{"id":5247,"date":"2025-08-15T21:52:00","date_gmt":"2025-08-16T01:52:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=5247"},"modified":"2025-08-30T23:56:09","modified_gmt":"2025-08-31T03:56:09","slug":"count-the-friends-at-a-party","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=5247","title":{"rendered":"Count Friends at a Party"},"content":{"rendered":"\n<p>You are invited by a host at a party with a total of $2025$ people, including the host.  Every pair of two people at the party have exactly one common friend. Among them, what is the maximum and minimum number of friends each person can have?<a href=\"javascript:toggle_visibility('count_friends_2025_08_15')\">\ud83d\udd11<\/a><\/p>\n<div id=\"count_friends_2025_08_15\" style=\"display:none\"><\/p>\n\n\n\n<p><strong>Solution<\/strong>: Let the party correspond to a graph on&nbsp;$2025$&nbsp;vertices connecting by edges as \u201cfriends\u201d. The condition \u201cevery pair of two people has exactly one common friend\u201d is the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Friendship_graph\">friendship condition<\/a>.<\/p>\n\n\n\n<p><a href=\"https:\/\/math.mit.edu\/~apost\/courses\/18.204-2016\/18.204_Elizabeth_Walker_final_paper.pdf\">The Friendship Theorem<\/a> says any finite graph with that property is a windmill (friendship) graph: there is one central vertex joined to every other vertex, and the remaining vertices split into disjoint pairs so that each pair together with the center forms a triangle.<\/p>\n\n\n\n<p>Because&nbsp;$2025=2\\times 1012+1$, the graph is the windmill with&nbsp;$1012$&nbsp;triangles glued at the center. Therefore:<\/p>\n\n\n\n<ul><li>the maximum number of friends, the center person, most likely the host, is&nbsp;$2025\u22121=2024$;<\/li><\/ul>\n\n\n\n<ul><li>the minimum number of friends, any non-center person, is&nbsp;$2$, i.e. the center and their paired partner.<\/li><\/ul>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>You are invited by a host at a party with a total of $2025$ people, including the host. Every pair of two people at the party have exactly one common friend. Among them, what is the maximum and minimum number &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=5247\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[19],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5247"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5247"}],"version-history":[{"count":10,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5247\/revisions"}],"predecessor-version":[{"id":5257,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5247\/revisions\/5257"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5247"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5247"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5247"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}