{"id":2944,"date":"2021-05-09T15:08:00","date_gmt":"2021-05-09T18:08:00","guid":{"rendered":"https:\/\/www.blogs.unicamp.br\/zero\/?p=2944"},"modified":"2022-05-23T17:48:24","modified_gmt":"2022-05-23T20:48:24","slug":"the-curious-circular-chess","status":"publish","type":"post","link":"https:\/\/www.blogs.unicamp.br\/zero\/2944\/","title":{"rendered":"The curious circular chess"},"content":{"rendered":"\n<p>Imagine an ordinary, rectangular chess board with 64 squares. So we started a process of rounding its edges, rounding all the squares so that we \u2018almost preserve\u2019 its structure, as shown below.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img fetchpriority=\"high\" decoding=\"async\" width=\"416\" height=\"416\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/7round-2.png\" alt=\"\" class=\"wp-image-2917\" srcset=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/7round-2.png 416w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/7round-2-300x300.png 300w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/7round-2-150x150.png 150w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/7round-2-24x24.png 24w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/7round-2-48x48.png 48w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/7round-2-96x96.png 96w\" sizes=\"(max-width: 416px) 100vw, 416px\" \/><\/figure><\/div>\n\n\n\n<p>Let&#8217;s put the pieces together, so it doesn&#8217;t look too strange to see.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img decoding=\"async\" width=\"707\" height=\"707\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/tabuleiro-1.png\" alt=\"\" class=\"wp-image-2918\" srcset=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/tabuleiro-1.png 707w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/tabuleiro-1-300x300.png 300w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/tabuleiro-1-150x150.png 150w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/tabuleiro-1-24x24.png 24w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/tabuleiro-1-48x48.png 48w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/tabuleiro-1-96x96.png 96w\" sizes=\"(max-width: 707px) 100vw, 707px\" \/><\/figure><\/div>\n\n\n\n<p>See that there is an immense similarity to the movement of the pieces, we will show this in a more isolated way to make it easier. Pawns, for example, can make their initial moves as follows.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img decoding=\"async\" width=\"1024\" height=\"527\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-08-224558_1366x768_scrot-1024x527.png\" alt=\"\" class=\"wp-image-2920\" srcset=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-08-224558_1366x768_scrot-1024x527.png 1024w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-08-224558_1366x768_scrot-300x154.png 300w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-08-224558_1366x768_scrot-768x395.png 768w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-08-224558_1366x768_scrot.png 1072w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure><\/div>\n\n\n\n<p>The horse is also easy on that board.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"529\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-08-224921_1366x768_scrot-1024x529.png\" alt=\"\" class=\"wp-image-2921\" srcset=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-08-224921_1366x768_scrot-1024x529.png 1024w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-08-224921_1366x768_scrot-300x155.png 300w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-08-224921_1366x768_scrot-768x397.png 768w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-08-224921_1366x768_scrot.png 1069w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure><\/div>\n\n\n\n<p>The bishops have no problem with this board format.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"966\" height=\"569\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-08-225518_1366x768_scrot.png\" alt=\"\" class=\"wp-image-2922\" srcset=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-08-225518_1366x768_scrot.png 966w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-08-225518_1366x768_scrot-300x177.png 300w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-08-225518_1366x768_scrot-768x452.png 768w\" sizes=\"(max-width: 966px) 100vw, 966px\" \/><\/figure><\/div>\n\n\n\n<p>The king&#8217;s movement is not affected as shown below:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"516\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-075137_1366x768_scrot-1024x516.png\" alt=\"\" class=\"wp-image-2924\" srcset=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-075137_1366x768_scrot-1024x516.png 1024w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-075137_1366x768_scrot-300x151.png 300w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-075137_1366x768_scrot-768x387.png 768w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-075137_1366x768_scrot.png 1042w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure><\/div>\n\n\n\n<p>At this point in the post, you may be thinking that the circular shape of the board is only aesthetic, thus not affecting game dynamics. But things will be fun with unlimited horizontal and vertical movements, like towers and queens.<\/p>\n\n\n\n<p>We will first look at a simple case of the movement of the tower. It has vertical and horizontal lines at its disposal to move, right? Not necessarily. Let&#8217;s look at a slightly less behaved scenario first to understand what this ability to move means. In the figure below, what restricts the tower&#8217;s movement?<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"563\" height=\"559\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-075757_1366x768_scrot.png\" alt=\"\" class=\"wp-image-2926\" srcset=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-075757_1366x768_scrot.png 563w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-075757_1366x768_scrot-300x298.png 300w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-075757_1366x768_scrot-150x150.png 150w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-075757_1366x768_scrot-24x24.png 24w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-075757_1366x768_scrot-48x48.png 48w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-075757_1366x768_scrot-96x96.png 96w\" sizes=\"(max-width: 563px) 100vw, 563px\" \/><\/figure><\/div>\n\n\n\n<p>Following the logic of its movement along horizontal and vertical lines, it could move as follows:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"567\" height=\"569\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-080225_1366x768_scrot.png\" alt=\"\" class=\"wp-image-2927\" srcset=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-080225_1366x768_scrot.png 567w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-080225_1366x768_scrot-300x300.png 300w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-080225_1366x768_scrot-150x150.png 150w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-080225_1366x768_scrot-24x24.png 24w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-080225_1366x768_scrot-48x48.png 48w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-080225_1366x768_scrot-96x96.png 96w\" sizes=\"(max-width: 567px) 100vw, 567px\" \/><\/figure><\/div>\n\n\n\n<p>After all, it \u2018just\u2019 followed the curved vertical line in front of her, or the curved horizontal line to his right. Now let&#8217;s look at a slightly more interesting scenario.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"949\" height=\"559\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-080548_1366x768_scrot.png\" alt=\"\" class=\"wp-image-2928\" srcset=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-080548_1366x768_scrot.png 949w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-080548_1366x768_scrot-300x177.png 300w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-080548_1366x768_scrot-768x452.png 768w\" sizes=\"(max-width: 949px) 100vw, 949px\" \/><\/figure><\/div>\n\n\n\n<p>In the same way as the tower furthest from the center, it would be able to follow a horizontal or vertical line in this curved plane, the tower in a position closer to the center, it would also be able to follow in the usual 4 directions (north, south, east and west), but also within your circle, two-way, clockwise and counter-clockwise. For in this circular board, we have our vertical and horizontal lines, but we also have that the movement within the circle in which the piece is found, would also fit as a horizontal or vertical movement.<\/p>\n\n\n\n<p>Just to pique your curiosity, about this \u2018singular\u2019 variant of chess. In a blog post I discuss why in traditional chess, it is <strong><a href=\"https:\/\/www.blogs.unicamp.br\/zero\/2019\/10\/30\/impossible-check\/\" target=\"_blank\" rel=\"noreferrer noopener\">impossible to perform a triple check<\/a><\/strong>. But in this circular variant we have the different characteristic of the classic version, that the same piece threatens another for more than one direction, this is no longer an impediment, as I show below.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"554\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-083940_1366x768_scrot-1024x554.png\" alt=\"\" class=\"wp-image-2934\" srcset=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-083940_1366x768_scrot-1024x554.png 1024w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-083940_1366x768_scrot-300x162.png 300w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-083940_1366x768_scrot-768x415.png 768w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-083940_1366x768_scrot.png 1052w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><figcaption>The rook threatens the king for its circular clockwise and counterclockwise movement.<\/figcaption><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"552\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-084148_1366x768_scrot-1024x552.png\" alt=\"\" class=\"wp-image-2935\" srcset=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-084148_1366x768_scrot-1024x552.png 1024w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-084148_1366x768_scrot-300x162.png 300w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-084148_1366x768_scrot-768x414.png 768w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-084148_1366x768_scrot.png 1055w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><figcaption>Here, the rook threatens the king by the vertical straight line and the circular straight lines in a clockwise and counterclockwise direction.<\/figcaption><\/figure><\/div>\n\n\n\n<p>Thinking for a while, we can even make a quadruple threat, as I show in the two bids below.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"542\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-084508_1366x768_scrot-1024x542.png\" alt=\"\" class=\"wp-image-2936\" srcset=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-084508_1366x768_scrot-1024x542.png 1024w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-084508_1366x768_scrot-300x159.png 300w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-084508_1366x768_scrot-768x407.png 768w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-084508_1366x768_scrot-390x205.png 390w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-084508_1366x768_scrot.png 1048w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"544\" src=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-084624_1366x768_scrot-1024x544.png\" alt=\"\" class=\"wp-image-2937\" srcset=\"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-084624_1366x768_scrot-1024x544.png 1024w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-084624_1366x768_scrot-300x159.png 300w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-084624_1366x768_scrot-768x408.png 768w, https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/2021-05-09-084624_1366x768_scrot.png 1052w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure><\/div>\n\n\n\n<p>Have you ever thought, having your King threatened in 4 different directions, scary and curious, don&#8217;t you think?<\/p>\n\n\n\n<p class=\"has-text-align-center\">Cover image credit to <a href=\"https:\/\/pixabay.com\/pt\/users\/mystraysoul-10217405\/?utm_source=link-attribution&amp;utm_medium=referral&amp;utm_campaign=image&amp;utm_content=5316902\">\u041f\u043e\u043b\u0438\u043d\u0430 \u0410\u043d\u0434\u0440\u0435\u0435\u0432\u0430<\/a> from <a href=\"https:\/\/pixabay.com\/pt\/?utm_source=link-attribution&amp;utm_medium=referral&amp;utm_campaign=image&amp;utm_content=5316902\">Pixabay<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Imagine an ordinary, rectangular chess board with 64 squares. So we started a process of rounding its edges, rounding all<\/p>\n","protected":false},"author":434,"featured_media":2919,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"colormag_page_container_layout":"default_layout","colormag_page_sidebar_layout":"default_layout","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"pgc_sgb_lightbox_settings":"","_vp_format_video_url":"","_vp_image_focal_point":[],"footnotes":""},"categories":[1218],"tags":[],"class_list":["post-2944","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-v-5-ed-2"],"jetpack_featured_media_url":"https:\/\/www.blogs.unicamp.br\/zero\/wp-content\/uploads\/sites\/187\/2021\/05\/figures-5316902_1920.jpg","_links":{"self":[{"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/posts\/2944","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/users\/434"}],"replies":[{"embeddable":true,"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/comments?post=2944"}],"version-history":[{"count":1,"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/posts\/2944\/revisions"}],"predecessor-version":[{"id":2945,"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/posts\/2944\/revisions\/2945"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/media\/2919"}],"wp:attachment":[{"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/media?parent=2944"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/categories?post=2944"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.blogs.unicamp.br\/zero\/wp-json\/wp\/v2\/tags?post=2944"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}