If you flew\u00a0Southwest Airlines in that nameless decade that preceded this one,\u00a0before they morphed into a more-or-less-generic mainstream megacarrier, perhaps you\u00a0miss\u00a0their reputation for funny flight attendants who made jokes about the safety instructions as you fastened your seatbelt low and tight across your waist.\u00a0 Or perhaps you long for\u00a0their stunningly philo-procrastinatic-policy of allowing one\u00a0to cancel one’s\u00a0flight, sans penalty, up to two hours AFTER the flight took off. \u00a0Yet when I look back on my dozens of disturbingly low-cost flights between BWI and Albany as I shuttled between my parents’ home in Washington, DC and my undergraduate studies at Williams College, I dream about Spirit, their old in-flight magazine. \u00a0At the back of each issue wedged between the bland articles and\u00a0complex\u00a0flight maps, the latter of which may have led me to study graph theory, was a puzzle I have never seen before or since…until now. \u00a0Shinro, a Japanese minesweeper-Sudoku hybrid, entertained me for countless hours…well, probably a dozen hours all told since there were only four puzzles per month and each took only 10-20 minutes. \u00a0I found it more relaxing and less frustrating than Sudoku, especially for those like me who prefer making big fat X’s and circles rather than trying to juggle dozens of tiny\u00a0digits. \u00a0Here’s how it works:<\/p>\n
Each arrow in the puzzle below points to one or more “holes”. \u00a0The numbers on the side indicate the number of holes in a given column or row. [Both images in this post are public domain, courtesy of Wikimedia Commons]<\/p>\n
If you look in the first row, you’ll note that there can be only one hole and furthermore there is an arrow pointing to it. Therefore, the first three boxes prior to the arrow can be crossed off (rather like clicking on a “safe” tile in minesweeper). \u00a0Similarly, the last three boxes in row two can be eliminated. \u00a0Once you’ve done that, it becomes\u00a0clear that the diagonal arrow at the end of row three must be pointing to a hole in the top row since you’ve crossed out the only other option. \u00a0Once that hole is marked, the rest of the boxes in the top row can be eliminated, and so forth. \u00a0The solution is\u00a0below: You can download Shinro to your Android<\/a>\u00a0or your iPhone<\/a>\u00a0here, though it goes by a couple other names. \u00a0For those like me who enjoy the satisfaction of marking up a puzzle with a pencil (or for the overly-confident, a pen), David Oranchak offers a free book of Shinro which you print out for free\u00a0here<\/a>\u00a0or order from Amazon<\/a>\u00a0for $12.95. \u00a0If you run out, never fear, Oranchak claims there\u00a0are over 1,000,000,000,0…[25 more zeros]…00,000\u00a0more combinations, some of which are on his website<\/a>. \u00a0If you’re interested in the mathematics behind the puzzles, he\u00a0also has a paper<\/a>\u00a0laying out an algorithm to generate Shinro puzzles with desirable qualities.<\/p>\n The number of possible distinct Shinro games is not obvious.\u00a0 Orenchak claims in the introduction to his paper that there are 64 choose 12 ways to place the holes and 9^52 ways to place the arrows, resulting in over 10^40 arrangements (not all of which are valid games). \u00a0Obviously, we can\u00a0reduce\u00a0that upper bound by identifying arrangements\u00a0that are isomorphic\u00a0under rotation and reflection. \u00a0How small can you get that upper bound to be? \u00a0Let us know in the comment section.<\/p>\n <\/p>\n![\"442px-Shinro_start.svg\"](\"http:\/\/blogs.ams.org\/mathgradblog\/files\/2015\/09\/442px-Shinro_start.svg_-300x300.png\")
<\/a><\/p>\n![\"442px-Shinro_end.svg\"](\"http:\/\/blogs.ams.org\/mathgradblog\/files\/2015\/09\/442px-Shinro_end.svg_-300x300.png\")
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