Jul 09 2013

Bioshock Infinite


How much of life can you evoke in a medium where the point of view is confined to that of a man walking around shooting things? Game designer Ken Levine has been trying to answer this questions for many years.

The first game of his I was aware of was System Shock 2, where you play a man trying to single-handedly stop a massive infestation of a space ship at the behest of a malevolent artificial intelligence. It's a first person point of view where you walk around and shoot stuff, hack terminals, and hoard stuff. System Shock 2 wasn't really what I would call good, really, and it certainly hasn't aged well---there really was no substance to the game beyond running around, hoarding objects, and shooting stuff.

Adventure under the sea

Later, and more famously, Levine created a game called Bioshock. That game takes place in 1960 in Rapture, a city built on the bottom of the ocean by a man named Andrew Ryan. Unsaid is that it was modeled after the principles of Ayn Rand. Ryan built the city to get away from taxation, progressivism, medical ethics, religion, and environmental conservation (each is cited as a reason in dialog), creating what would be basically recognized as a Libertarian (big L) city. The city is clearly modeled after the gulch that John Galt takes his fellow characters to in Atlas Shrugged that they might escape all government intervention.

The player character in that game is Jack, and we know all of nothing about Jack other than he crash lands above Rapture and takes a bathysphere down to it, finding the city in shambles. Rapture has devolved into chaos and murder after a criminal cartel stokes the fires of literal class warfare and builds an army on the back of cheaply available genetic alteration. Soon total anarchy reigns as mindless addicts roam around in search of ADAM, the material that powers these genetic alterations.

Jack is just trying to survive at first, but soon becomes embroiled in the war for Rapture itself between Andrew Ryan and gangster Frank Fontaine. At no point past the opening scene does Jack speak or express himself in any real way, and as such Bioshock has no main character. This is the curse of the silent protagonist, which gaming has yet to shed.

Bioshock's setup by itself could have been great. This is high-concept stuff, and Levine doesn't necessarily go for the most obvious problems with a purely libertarian society, instead allowing a more specific and organic plot to establish itself (to wit, the conflict between organized crime that develops on Rapture and Ryan, who ultimately becomes a government entity). It highlights the obvious issues with such a society, but doesn't belabor the points. Coming out amid the fall of libertarianism in many circles (2007), it was also timely.

But Bioshock in the end has a story that's just as hacky as any by M. Night Shyamalan.


Jack has had his memories replaced and is actually Andrew Ryan's son, kidnapped by gangster Fontaine and programmed to kill for him. He highjacks the plane that crashes above Rapture so that he can enter the city and assassinate Ryan. After he does so, he is saved by a doctor on Rapture and goes on to kill Ryan.


It's not clear why Levine needed to rely on such a silly thing---wasn't the setup alone enough to go on? Add to this the concept of the victims of childhood experimentation, the Little Sisters, and you have more than enough grist for any mill.

So, Bioshock ended up not being a very good game (note, though, that Metacritic lists it as the 9th best reviewed game of all time, with 96% positive reviews). None of the characters is the least bit interesting, and you never interact with them anyway, besides showing up to shoot them in the face.

Up Up and Away

All of this leads me up to Bioshock Infinite, which really has nothing to do with Bioshock, other than it being developed by Ken Levine and having many of the same gameplay elements as the other Shocks. Infinite takes place long before its predecessor, in 1912, in the equally outrageous floating sky city of Columbia. (The city appears to float on dirigibles, but the game goes on to claim that quantum physics is involved.) Columbia was built by the USA, but was taken over by Zachary Comstock, who as its leader secedes from the US (and as it floats far above them, is not so easy to invade). Columbia is a microcosm of the postbellum South, demonizing Abe Lincoln while hailing Jefferson Davis, dehumanizing blacks and promoting strict Christianity (Comstock is called the Prophet).

Into Columbia is dropped our non-silent protagonist Booker DeWitt. DeWitt says he's been enlisted to go to Columbia to retrieve a woman, Elizabeth, who's being held captive there. DeWitt was a soldier with a checkered past, having participated in the massacre at Wounded Knee and been fired from the Pinkerton Detective Agency. He owes gambling debts, and takes this job to wipe them out.

When DeWitt arrives in Columbia he is baptized, and nothing about the game tries to hide its references to (and criticisms of) Christianity, especially Baptist Christianity. Afterward Booker ... walks around for awhile, takes in the sights. This is a first person shooter where you don't shoot anything, hit anything, or do anything except walk around, see the sights, eat hot dogs and candy, for the first, say, 30 minutes. It's marvelous, it's beautiful, and it's amazing. In this alternate universe a barbershop quartet in 1912 sings the Beach Boys' God Only Knows while floating on a moving stage; a turn-of-the-century couple enjoys the novelty of iced cream while chattering about this and that. The wonder of Rapture from Bioshock lasts only a few minutes before you're forced to start hitting guys with a wrench in confined spaces, but in Infinite's Columbia we get an open, bustling, living city to walk around in and explore. Nobody will try to fight you. You are unarmed. It's great.

Soon enough, DeWitt finds Elizabeth and frees her from her prison, narrowly escaping the clutches of a gigantic mechanical bird that serves as her jailer. After the escape they arrive on an artificial beach (floating in the sky, of course). And then ... you and Elizabeth walk around, look at it all. Elizabeth has never been out of her jail, but she has read and seen pictures all her life, and she is a marvel to behold. As the player (as DeWitt) walks around, she goggles at cotton candy, at music and dancing on the pier, at the promise of an airship that goes to Paris. Now you're not only enjoying the fantastical surroundings, but the joy of seeing a person experiencing life for the first time. After a few more fighting segments, you explore a boardwalk with even more of this. All of it's optional, all of it's missable if you just want to get through the game. Go ahead, don't walk by the stage with animatronic puppets that Elizabeth is entranced by. You monster.

In short, Bioshock Infinite's first two hours is what I would call pretty much the finest experience gaming has ever offered. It has more heart and creativity than I've ever seen in the medium. It's a joy.

The curse of J.J. Abrams

It's a sad fact that some of the greatest pieces of art are harbingers of the awful tropes and tricks that you will be seeing for the next 20 years. The magnificent Star Wars heralded the age of mindless blockbusters which rely on special effects to woo equally mindless viewers. Batman Begins and Casino Royale (both wonderful films) began the current spate of "gritty reboots", remakes of older movie franchises with more serious tone which Hollywood cannot currently get enough of.

And Lost began the trend that every god damn thing has to have time travel and multiple dimensional chicanery. The problem here is that J. J. Abrams's forays into time travel and multidimensional stuff (in Lost, Felicity, and in Fringe) were fun as hell, but handled quite deftly, and then mainly let go. In Star Trek the time travel sets up the movie, but after that it is not a factor. Felicity doesn't introduce the time travel until the final 4 episodes of the final season, and even then it's a one-time deal that motivates the action.

In Bioshock Infinite, the setup was already there. The gray protagonist anti-hero Booker, the learned and spunky Elizabeth, the odious racist and megalomaniac Comstock, the floating city and its denizens---all of this adds up to something special and complete. Make your story, it would be great. But that wasn't enough for Levine. Just like Bioshock had to have a memory-loss twist Luke-I'm-Your-Father moment, Infinite suffers from the Lostification of everything. Namely, Elizabeth can open portals to other dimensions. Also


Booker's memory is also incomplete and he has a connection to his place


As such, Bioshock Infinite falls back on tropes for its last couple hours, tropes that are so prevalent (due to J.J. Abrams, in part) that they are boring. They overshadow the real drama, and sully the joy from the beginning. Before long, you and Elizabeth are jumping in between timelines, so that the events that you experienced are expunged, and maybe I'm just an old person, but that kind of thing is dumb. It's cheap, it's blunt, it's an easy way out of having deal with hard consequences of other things written into your story. And if it's not in the service of rebooting something, what the hell is the good of it?

Bioshock Infinite is a good game, a very good one. But it could have been a nearly perfect one, one which elevated the medium, one which said that shooting people in the head and doing sci-fi gobbledygook aren't the only things gamers want to do. Gamers can handle pathos, and suffering, and redemption, and wonder, without any stupid nonsense. They can handle a stark look at racism and Christianity, at the deification of our slave-holding American forefathers, at the possessiveness of a father who's driven by vanity, and the mixed morality of a degenerate gambler-cum-savior.

At one point in the narrative, Elizabeth spies a guitar sitting in a storeroom, with a terrified little boy hiding under the stairwell. Booker can, at the player's behest, go to the guitar and strum chords as Elizabeth softly sings the spiritual May the Circle Be Unbroken, while handing the child an apple from a nearby barrel. The child recoils, but then relaxes a bit as he sees the gift and the apparent kindness of the people in front of him. What a lovely, inspired moment. I would pay an awful lot of money to play more of that game.

May 27 2013

When will you come back home?

As someone who becomes obsessed with things generally, and musical ideas in particular, I have to a large extent inhabited the Ryan Adams song "When will you come back home?" over the past couple weeks. Often, recording a song helps me kick an obsession. Once I've totally dissected a song to its component parts, re-recorded, mixed, mastered, edited, and posted a film of the song, I can finally put it to bed. But this time, no such luck.

What makes this song so great? It comes on the first disc of the otherwise unremarkable Cold Roses double album. The chord progression is nothing special:

Verse (A part):
E                        B                 
Something in the way she eases my mind and
A                            F#m
lays me 'cross the bed til I close my eyes
A                              B                  E
stirs me in the morning till I can't ever be satisfied [repeat]

Verse (B part):
           A                B                   E     A
If I could find my way back home, where would I go?
when everything about me I used to be
shivers in the sheets an the blankets of snow
A                       B
out there in the woods, looking for me

E                             A
When, when will you come back home? [repeat]
F#m                         A
No one leaves the lights on    in a house where
nobody lives anymore

C#m              A                    E       A    E  A
Everything about me that you liked is already gone
C#m              A                   E  A
Everything about me we loved is gone

As usual, the chords above differ from the transcriptions available online, because online transcriptions often kind of suck. The song goes from the tonic (E) to the V, IV, and ii. It's all quite standard and simple, but that makes it durable. The melody is simple and wonderfully melancholy in a way that modern country for me is typically incapable of.

Close reading

Can the lyrics make a song great by themselves? No, but they can carry a lot of the weight, and the lyrics here do. Adams is a flawed songwriter who tends to work on quantity rather than quality. But here, the songwriting is top notch.

Something in the way she eases my mind

and lays me 'cross the bed til I close my eyes

stirs me in the morning til I can't ever be satisfied

The song begins as an expression of the past, though we don't initially know it. He uses the present tense; of course, the woman still exists, and she could still do these things. And in his reverie, perhaps he forgets that. He's dwelling on the good times, but we're in for a crash.

I leave Carolina every night in my dreams

like the girls who try to love me that I only leave

rock me like a baby doll and hold me to the chest

but I'm always running too fast

We're caught in a bit of a cliche. The speaker is the vision of a rambling man, a malcontent who wants love but is to various degrees incapable of accepting it. He's never content, he ruins things and creates chaos, thrives on it, even. He knows what he has, and knows he's going to lose it.

We're still not in the now; we haven't heard about "her", this particular her. Are we still with her? Is this a worried song or a lamentation?

 If I could find my way back home, where would I go?

when everything about me I used to be

shivers in the sheets and the blankets of snow

lost out in the woods, were you looking for me?

The first line of this passage is almost too cute. Obviously if you can find your way back home, the implication is that you would go home. But that isn't the implication here: he doesn't know how to maintain a relationship, but even if he did, would he? Is that ultimately what he would want, something stable, something healthy? Or is the need for chaos too great? Here we finally get a glimpse of why: "everything about me I used to be" is lost, maybe gone. The speaker isn't committal, but he's open to the idea that he's unlovable, a broken man who breaks things around him. And we still don't know if they're still together!

When, when will you come back home?

When, when will you come back home?

No one leaves the lights on in a house where

nobody lives anymore

The monologue ends, and the speaker finally addresses his lover. She's left him, their house, their home. He's still there, but nobody lives there. Is he sincere when he says he's nobody without her? The previous lyrics don't seem to describe that in so many words. We're left thinking this is merely a plea, rather unlike the frank and honest words that came before. He is done conversing and only begs.

Loaded like the boxes up in the bedroom

coming off the hinges like the door

the shadows dancing up in the windows

they're not who we are—they're who we were

The image of a couple, shadows seen through the canvas of a window drapery, one of them packing her stuff up to leave for real, everything falling apart. But why does he mix tenses? They are who we were: what does that mean? Are we back in his memory, or is he alluding to the actual couple in the here and now? If the latter, who we were perhaps refers to when they were strangers, rather than before they were close. The more obvious interpretation, absent this, is that he remembers all the good times up in the bedroom, but these lines indicate perhaps not.

I'm not gonna break, but if I do

I'm gonna shatter like the glass I turned your heart into

I'm broken like the windows in the house where I used to live

This is kind of an interesting internal conversation. First "if" he were to break, but then "I'm broken". It's interesting in the first place because of the obvious falsehood that he's not going to break, which clearly has already happened. Ryan Adams to a certain extent still exists in the world that Tom Waits does, that Bob Dylan does, a time where men are over-feeling but under-emoting. They don't change their status to "it's complicated" on Facebook and then blog about it on their Livejournal—they drink alone and suffer and ramble around, wallowing in heartache and directing it to creativity when not hurting themselves with drugs.

I'm not sure what "I'm broken like the windows in the house where I used to live" is supposed to indicate. It's almost a throwaway, something to fill in the verse and strain the metaphor which was already strained in the prior line. One is left to speculate whether the house he used to live in is their shared house, or just a nostalgic memory of an old shitty place he used to live in.

Everything about me that you liked is already gone

Everything about me we loved is gone

In the bridge, he acknowledges, or at least posits, that she should stay away. In a song whose chorus is an open plea to come back, it's wonderfully out of place (it also proves he is talking to her, not himself). Everything about me "we" loved is gone. He hasn't just grown into a person that's incompatible with her, he's actually become a wretch, in his own eyes and hers. The song's title has an obvious answer: never. Nor does he think the woman should return. But knowing something is bad for another person doesn't make us not selfishly want it. But in fact, we gather that he more or less deliberately pushed her away.

What a brilliant piece of writing. The inner thoughts mixed in with memories of conversations of the two people, the nostalgia and the self-hatred. The images conjured are wonderful and sad.

Transcription notes

By far the hardest thing to do in covering a song is listening to it. Listening is a skill I learned long after I was able to play. Playing is mechanical, brainless even, but listening is tough as hell. The toughest thing of all to listen to and unpack is vocal harmony. Although I did it by ear, a little trickery allowed me to actually isolate the backup vocal of this song from the lead vocal during the chorus, verifying that I had it right.

This was only possible because while the lead vocal is mixed into both channels, the backup vocal shows up mainly in the right channel. By separating the waveforms for the two channels, inverting the left channel, and adding to the right one, it is possible to hear Ryan singing just the backup:

The guitar part in the intro wasn't posted anywhere on the net. In the recording above I played it in hybrid style.


(Click to enlarge.)

Mar 05 2013

Georgia drivers die mostly because of yokels

A huge traffic sign adorns the highway near my home, announcing the number of fatalities on Georgia roads this year so far (it's about 150). Moving here from California, knowing this is a small state, that seems like a lot, and it is. Here is the data from 2008 for both states (rates per million people in parentheses)

California Georgia
Total population 38,049,462 9,544,750
Crashes 452,595 (11900) 306,367 (32100)
Injuries 170,496 (4480) 115,797 (12130)
Fatalities 3113 (82) 1508 (158)
Deaths per 100 million
miles driven
1.04 1.37

Basically, if you are in Georgia you are 2.7 times likelier to be in a crash and 1.9 times likelier to die than in California.

Going through the National Highway Traffic Safety Administration data on fatal crashes does yield some interesting tidbits, though none reveal the reason for this disparity. 8% of fatal crashes in GA were due to trees, vs 4% for CA. 11% of fatal crashes in GA were due to ditches or embankments, whereas CA were likely to hit curbs (8%) before dying. Californians hit pedestrians a lot more (22% of incidents vs 13% for Georgia).

Anyway, most of the reason for the difference in rates is simple:


In 2008 the fatality rate in Georgia for rural areas was 1.82/100 million miles and only 0.97 in urban areas (for overall 1.37). 94.4% of Californians live in urban areas, but only 71.6% of Georgians do. Both states have broken down their fatalities by rural/urban areas (data from 2007):

California Georgia
Urban 2481 703
Rural 1493 (!) 805

We can calculate the rates by dividing by the populations of each region (rates are per million people):

California Georgia
Urban 69 102
Rural 700 (!) 297

These numbers are a tad rough, since the number of fatalities fluctuates more than a little bit from year to year. Nevertheless, the conclusion is that if you're in Atlanta, you're not in twice as much  danger as you are in California (only around 47% more). Part of that is that Georgia drivers drive drunk more often (3.1 alcohol related fatalities per 100k population vs 2.1 for California). Part of it, as far as I can tell, is that they kinda suck at driving.

But if you're driving around our state, you might want to stick to the city.

By the way, what the fuck with rural California drivers?!?! Maybe it's a glitch with how the state counts rural population vs rural crashes. But god damn. They actually manage to be twice as bad as Georgia rural drivers.

Mar 04 2013

Sudoku: advanced techniques

Sudoku puzzles are classified according to their difficulty. My old newspaper, the LA Times, did so as Gentle, Moderate, Tough, and Diabolical. Sites like Daily Sudoku have basically the same division, though the names are different (hard and very hard instead of tough and diabolical). Difficulty is determined by the techniques needed to solve the Sudoku. Simple elimination techniques, naked twins, and indirect inferences come up in Gentle and Moderate. Tough and Diabolical additionally require more advanced stuff.

Hidden twins is an advanced technique I've already talked about, where candidates are eliminated based on a twin that isn't readily apparent. This is a bit tricky, since you're actually adding a candidate to a triplet that isn't there so that it matches two other squares. With practice, though, hidden twins are easy to spot.

However, we need to go beyond those techniques to solve a very hard puzzle like the following that I got from Daily Sudoku:


Using a combination of box, row, and column searches, twins, and indirect inferences, we can get to this point with the puzzle (I've put in all pencil marks):


To see how I got this, see the end of this post. You can convince yourself that the puzzle cannot be solved further using any basic techniques.

The first thing I notice is that the first row has hidden twin in 1 2. See how squares two and four of that row are the only squares with those numbers? Thus, we can remove the candidate 4 from Box 2 square 1.


This lets us solve for the 4 in Box 2, and eliminate a few other candidates. Here is the puzzle as it stands:


Not much progress. However, I think of a Sudoku like this as a delicately balanced thing. If the puzzle wasn't just so, it would collapse. Solving for even one more square could lead to many solutions. Finding that is the key to finishing tough puzzles like this.

What I'm looking at is squares that contain a cycle of three digits that can "see" each other. So, for instance, the 3 7 in Box 1 can see the 1 7 in Box 3. The common digit is 7. So the question is, can either of these squares see the other pair, 1 3? It just so happens that the 1 7 can see the 1 3:


How does this help us? If the leftmost highlighted square were 7 then the lower square must be 3. If the leftmost square is 3, then the lower square is also 3. Thus, the lower highlighted square is 3.

We actually got a little bit lucky. Under other circumstances, this would only have allowed us to eliminate 3 as a possible in Box 3 square 5. That itself would have led to a number of solves, though. Like I said, it's a fragile structure.

Because the structure of the highlighted box vaguely resembles a Y, this tends to be called a Y-wing elimination. I think it's kind of a silly term, and rather like to think of it as a cycle rule. In this case it was 3 7 sees 1 7 sees 1 3. The cycle is complete. You start with a square with two possibles, and look around for any square with a pair that has one number in common. If you find one, see if that square sees any squares with the complementary pair. If so, you have found a Y-wing. Then just use basic logic to determine what candidates can be removed.

With just this one technique, the puzzle is solved very quickly. The finished puzzle is below, along with the solution to get to the second step.


Solution to get to step one: The labeling scheme is box,square, where numbering runs from left to right.

5,6 = 8
4,1 = 1 (indirect from box 5)
6,2 = 6
7,4 = 9 (column)
9,9 = 9
8,1 = 9
5,3 = 9
6,6 = 3
2,9 = 5 (enumeration)
9,1 = 4
9,2 = 2
7,3 = 5
7,8 = 6
4,3 = 2
6,1 = 5
6,4 = 2
8,9 = 7
9,7 = 8
9,8 = 5
6,7 = 7
6,9 = 8

Mar 03 2013

Sudoku: medium difficulty methods

A simple elimination algorithm that searches box-by-box, row or column will solve most Gentle and maybe even some Moderate puzzles. The final simple technique knocks out pretty much any remaining Gentle:

Technique 4: Enumeration

If the previous searches fail, try checking each square on an individual basis to see what numbers it can "see". For each square run through the numbers 1 through 9. You may find that a given square sees every number but one of them.

This doesn't really need its own example. I've used the word "see", which is common, to indicate that the square you're looking in is in the same box, row, or column as the number of interest. That is, a number that excludes a possibility is something the square can see. Enumeration looks at a square and its relation to all three groups (box, row, and column), starting with the premise that "well, some number has to go in this box".

Technique 5: Indirect inferences

The key to solving every Sudoku is to make the logical inferences required by the basic rules, obviously. But the complexity of these inferences goes up with the difficulty. The techniques used by advanced players reduce the complexity to a simple and recognizable pattern that can be searched for.

Consider the following puzzle:


Focus on Box 4 (far left box in the middle row of boxes). What do we know about where the 1 goes? The 1 from the box above it means that it's not in column 2, and the 1s to its right mean that it is either in square 7 or square 9. But now look at Box 7 (far left box in the bottom row of boxes). The 1 in Box 1 excludes the 1 from being in the middle row, but it could be in any one of the squares in the left column. But it has to be in the left column! So we know that the 1 in Box 4 must be in square 9:SudokuIndirectTwinsEnum02 SudokuIndirectTwinsEnum02b

This is an indirect inference. We don't have to know where the 1 is exactly, we only have to know where it is in relation to the box we're interested in. The same works for the 3, if we notice two simultaneous indirect inferences:SudokuIndirectTwinsEnum03

The 3 in Box 1 means that 3 appears in the left column of Box 7, and the 3 in Box 6 means that a 3 occurs in the middle row of Box 5. Thus the 3 in Box 4 must be in square 3. These two solves are not possible without this kind of inference. And, fair warning, this does come up in Gentle puzzles (though not usually two simultaneous ones like the latter example).

Implementing this kind of reasoning in your head takes a bit of practice. While I'm doing a basic search pattern, I note any such grouping that puts a number in a row or column without knowing exactly where it is. If I see one, I quickly look at adjacent boxes to see if that helps solve something. If not, I leave it. Once you've done it enough times, it isn't hard. Just remember that you can know something about the placement of a number without knowing everything about it.

Pencil Marks

Once a puzzle gets above Moderate, and sometimes even Moderate itself, you'll sometimes want to start listing the possible candidates for a given square in the square itself. These are called "pencil marks", since obviously you'll want to erase them later. So, continuing the puzzle above, I solve for a couple more numbers, and then I list in Box 9 the possible candidates for some of the remaining squares:SudokuIndirectTwinsEnum04

For a puzzle that's not too complex, it isn't worth going through and doing the pencil marks for every square. The pencil marks help us cut down on the amount of complexity we have to do mentally. They can, if you will, "store" a large number of inferences we've made that are hard to remember all at the same time. In this way, we break the problem down into smaller parts. For now, I'd rather not explain why I did the pencil marks in these specific squares (I do have some guidelines for when to do so, but let's leave it for later). Listing the pencil marks is as simple as carrying out Enumeration. Instead of mentally running through the numbers, you list any number that isn't excluded (any number that isn't seen by the square).

Now, in the figure above, look at the bottom row. The 2 in the bottom row has to appear in the seventh or eighth squares (Box 9 square 7 or Box 9 square 8). Thus, it cannot occur in Box 9 square 6. We have that listed as a possible candidate, but we can now remove it from there:SudokuIndirectTwinsEnum05

Notice that in the far right column, two of the squares have their candidates listed as 6 7. Those squares can only be 6 or 7. If one is a 6, the other is a 7, for sure. That means that the square at the top of this column cannot be 6 or 7. We have found a twin.SudokuIndirectTwinsEnum05b

In this case, it would normally be called a naked twin, because you can just look and see it plainly. As we'll see, this isn't always so. Now, the 6 7 twin also allows us to remove 7 as possible candidates from the other squares in Box 9, because those cannot possibly be 7 either:SudokuIndirectTwinsEnum06

From this, we see that we have another twin, 1 2 in the bottom row. This forces the far left square in the bottom row to be 7, since 1 must be removed.SudokuIndirectTwinsEnum07

Now, to find the naked twin, we had to notice that the 2 was required to be in the bottom row of Box 9. In fact, the squares in the bottom row in that step all had possibilities 1 2 7. The left-most square had 2 excluded, but nonetheless the pencil marks match if we ignore that. This is also a twin (or a triplet, if you like). However, it is a hidden twin, in that you can't just tell at a glance that it is a twin. The inference that the two had to be in the bottom row of Box 9 needn't have come from recognizing the triplet. Such an observation is a kind you can make any time, and it won't necessarily correspond to any of the common techniques.

With this in place, the puzzle can now be solved with basic methods already discussed. The answer is below.

Now, back to pencil marks. If I think a puzzle is "hard" (as in unyielding), I will normally start filling in pencil marks. As you can see from the above, I had gotten all the way through the three major search patterns and only come up with four solves---and two of those required an indirect inference! So, when I see that, I just resign myself that this one is going to be a bit tough and start listing. A good place to start listing is where you see a given box intersects a lot of different numbers. When you try Enumeration on it, you will get only two or three candidates. And then, obviously, you want to search for twins, so do the pencil marks on adjacent squares. This is how I came to list the pencil marks you saw above. Had I not started there, the puzzle might have taken a bit longer.SudokuIndirectTwinsEnum08

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