It occurred to me only recently that I have a revealing habit: when I am under a lot of stress, I find myself doing Sudoku puzzles. I suppose what I crave is the reward of completion, and the illusion (it is only an illusion, I’m sure) that I am a successful problem-solver. There have been moments in my life when I have clung to my iPad Sudoku app like an alcoholic clinging to a bottle.
There is something magical about the process of solving a Sudoku puzzle, and the magic may reside in the fact that a new puzzle contains a lot of empty spaces with a few numbers filled in, which corresponds to a solved puzzle with all the numbers filled in. The new puzzle may look like this:
The corresponding solved puzzle looks like this:
The complete set of arranged numbers in the solved puzzle can be completely deduced from the mostly blank new puzzle. The digits are all there in the new puzzle, as surely as if they were written in invisible ink. The job of the puzzle-solver is to reveal the answer that already exists.
To an external observer, a Sudoku-solver may appear to be engaged in repetitive and mechanical tasks, but the solver’s experience is a richer one, and it’s all about the use of tools. There is, for instance, the tool I think of as the “double exclude”. One of the basic principles of Sudoku is that every vertical column and horizontal row of 9 squares must contain each number from 1 to 9 once and exactly once, and that every large square of 9 squares must also contain each number once and exactly once. So, if I know where the 1, 2, 4, 5, 8 and 9 are placed in a row, I therefore know that the three blank squares must contain a 3, 6 and 7. Let’s say I determine, using any evidence available, that one of three squares can contain a 3 or a 7 but not a 6, and then I discover that another of the three squares can also contain a 3 or a 7 but not a 6. I have now proven that the remaining square must contain a 6.
This “double exclude” is one of the simplest and most common tools used by a Sudoku-solver. We may not actually refer to it as a “double exclude” (indeed, I do not actually have names for my Sudoku-solving tools, and I just made that phrase up). But we all use the tool, or we would never be able to solve a puzzle.
I have gradually advanced in Sudoku skills from Easy to Medium to Hard to low-level Expert (which can be a real bitch). The reason I have advanced is that I have built up my toolkit, and discovered new tools. I use triple excludes and quadruple excludes, cross-reaches and box-transposes; if all else fails and I am really at wit’s end I might reach for the ugliest tool in my toolkit, which is the “erase conjecture”. This involves making a wild guess: I don’t know if this box contains a 2 or a 9, but goddammit, I’m going to put in a 2, and then I’m going to see if everything checks out. If it does, I’ve solved the puzzle; if it doesn’t, I hit “erase” and start over and I know it’s a 9.
I’ve lately mostly switched to a variant of Sudoku called KenKen, which is the exact same idea but offers more variety in terms of addition, multiplication, division and subtraction. I only recently moved up to the Hard level of KenKen, and I faced a really dire situation with one of my first Hard-level 6×6 KenKens. I kept making mistakes, and couldn’t figure out why. I would erase and start over, but couldn’t figure out the answer at all, and eventually realized I’d been staring at this one unsolvable puzzle for five agonizing hours without success. Even my attempts at the “erase conjecture” move were only leaving me more confused. I stared at my iPad in utter frustration and self-disgust, wondering if I’d reached my limit, if I was truly out of ideas and would never advance from Hard to Expert 6×6.
At that moment, I suddenly managed to invent for myself a new tool. It was based on an old one I already had. One very simple tool in 6×6 Ken-Ken is the “21-add”: the six digits from 1 to 6 add up to 21, which means that all the digits in every row and column must add up to 21. If the initial puzzle happens to offer the helpful information that, say, one set of three digits in a row adds up to 10 and another set of two in the same row adds up to 5, then the remaining digit must be a 6. I now stared in puzzlement at this infuriating KenKen puzzle, running through all the tools in my toolkit, checking one last time to see if I had possibly missed any opportunity to apply a 21-add. I had not.
But then I discovered something amazing: yes, there was no opportunity for a 21-add, but if I considered two adjacent columns of this puzzle together, I did have an opportunity to apply a “42-add”. There were no simple cases of contiguous rows in a single column that would work, but there were two columns next to each other in which I did know all the additive totals except for a single square, and the total without this square was 39. Therefore, the remaining square had to contain a 3. I put in the 3, and I had the entire puzzle solved within minutes.
I’m convinced that this particular puzzle was unsolvable without the use of the 42-add. There was no other way to solve it, and even though the tool was simple and obvious once I had thought of it, I realized that I might very well have given up before reaching that point, and would never had the pleasure of putting the 42-add into my toolbox (where it now remains, and where I can now use it frequently).
If somebody had been watching me suffer over this puzzle, they would not have been able to understand the powerful force of revelation at the moment that I discovered this new tool. They might have seen that I’d solved the puzzle and would have guessed that I’d just applied some impressive logic. What they would not have been able to understand is that I’d just done something more than apply logic: I’d discovered a concrete technique of permanent value, something I’d be able to use again, thus permanently increasing my skill level at KenKen.
So, what does this all have to do with the philosophical question of ethics and politics and morality? I’m sure you were about to ask.
I think there are important moral/ethical lessons to be learned from puzzle-solving, but these point should not be misunderstood as a suggestion that the simplicity and certainty of puzzle-solving can ever exist in the complex and uncertain realm of real-world existence. Of course it can’t; every Sudoku or KenKen puzzle is a perfect equation with a complete solution. It’s doubtful that any ethical or moral puzzle will ever be a perfect equation with a complete solution.
But that doesn’t mean we can’t transfer important ideas from mathematical puzzle-solving to real-world problem solving, and I think the basis of this transfer can be found by asking this question: what tools are in our toolkit? What is the ethical equivalent of the double-exclude, the triple-exclude, the box-transcription, the erase-conjecture? Do we have a 21-add? Do we have a 42-add? If we don’t, can we get one?
When I think about the political and ethical debates I hear swirling around me — at work, at social gatherings, on TV news and online and in magazines — it seems to me that we are dealing with a very limited toolkit.
When debating questions of war and foreign policy and problems of global violence, for instance, we see the same lame, weak tools being put into play over and over. There is the “what about Hitler?”. There is the “best defense is a strong offense”. Likewise when debating, say, the ethics of gay marriage, there is the tool known as “traditional definition of marriage”, countered by the tool known as “equal rights for all”.
We tend to get lost in our wearying emotional reactions when we try to discuss these difficult questions — but if we take a step back and examine our patterns of discussion, we’re likely to discover a severe lack of sophistication and complexity in the tools we are using. On the familiar Sudoku scale of “Easy/Medium/Hard/Expert”, it seems apparent that most of our public debates about vital political and ethical questions are being conducted, sadly, at the “Easy” level. We’re not using very complex tools at all.
I’d like to further develop the idea I’m presenting here in future Philosophy Weekend blog posts. For now, I’d like to leave it at here and simply point out that, when attempting to engage in any complex debate, there are likely to be tools available that are not apparent on first glance. Again, it took me five hours with a single KenKen puzzle to realize that I could extend the concept of the “21-add” to two columns together and call it a “42-add”. What the hell was I wasting my time doing during these five hours, before I thought of this obvious device?
I don’t know what the hell I was wasting my time doing, but I think it’s the same type of thing many frustrated people are wasting their time doing when they currently debate difficult ethical, moral and political issues at a low level of intellectual sophistication, getting lost in their feelings of frustration and anger instead of realizing that there are new tools that might actually help us clear things up.
Let’s take one of my own frequent fascinations: the apparent necessity of war and the possibility of lasting world peace. Just for the hell of it, even though there is absolutely no concrete meaning to what I’ve drawn, I’ve come up with a mock Sudoku puzzle on the top of this page, in which I’ve filled in a few squares with the names of a few major historical wars, events and controversial leaders (“World War II”, “Fall of Soviet Union”, “Mao”, “September 11”). The only thing each of these values have in common is that they are all familiar historical touchpoints that often come up in geopolitical debate. Let’s pretend this is a Sudoku puzzle that we are going to try to solve.
I’m not going to ask us to try to solve this puzzle today (and, of course, there is no validity to the placement of the terms in this puzzle at all, so there is nothing here to actually solve). What I am going to ask us all to consider, though, is this: if we were trying to solve this puzzle together, what level of expertise would we rate ourselves at? What tools do we have that we know how to use? Are we at the level of Easy, Medium, Hard or Expert?
I think this might be a fruitful way to think differently about the eternal questions of global politics that have had us all frustrated and aggravated for centuries. My guess is that we’d be flattering ourselves to even call ourselves puzzle-solvers at the Easy level when it comes to questions of war and peace. I don’t think we’re even at the level yet of seeing the same squares.
I’d like to leave this here today, and return to further develop this theme later. I’m curious if the strange words I’ve written here (it seems strange even to me, to compare ethics and politics to puzzle-solving) make sense to anybody but myself, or if I’m totally crazy to think there might be some value to considering difficult real-world issues in this way. Please let me know what you think.