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<p>In the figure below, the surface of a modern football is made of <strong>20</strong> white hexagons and \(x\) black pentagons, where each pentagon is surrounded by <strong>5</strong> hexagons and each hexagon is surrounded by <strong>3</strong> pentagons.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767769381727-jovzf50mpm.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NjkzODE3Mjctam92emY1MG1wbS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyOTI2MzQxLCJleHAiOjE3OTI5MjY2NDF9.EKn2KKiOOy7NRI9JZfU_62KAnGbjmZmWDMC78mhhNvI" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Find the value of \(x\).</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Find the number of vertices on a football, i.e., the number of points where the angles of black pentagons meet with the angles of white hexagons.</div></li></ol>
<p>For the football pattern shown, exactly one pentagon and two hexagons meet at every vertex.</p>

In the figure below, the surface of a modern football is made of **20** white hexagons and \(x\) black pentagons, where each pentagon is surrounded by **5** hexagons and each hexagon is surrounded by **3** pentagons.

Find the number of vertices on a football, i.e., the number of points where the angles of black pentagons meet with the angles of white hexagons.

For the football pattern shown, exactly one pentagon and two hexagons meet at every vertex.

Counting Faces and Vertices by Incidence

Polygons

<p>The surface of a modern football is made of \(20\) white hexagons and \(x\) black pentagons, where each pentagon is surrounded by \(5\) hexagons.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2-1" data-layout="2-1"><div class="question-column" style="flex: 2; align-self: flex-start;"><ol class="i-list"><li data-list-item-id="item-1780819119824-vqyn2hm8e" data-list-item-path="item-1780819119824-vqyn2hm8e"><div class="list-item-paragraph">Find the value of \(x\).</div></li><li data-list-item-id="item-1780819120352-s2zqc6bbm" data-list-item-path="item-1780819120352-s2zqc6bbm"><div class="list-item-paragraph">Find the number of vertices on a football, i.e., the number of points where the angles of black pentagons meet with the angles of white hexagons.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760864276481-igvizimuqi.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA4NjQyNzY0ODEtaWd2aXppbXVxaS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyOTI2MzQxLCJleHAiOjE3OTI5MjY2NDF9.F2--bnNpBcn4e5ICzIoTw5MMwrI81UkN1582N6vlmEs" style="max-width: min(300px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>
<p>Editorial clarification of the intended football pattern: each hexagon borders exactly three pentagons, and exactly one pentagon and two hexagons meet at every vertex.</p>

The surface of a modern football is made of \(20\) white hexagons and \(x\) black pentagons, where each pentagon is surrounded by \(5\) hexagons.

Editorial clarification of the intended football pattern: each hexagon borders exactly three pentagons, and exactly one pentagon and two hexagons meet at every vertex.

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