME as physically being a prisoner of the ice, putting on plays in the eternal darkness of the winter months and praying to whatever deity is willing to listen that the ice will break up in the summer. It is not as cold, for one thing. The risk of scurvy is also greatly reduced. The isomorphism is not perfect, but I think you will agree that it is rather satisfying.

Yves Tourigny
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