{"id":2024,"date":"2017-05-11T07:06:21","date_gmt":"2017-05-10T22:06:21","guid":{"rendered":"http:\/\/mathcs.ksa.hs.kr\/?p=2024"},"modified":"2017-05-11T07:06:37","modified_gmt":"2017-05-10T22:06:37","slug":"2017-09-pow","status":"publish","type":"post","link":"https:\/\/mathcs.ksa.hs.kr\/?p=2024","title":{"rendered":"2017-09 POW"},"content":{"rendered":"<p>Consider the first quadrant in the Cartesian plane divided into unit squares by horizontal and vertical lines at the positive integers.<\/p>\n<p>(1) Place $latex 3$ dots (<em>clones<\/em>) in the shape of an $latex L$-tromino in the bottom left-most squares, and draw a &#8220;barbed wire fence&#8221; enclosing the dots and their $latex 3$ respective squares: this is the red fence in following figure.<br \/>\n<a href=\"http:\/\/mathcs.ksa.hs.kr\/wp-content\/uploads\/2014\/05\/2014-07.jpg\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/mathcs.ksa.hs.kr\/wp-content\/uploads\/2014\/05\/2014-07-300x300.jpg\" alt=\"2014-07\" width=\"200\" height=\"200\" class=\"aligncenter size-medium wp-image-1168\" \/><\/a><br \/>\nAt each step you can erase a dot and replace it with two copies in adjacent squares, one directly above and the other directly to the right, as long as those squares are currently unoccupied.<\/p>\n<p>Prove that it is <strong>impossible<\/strong> to free all clones from the prison.<\/p>\n<p>(2) A single clone is placed in square $latex (0,0)$, and the prison encloses<\/p>\n<ul>\n    (a) the $latex 10$ squares $latex (i,j)$ with $latex i+j\\leq 3$;<br \/>\n    (b) the $latex 6$ squares $latex (i,j)$ with $latex i+j\\leq 2$.\n<\/ul>\n<p>Show that there will always be at least one clone in the prison.<\/p>\n<p>Due day is May 24.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Consider the first quadrant in the Cartesian plane divided into unit squares by horizontal and vertical lines at the positive integers. (1) Place $latex 3$ dots (clones) in the shape of an $latex L$-tromino in the bottom left-most squares, and&hellip; <\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[],"tags":[],"_links":{"self":[{"href":"https:\/\/mathcs.ksa.hs.kr\/index.php?rest_route=\/wp\/v2\/posts\/2024"}],"collection":[{"href":"https:\/\/mathcs.ksa.hs.kr\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathcs.ksa.hs.kr\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathcs.ksa.hs.kr\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/mathcs.ksa.hs.kr\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2024"}],"version-history":[{"count":1,"href":"https:\/\/mathcs.ksa.hs.kr\/index.php?rest_route=\/wp\/v2\/posts\/2024\/revisions"}],"predecessor-version":[{"id":2025,"href":"https:\/\/mathcs.ksa.hs.kr\/index.php?rest_route=\/wp\/v2\/posts\/2024\/revisions\/2025"}],"wp:attachment":[{"href":"https:\/\/mathcs.ksa.hs.kr\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2024"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathcs.ksa.hs.kr\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2024"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathcs.ksa.hs.kr\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2024"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}