The Stack

There are really two opposite ways to be stuck on a logic puzzle. The more usual way is to be unable to eliminate possibilities, so that the number of solutions under consideration is too large to simply iterate through. The other way is to eliminate all the possibilities. If you can do that, and the puzzle actually does have a solution, it means there’s an error in your reasoning somewhere, a variation that you’re forgetting about. But knowing that there’s an error doesn’t help much in finding it.

That’s the position I’m in with the last Battleships puzzle. There’s one four-long ship in every puzzle, and this puzzle has only one row or column that can hold it. Since the grid is 10×10, that means there are only seven positions this ship can go in. And I can prove every single one impossible.

There must be a mistake. That much is clear. But is the mistake mine or theirs? I actually think it’s probably mine in this instance. Battleships has a certain amount of error-detection built in. There are exactly 20 tiles worth of ships in every grid, so you can tell if there’s a mistake in one of the row or column numbers by adding them all up and seeing if you get 20. This would fail if there are two or more errors that cancel each other out perfectly, but how likely is that? So I think I’ll have to keep on looking for a solution here.

I can’t say the same for the last remaining Cross Math puzzle. Here, a similar summing technique proved that the puzzle is wrong. In these puzzles, you have to fill the numbers 1 through 9 in such a way to satisfy all six horizontal and vertical equations. Now, observe that the horizontal equations in this one have no multiplication or division. If they had nothing but addition, then the sum of all three rows would have to add up to the sum of all the numbers from 1 through 9, which is to say, 45. But some of the numbers are subtracted. The effect of subtracting a number instead of adding it is to subtract that number from the total sum twice: once for removing it as an addend, and once for the actual subtraction. So no matter what numbers have their signs flipped, the result will differ from 45 by an even number. The sum of all three rows therefore has to be odd. But this puzzle gives us rows that sum to 18.

Having thus convinced myself that at least one of the sums here was wrong, I told the game that I was done with the puzzle, and was pleased to be vindicated: in the game’s solution, the top row makes 8, not the 7 claimed. Proving a puzzle incorrect in this way is a sort of metapuzzle, and at least as satisfying as solving the puzzle itself.

Yeesh, did I really just make a “definition of ‘game'” post? I must really be running out of steam. Doubtless this has to do with my continued inability to solve the very last Battleships puzzle. (And I do mean the very last — I’ve solved the rest of the set.)

To counteract this, I’ve moved on to the Trivia section for a bit. It turns out to be annoying — enough so that I didn’t finish it in one sitting like I had planned, even though it’s half the size of the other sections. Partly this is due to the design. You may have noticed from my screenshots that every puzzle type in Games Interactive 2 has its own logo, with a picture labeled in a zany font. Usually this is something you can ignore, but for Trivia, they animated it and made it part of the game. The picture is a caricature of a game show host, which makes sense, I suppose, but honestly isn’t the feel I’d have gone for. He wiggles his head back and forth while waiting. His facial expression changes with your guesses. When you get something wrong, a buzzer sounds and the light on his podium turns red; when you get something right, a brief latin-rhythmed victory ditty plays. If you have sound turned on, you hear this enough for it to become irritating. If you don’t, you just get a pointless pause for a few seconds before it goes on to the next question.

But the worst part of the Trivia section is the questions where you have to type in an answer. This isn’t always the case — most of the questions are multiple choice. But some are type-ins, and a lot of them aren’t really designed for this game. For example, at one point it asks you for the two ways, other than Lisa’s saxophone tune, that the opening credits sequence from The Simpsons varies from episode to episode. Nobody is going to get that one right, because nobody, even if they know the answers, is going to phrase it in exactly the ways that the computer is checking for. But let that slide for the moment; a lot of the questions are reasonably answerable. The more consistently annoying problem is that the game isn’t very responsive to typing. You pretty much have to hold keys down for half a second or so to make sure they register. This is basically intolerable, and I don’t remember the first Games Interactive having this problem.

The English language is unusual for having two unrelated words for “play” and “game”. Compare Spanish jugar/juego, German spielen/Spiel, Russian играть/игру. The seminal ludology text Les jeux et les hommes is usually rendered in English as “Man, Play, and Games”, using two words to express the full extent of what Caillois means by jeux. The very idea that playing and games are two separate concepts seems to only make sense to people whose native language supports that notion, and perhaps not even them, if they think about it. From the right perspective, games seem like a subset of play, comprising play that uses fixed rules.

Games Magazine goes against this by generalizing “game” to the point where it may even be a superset of play — crosswords and cryptograms and similar puzzles are leisure activities, sure, but I hesitate to categorize them as play. You don’t play a cryptogram, you solve it. (Solitaire Hangman is kind of a special case here.) But the two volumes of Games Interactive don’t hesitate to call them “games”, asking you to “Select Games” from the main menu and such. I call them “puzzles”, but the word “puzzle” is largely avoided in the games themselves, and when it is used, it’s to refer to an individual unit within a multi-part “game” like a Battleships set; crosswords, which are served up one at a time, are never called “puzzles”. The one big exception is the “Visual Puzzles” category in Games Interactive 2, which seems like a mistake. The first Games Interactive called the category just “Visual”.

It’s almost as if they’re using the word “game” not in its customary sense, but as a translation of “jeux“. I mean, Caillois did specifically give crosswords as an example in Les jeux et les hommes. But then, he also mentioned dancing, and that seems over-broad even for the magazine.

The strange part is that, while I’m reluctant to classify traditional print puzzles as games, I have no problem regarding Games Interactive and its sequel, which are composed of traditional print puzzles, as games. I suppose part of it is that they were packaged and sold as a games. But also, they’re not very different from other familiar puzzle games, and the interactivity is a big factor. The ability to place your battleship pieces arbitrarily and experimentally and delete them without consequence or erasure marks when they don’t pan out, or to try out different letter mappings in your cryptogram and let the computer instantly show you the consequences, or to click on parts of a picture in a visual puzzle and be instantly told whether each click was right: these things change how you engage with the puzzles, making it more, well, playful. There are puzzle types here — chiefly Logic and Visual one-offs — that don’t support playful interactivity, instead essentially just giving you a question and prompting you for an answer. Those are the least successful puzzles here, barring bugs. As for the rest, even if you don’t play the puzzles per se, at least you play with them. And that’s a distinction I’d be surprised to see in any language other than English.

Among their many other flaws, the two Gameses Interactive are lopsided. There’s a good variety of different puzzle types, but if you want to play ’em all, you’re going to spend the bulk of your time on the Battleships and Paint By Numbers, as my posts here will attest. It’s like one of those Mixed Nuts assortments that’s mostly peanuts.

I feel like it’s gotten worse in Games Interactive 2, too. It definitely has in one respect: it has six Paint by Number sets where the first game had only four. But it seems to me that the Battleships have gotten harder. I said in my first post about Battleships that it took me about an hour to complete a set. I think that was an understatement, but now, I’m wishing that a set of Battleships took me so little time. I think there are some individual puzzles that take me that long. I don’t have exact numbers, though, because the in-game clock maxes out at 99:59.

The thing is, though, I could well be wrong. That is, even if the puzzles are taking me longer, it doesn’t necessarily mean that they’re actually more difficult. It must be remembered that back in the first game, I got a number of the Battleships puzzles wrong. By now, I’ve learned from experience how to quickly and accurately check the correctness of a solution. I start by checking the ship counts — one four, two threes, three twos, four ones. It gets so you can do this at a glance. Then a sweep over the numbers in each row and column, and finally make sure that none of the ships are touching at their corners. Thanks to my thoroughness, I haven’t submitted an incorrect solution in a long time. But it takes a lot more time to find a correct solution than a solution that’s almost correct and fails in just one respect that you failed to notice before submitting. And the almost-correct solution is just as useful for completing the game, by both my personal criteria, which basically just consist of “Submit solutions in good faith”, and those required for unlocking the final puzzle in Games Interactive 2, which don’t even require that.

It is with some relief that I report that I have finished the Paint by Numbers puzzles, and made some progress on the Battleships. I’ll be closing out the Logic section entirely before long, and that will leave just the three most lightweight categories: Trivia, Visual, and one that’s new to Games Interactive 2: Word Search. I haven’t mentioned the word searches yet, because they’ve never been the focus of my play sessions. And yet I’ve almost finished them.

Let me get this out of the way: Word searches are a joke. They’re the puzzle type that fans of all other types of puzzles look down on. Logic puzzles require real thought, crosswords test your knowledge and vocabulary, and cryptograms combine all of the above, but all a word search requires is the ability to not be bored by word searches. The reason I’ve gotten so many of them done is that I’ve been doing them one at a time between other puzzles in order to avoid having to do them all at once. It’s one thing for a variety puzzle game to have one or two word searches for variety’s sake — one per issue of the magazine was pretty much the right amount, if you ask me — but this game has twenty. That’s more than any other single puzzle type except crosswords.

The word searches from Games Magazine have one real point of interest: The unused letters spell out sentences, usually some quotation linked to the puzzle’s theme. As far as I know, this was never stated outright in the magazine. It was just a little bonus for those who noticed it. The closest they came to spilling the beans was in a March issue with an Irish-themed word search for St. Patrick’s Day: the editor’s message for the issue mentioned that the editor had suggested slipping some Irish names into the unused letters, only to be told “Wait til you see what we’ve already got going on there”. So at that point, not even the editor of the magazine was in on the secret. But that anecdote was enough for me to catch on, and now, the information proves useful: looking for the hidden sentences provides a welcome shortcut to finding all the words, in addition to making the whole process less tiresome.

As you’re probably expecting if you’ve been following these posts, there are problems. There are occasional typos in the word lists: SUBSITUTE for SUBSTITUTE, BARBEQUE where the grid contains BARBECUE. Such words cannot be marked in the grid; the game only accepts what’s in the word list. Occasionally there’s a word that’s left out of the word list entirely. You can tell that it’s supposed to be part of the solution because it interferes with the secret unused-letters text, but it’s just not there. One particular puzzle is missing something like half of its word list. I only have a vague sense of what the unused letters were supposed to spell out in that one. In one of the puzzles, part of the grid was actually misplaced, showing letters one space to the right of where they should have been. It took me a while to figure out what was going on there. Once again, I find myself thinking that some of these bugs could make good puzzles in their own right if deployed deliberately.

The UI is mainly reasonable, but has one biggish problem: words are marked with ovals around them, and these ovals are sometimes wide enough to intersect with adjacent letters, especially when the ovals are diagonal. Now, the puzzles come in different sizes, and the larger grids are fit on the screen by using a smaller font. Throw a couple of lines across the smaller font and it becomes illegible. It’s not hard to think of solutions to this. Like, instead of circling the words, put a line through the marked letters! Ideally, make the line a light color and display the letters on top of it so you can still read them. But I suppose you can’t do that in the print version, so it violates the game’s prime directive.

Unexpectedly, I seem to have found a new technique for use in Paint by Numbers: eliminating possible positions for a run along an edge (either the natural edge of the grid, or any line past which you’ve eliminated any possibility of occupied squares) by considering how the perpendiculars affect the next tiers inward. For example, suppose the leftmost column in the grid says just “7”, and the column adjacent to it says “5”. It follows that the run of 7 must either have no rows in common with the 5, in which case it coincides with seven rows that start with 1, or have enough rows that begin with something other than 1 to accommodate the 5. That is, if you have seven rows that start with 1 except for one in the middle that starts with 2, that can’t be where the 7 goes, because it would place an isolated square into the second column where it can’t be part of the 5.

It’s obvious when you think about it. (If it doesn’t seem obvious from my description, my description is to blame. It would probably be clearer from an illustration.) But it’s a thing that’s easy to not think about, especially if you’re plugging away at the more usual techniques. The only thing that made me start thinking about it in this way was a sequence of puzzles that pretty much relied on this sort of reasoning, having lots of one-thick outlines around the edges.

I understand that popular logic puzzles, like nonograms and sudoku, have names for specific techniques. I don’t know the names, though, because I haven’t studied the theory. To me, finding the techniques yourself is part of the fun. Which I suppose marks me as a programmer.

While I’ve been stuck on the Battleships, I’ve been progressing through the Word Puzzles. I finished them today. I haven’t been posting about them because I don’t have much to say about them. It’s exactly the same assortment as in the first Games Interactive: Bulls Eye, Mind Flexers, Quote Boxes, and Solitaire Hangman. But I do have some complaints, which I’ll make now.

One of the Bulls Eye puzzles calls for a word that can be formed from the first letters of a sequence of words in the instructions, but the instructions are apparently not the same ones this puzzle was originally printed with, so that clue is impossible.

One of the Hangman sets is composed of highly unusual words, like “ouabain” and “buprestid”, which is basically cheating. I remember playing Hangman as a child: the other children would say “Oh no, it’s Carl! He has a large vocabulary, and will doubtless choose an obscure word none of us know!” But I recognized even then that this would be a cheap victory. The true triumph is in choosing a word that’s perfectly common, but that they still wouldn’t guess, like “shoebox”. And that’s what most of the Solitaire Hangman sets are like, apart from this one. I can’t say that it was completely impossible, though, because I managed to get the word “siphuncle” right, despite not knowing it, just from guessing likely letters. The twelve most frequent letters in the English language are ETAOINSHRDLU, and applying those to “siphuncle” yields SI_HUN_LE with only five wrong guesses. And of the remaining letters of the alphabet, the ones most likely to appear before an H are C and P — which just happen to be the letters we need in the word.

And finally, there’s a certain amount of repetition. Two of the Quote Boxes sets use the same Elizabeth Taylor quote about people with no vices. Even worse, Hangman set 6 is simply a repeat of set 5 in a different order. I can believe that they’re just copying stuff from the magazine, and that the magazine repeated puzzles occasionally. But it’s definitely something that should have been caught before publication.

Once again, I seem to have gotten stuck in a pattern of starting Battleships sets and not finishing them. It’s funny: when I finished the last of the Battleships in the first Games Interactive, my reaction was one of relief that I would never have to do this again, even though I knew there was an entire additional volume of the things waiting on the Stack. The Paint by Numbers sets may take a great deal longer than the Battleships, but I’m finding the Battleships far more onerous. I think this is because Paint By Numbers is marked by near-continual progress. It can get slow, but you’re always improving your knowledge of what’s in the grid. Whereas in Battleships, I can just plain get stuck. It makes me think there’s got to be some approach I haven’t discovered yet, some way of looking at the grid that simplifies the impenetrable.

I do think I’m doing better than I did in the first game. If nothing else, I’m managing to check for mistakes before submitting my solutions pretty consistently. (The UI changes help here.) But the fact that I’m aiming for real solutions in the first place is pure stubbornness on my part, given how contemptible a production it is, and especially given that I’ve already cheated. Finishing this game obviously means solving the Final Puzzle, and, lest we forget, you don’t actually have to solve any other puzzles to do that. You just have to attempt them. Actually getting them right is effectively a side quest.

But if I were the sort of person who’s willing to just skip over the content to get to the end, I probably wouldn’t be writing this blog. Look: It bothers me that I cheated my way past the final boss in the South Park FPS. I recently got a bunch of Sonic the Hedgehog games in a Humble Bundle, and I’ve been contemplating putting Sonic CD back onto the Stack. That’s is a game I played before starting this blog — at the time, it was the only Sonic game for PC. I played through every level and reached an ending, but it wasn’t the good ending, and I’m thinking lately that it shouldn’t count. I have similar twinges about my suboptimal completion of Police Quest 3. So I have good reason to believe that if I don’t do my best to complete Games Interactive 2 for real, it’s just going to haunt me later.

I find that there are two distinct modes of thought involved in solving Battleships puzzles. It’s not quite the “logic vs intuition” thing again, though.

On the one hand, you have the same sort of approach as Paint by Numbers puzzles: proving things about the grid. You have certain constraints, including puzzle-specific ones like “this row has exactly three occupied spaces in it” and general ones like “there is exactly one set of four consecutive occupied spaces”, and from these constraints you construct chains of reasoning about what must be in specific spaces. “There are two three-tile long ships and only three places where they can fit. So at least one of them must be in one of these two places, both of which border on this square. Therefore, this square is unoccupied.” You then mark the square as unoccupied, and start using that as a constraint in further proofs.

On the other hand, you can also think of it as an assembly puzzle, like pentominos or tangrams. You have ten pieces of varying length, and you have to put them together into a shape that fits certain constraints. True, most assembly puzzles use a target shape where the pieces touch each other, and that’s specifically forbidden here, but that doesn’t really make much difference. You’re still thinking about it in terms of moving pieces around. There’s typically a very limited set of places where the four-long ship can go, so you try it out in one of those places, and when you see that it prevents the rest of the pieces from fitting in nicely, you move it somewhere else. This is really equivalent to reasoning from trial-and-error, but I find that conceptualizing it as moving pieces around makes it a lot easier. Sometimes I’ll mark in all the pieces in an way that doesn’t fit the numbers, because it helps me to see how to fix the problem by moving something.

Usually my trajectory through a puzzle starts with the former approach, and switches to the latter when progress fizzles out. In some puzzles this happens very quickly. Notably, the UI discourages the assembly model. I can think about it in terms of moving pieces around, but I can’t actually remove an entire ship from one place and put it somewhere else as an action. The only form of interaction is marking and erasing individual tiles, which fits better with the other approach. This sways how I want to solve the puzzle. I can’t blame Games Interactive for this, though. The same applies even more strongly to the print version.

As before, Paint by Numbers is weekend work. The game has six sets. I’ve done two so far, and expect to manage a third today, leaving the other half for next week. The palette this time is missing the dithered grey that I previously used to mark uncertain squares, so I’ve been using yellow for that purpose instead. At least the two-semantically-distinct-whites problem is gone.

The chief characteristic of Paint by Numbers is that it’s methodical. I described the basic solving process in a previous post: you look for a thing that just barely fits in the space available, which gives you information about its position, which imposes constraints on other things, and so on. It takes time, but you can keep on extending the chain of logic for as long as you keep finding leads. Every once in a while, though, the chain seems to end. Maybe you reach a point where the necessary reasoning takes more steps before yielding concrete results. Maybe there really is a simple next step, but it’s hard to spot. Either way, I wind up marking, in a different color, the things that I suspect rather than just the things that I know.

I don’t like taking this step, because it seems so much less certain than the plodding and methodical reasoning. But is it really? In theory, logic should never steer you wrong, but it’s being applied by a fallible human mind. I frequently do make mistakes. Occasionally, I even make mistakes so severe that I have to correct them by wiping the entire grid and starting over. Intuition may well be a better guide, especially as I train it up by solving these things. It would certainly be faster. I remember struggling with anagrams in crosswords as a child, seeing no good way to solve them other than by writing down every possible arrangement of letters to the point where they stopped making sense. Nowadays, all I usually have to do is look at the letters and, if I have enough constraint from the cross-letters and clue, the solution just pops out at me unbidden. Could a human mind get to the point of solving nonograms the same way? I suspect so, given the prodigious mental feats that “Human Calculators” have trained themselves to do. Heck, after all this intensive practice, I may well be closer to that point than I’ve been allowing myself to acknowledge.