{"id":18622,"date":"2025-06-13T07:55:57","date_gmt":"2025-06-13T07:55:57","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=18622"},"modified":"2025-06-13T07:56:03","modified_gmt":"2025-06-13T07:56:03","slug":"on-a-board-measuring-1x100-each-square-is-numbered-from-1-to-100","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/on-a-board-measuring-1x100-each-square-is-numbered-from-1-to-100\/","title":{"rendered":"On a board measuring 1&#215;100, each square is numbered from 1 to 100"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">On a board measuring 1&#215;100, each square is numbered from 1 to 100. Three colors are used to paint the squares from left to right. The following pattern is repeated: 1 blue square, 2 red squares, 3 green squares, 1 blue, 2 red, 3 green, and so on. What is the highest numbered square that is painted blue?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\"><strong>The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We are given a 1\u00d7100 board with squares numbered from 1 to 100, painted in a repeating pattern:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>1 blue<\/li>\n\n\n\n<li>2 red<\/li>\n\n\n\n<li>3 green<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">This pattern repeats indefinitely: 1 blue, 2 red, 3 green, 1 blue, 2 red, 3 green, etc.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Understand the pattern<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The pattern covers <strong>1 + 2 + 3 = 6 squares<\/strong> in each full cycle:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Blue at position 1<\/li>\n\n\n\n<li>Red at positions 2 and 3<\/li>\n\n\n\n<li>Green at positions 4, 5, and 6<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">This 6-square cycle repeats across the board.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Determine how many full cycles fit into 100 squares<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">We divide 100 by 6: 100\u00f76=16&nbsp;full&nbsp;cycles&nbsp;with&nbsp;a&nbsp;remainder&nbsp;of&nbsp;4100 \\div 6 = 16 \\text{ full cycles with a remainder of } 4<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">So:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>16 full 6-square patterns = 16 \u00d7 6 = 96 squares<\/li>\n\n\n\n<li>Remaining: 100 \u2212 96 = 4 squares<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Count how many blue squares appear<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Each full cycle has exactly <strong>1 blue square<\/strong>, and the blue square is <strong>always the first square<\/strong> in the cycle.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">So:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>From 16 full cycles \u2192 16 blue squares<\/li>\n\n\n\n<li>The 17th cycle begins at square 97<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Now check the remainder:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Square 97: blue<\/li>\n\n\n\n<li>Square 98: red<\/li>\n\n\n\n<li>Square 99: red<\/li>\n\n\n\n<li>Square 100: green (if needed, though only 4 squares remain in this partial cycle)<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, square <strong>97 is also blue<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final count: 16 (from full cycles) + 1 (from the start of the partial cycle) = <strong>17 blue squares<\/strong><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The <strong>positions<\/strong> of blue squares occur every 6 squares, starting at 1: 1,7,13,19,&#8230;,971, 7, 13, 19, &#8230;, 97<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This forms an arithmetic sequence:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>First term: 1<\/li>\n\n\n\n<li>Common difference: 6<\/li>\n\n\n\n<li>General term: an=1+(n\u22121)\u22c56=6n\u22125a_n = 1 + (n &#8211; 1) \\cdot 6 = 6n &#8211; 5<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Set 6n\u22125\u22641006n &#8211; 5 \\leq 100: 6n\u2264105\u21d2n\u226417.56n \\leq 105 \\Rightarrow n \\leq 17.5<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">So the maximum integer n=17n = 17 a17=6(17)\u22125=102\u22125=97a_{17} = 6(17) &#8211; 5 = 102 &#8211; 5 = \\boxed{97}<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 Final Answer: <strong>97<\/strong> is the highest numbered square that is painted blue.<\/h3>\n","protected":false},"excerpt":{"rendered":"<p>On a board measuring 1&#215;100, each square is numbered from 1 to 100. Three colors are used to paint the squares from left to right. The following pattern is repeated: 1 blue square, 2 red squares, 3 green squares, 1 blue, 2 red, 3 green, and so on. What is the highest numbered square that [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-18622","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/18622","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=18622"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/18622\/revisions"}],"predecessor-version":[{"id":18623,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/18622\/revisions\/18623"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=18622"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=18622"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=18622"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}