The last three installments of this series have been based, one way or another, on playing solitaire.  Some of you might be thinking that I spend all my free time playing this electronic card game on my computer.  It’s not true.  I also play Minesweeper, and I’ve learned a few things from that, too.  Interestingly, one of the things I learned involved my brother Roy, whom I mentioned last week.

  Roy is no intellectual lightweight.  He had a graduate level fellowship in philosophy for a while.  Long before that, there was a moment when I said to Roy and our common friend David (whom I know I mentioned in the article called David, and probably since then as well) that one of them, and I wasn’t certain which, was the smartest person I knew.  It wasn’t just my opinion; David at that moment said that indeed, he thought the same of Roy and me, that one of us was the smartest person he knew.  Roy indicated that he, too, thought that either David or I was the smartest person he knew (although he adds to his telling of this story that thereafter he thought perhaps we were not so smart as he had estimated if we thought he might be as smart as that).  Roy is sharp, and he studied logic, so he knows how to think.

  He also plays Minesweeper.  I think he was the first person to suggest to me playing it without the flags, the markers you use to show yourself where the bombs are and prevent accidental detonation.  I always play without flags on the beginner and intermediate levels, although usually use them on the advanced level.  It’s an interesting added level of challenge to the game.  He also called my attention to the game functions that were accessed by pressing both mouse buttons at once, a technique that improves game play time significantly.  In talking about playing this electronic solo game of inverted battleship, he speaks of running out of logic.  By this he means coming to the place where there isn’t any information from which you can determine whether any unknown space on the board is safe or mined.

  One day he was watching someone play the game, not to tell them what to do (that annoying practice we recently noted in Togetherness) but merely because he happened to be talking to them while they were playing.  The player then started making moves Roy had not previously noticed.  Looking at groups of blocks, this player would note that one known block indicated that one of two adjacent blocks had to be a bomb, and that another known block indicated that one of three blocks had to be one, two of which were the same two previously noted; from this he would conclude that the third block of the three could not be a bomb, and would mark it as safe.  Similarly, but perhaps more surprisingly, he would note that one square required one of two blocks and the other two of three, and that meant that that third block had to be a bomb.  Even more interestingly, when one of three blocks was a bomb and two of three blocks was a bomb, and those two sets of three blocks overlapped by two blocks, the player deduced that the third block of one set was not a bomb and the third of the other set was, since in that circumstance the two remaining squares had to contain exactly one bomb between them.  The same techniques worked with more squares and more bombs, as long as the number of non-overlapping squares was not sufficient to account for all the unidentified bombs.

  If you’re a Minesweeper player, I’ll leave those for you to unravel.  I’m not sure from the story which of them he had not adduced for himself, and it was certainly clear that as soon as he saw it done he knew how it was done.  Yet those techniques (at least one of them) opened new approaches to play in the game.  They demonstrated that it was still possible to draw information from the data which he had never drawn before; that when he had thought he had “run out of logic” there were still ways to bring logic to bear on the game, to find answers to the critical questions.

  I use those techniques when I play, and win fairly often.  I carry it another step as well.  If I truly am entirely out of information from which I can find the next move, I apply another sort of logic to the problem.  I could, I suppose, hit a random unknown square and hope for the best.  Yet in general there are two things true about the squares on the board which are still unknown.  The first is that some are more likely to be bombs than others.  If I’ve got ten unknown blocks and three bombs, that means each block seems to have a thirty percent chance of being a bomb; but if I also know that two of those four blocks at the bottom edge have to be bombs, that means only one of the other six can be, and I’m safer choosing one of them than one of those four.  The second truth is that some of those blocks, if they are not bombs, will give me more information than others.  If I pick a square next to exactly two of those four unknown squares along the bottom, and it proves safe, it might just tell me whether I can eliminate those two squares and make the other two the bombs.  I don’t have enough data, enough logic in Roy’s terminology, to know which squares are safe; but I do have enough to know how to pick a square that’s more likely to be safe and more likely to give me the information I need to proceed.

  One of the great logic problems shared by those of us who love such things is known as the Napoleonic Hat Trick.  Although there is a simpler version, the more difficult is attributed to him.  He wanted to appoint a commander, so he brought three candidates into the room, blindfolded.  He explained to them that he was opening a case in which he had three black hats and two red hats; that he was going to place one hat on each man’s head and then close the case; and that once the blindfolds were removed, the first man who could correctly tell him the color of the hat on his own head without seeing it would be his chosen commander.  You need to know that at the moment the blindfolds came off, no man knew what color hat was on his head because no man saw what he needed to see.  After that, the second level of the logic is that everyone knew that no one else saw what he needed to see.  At the third level, one of the men knew what color his hat was.  He could not know that from what he saw; he could not even know it from his companions’ failure to answer.  He could only derive it from recognizing their thoughts before they recognized his.

  Whatever puzzles you must solve, take your logic to the next level.

  Next week, something different.

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M. Joseph Young is co-author of Multiverser and Vice President for Development at Valdron Inc.  His many contributions to online literature are indexed for convenience, and he looks forward to discussing these things by e-mail or on our Gaming Outpost forums.