{"id":3514,"date":"2026-04-25T16:55:44","date_gmt":"2026-04-25T16:55:44","guid":{"rendered":"http:\/\/www.robertandrews.org\/LIFE\/?p=3514"},"modified":"2026-04-25T16:58:30","modified_gmt":"2026-04-25T16:58:30","slug":"kings-aside","status":"publish","type":"post","link":"https:\/\/www.robertandrews.org\/LIFE\/kings-aside\/","title":{"rendered":"Kings Aside"},"content":{"rendered":"<h1><strong>Kings Aside<\/strong><\/h1>\n<p><a href=\"http:\/\/www.robertandrews.org\/LIFE\/card-tricks\/\">RETURN TO CARD TRICK MENU<\/a><\/p>\n<hr>\n<ol>\n<li>Remove the kings from a deck of cards<\/li>\n<li>Deal out two piles of 15 cards face down &#8211; an upper and lower pile<\/li>\n<li>Set remaining cards aside.\u00a0 We don&#8217;t need them.<\/li>\n<li>Cut the cards anywhere you like, setting the selected cards next to the source pile<\/li>\n<li>Put the king on one of the upper piles<\/li>\n<li>The Mark selects which of the other two piles from the lower piles and put it on top of the King<\/li>\n<li>Place the remaining king FACE UP on the untouched lower pile<\/li>\n<li>Take the upper pile without the king, and place it on top of the face up king<\/li>\n<li>Take that pile and place it face down on the upper pile<\/li>\n<li>Deal out two piles, taking note of which pile has the face up red king in it<\/li>\n<li>Throw away the pile without the king<\/li>\n<li>Repeat:\u00a0\u00a0Deal out two piles, taking note of which pile has the face up red king in it<\/li>\n<li>Throw away the pile without the king<\/li>\n<li>Repeat until you have only two cards remaining &#8211; that will be the other king<\/li>\n<\/ol>\n<input type='hidden' bg_collapse_expand='6a20283e33b0d0094953403' value='6a20283e33b0d0094953403'><input type='hidden' id='bg-show-more-text-6a20283e33b0d0094953403' value='Gemini&#039;s Analysis'><input type='hidden' id='bg-show-less-text-6a20283e33b0d0094953403' value='Show Less'><button id='bg-showmore-action-6a20283e33b0d0094953403' class='bg-showmore-plg-button bg-orange-button  '   style=\" color:#4a4949;\">Gemini&#039;s Analysis<\/button><div id='bg-showmore-hidden-6a20283e33b0d0094953403' >\n<p data-path-to-node=\"0\">This trick is a classic example of <b data-path-to-node=\"0\" data-index-in-node=\"35\">Binary Elimination<\/b> (also known as the &#8220;Automatic Placement&#8221; or &#8220;Under-Down&#8221; principle). It uses a mathematical process called a <b data-path-to-node=\"0\" data-index-in-node=\"163\">monotonic decimation<\/b> to ensure that the face-up King and the face-down King always travel together during the deal.<\/p>\n<p data-path-to-node=\"1\">Here are the mechanics behind the illusion:<\/p>\n<h3 data-path-to-node=\"2\">1. The &#8220;Symmetry&#8221; Setup<\/h3>\n<p data-path-to-node=\"3\">By dealing two piles of exactly 15 cards, you are creating two equal sets. When you &#8220;cut&#8221; these piles, you are essentially creating four smaller piles, but the <b data-path-to-node=\"3\" data-index-in-node=\"160\">total number of cards<\/b> in each section (Upper and Lower) remains constant at 15.<\/p>\n<p data-path-to-node=\"4\">Let\u2019s label the piles:<\/p>\n<ul data-path-to-node=\"5\">\n<li>\n<p data-path-to-node=\"5,0,0\"><b data-path-to-node=\"5,0,0\" data-index-in-node=\"0\">Upper A and Upper B<\/b> (Total: 15)<\/p>\n<\/li>\n<li>\n<p data-path-to-node=\"5,1,0\"><b data-path-to-node=\"5,1,0\" data-index-in-node=\"0\">Lower A and Lower B<\/b> (Total: 15)<\/p>\n<\/li>\n<\/ul>\n<h3 data-path-to-node=\"6\">2. The Sandwiching Mechanic<\/h3>\n<p data-path-to-node=\"7\">When you place a King on one pile and cover it with a pile from the <i data-path-to-node=\"7\" data-index-in-node=\"68\">other<\/i> set, you are performing a <b data-path-to-node=\"7\" data-index-in-node=\"100\">variable displacement<\/b>.<\/p>\n<ul data-path-to-node=\"8\">\n<li>\n<p data-path-to-node=\"8,0,0\">You place King #1 (Face Down) on <b data-path-to-node=\"8,0,0\" data-index-in-node=\"33\">Upper A<\/b>.<\/p>\n<\/li>\n<li>\n<p data-path-to-node=\"8,1,0\">You place <b data-path-to-node=\"8,1,0\" data-index-in-node=\"10\">Lower B<\/b> on top of it.<\/p>\n<\/li>\n<li>\n<p data-path-to-node=\"8,2,0\">The total number of cards above that King is now equal to the size of <b data-path-to-node=\"8,2,0\" data-index-in-node=\"70\">Lower B<\/b>.<\/p>\n<\/li>\n<\/ul>\n<p data-path-to-node=\"9\">Next, you place King #2 (Face Up) on <b data-path-to-node=\"9\" data-index-in-node=\"37\">Lower A<\/b> and cover it with <b data-path-to-node=\"9\" data-index-in-node=\"63\">Upper B<\/b>.<\/p>\n<ul data-path-to-node=\"10\">\n<li>\n<p data-path-to-node=\"10,0,0\">The total number of cards in the final combined stack is 32 (15 cards + 15 cards + 2 Kings).<\/p>\n<\/li>\n<\/ul>\n<h3 data-path-to-node=\"11\">3. Mathematical Parity (The Secret)<\/h3>\n<p data-path-to-node=\"12\">Because the piles were divided symmetrically, the face-down King and the face-up King are now separated by exactly <b data-path-to-node=\"12\" data-index-in-node=\"115\">15 cards<\/b>. In a 32-card deck (which is <span class=\"math-inline\" data-math=\"2^5\" data-index-in-node=\"153\"><span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>), dealing into two piles acts as a <b data-path-to-node=\"12\" data-index-in-node=\"192\">Binary Filter<\/b>.<\/p>\n<h3 data-path-to-node=\"13\">4. The Power of Powers of Two<\/h3>\n<p data-path-to-node=\"14\">The &#8220;Deal and Discard&#8221; phase works because you are effectively dividing the deck by 2 each time:<\/p>\n<ul data-path-to-node=\"15\">\n<li>\n<p data-path-to-node=\"15,0,0\"><b data-path-to-node=\"15,0,0\" data-index-in-node=\"0\">Round 1:<\/b> 32 cards become 16.<\/p>\n<\/li>\n<li>\n<p data-path-to-node=\"15,1,0\"><b data-path-to-node=\"15,1,0\" data-index-in-node=\"0\">Round 2:<\/b> 16 cards become 8.<\/p>\n<\/li>\n<li>\n<p data-path-to-node=\"15,2,0\"><b data-path-to-node=\"15,2,0\" data-index-in-node=\"0\">Round 3:<\/b> 8 cards become 4.<\/p>\n<\/li>\n<li>\n<p data-path-to-node=\"15,3,0\"><b data-path-to-node=\"15,3,0\" data-index-in-node=\"0\">Round 4:<\/b> 4 cards become 2.<\/p>\n<\/li>\n<\/ul>\n<p data-path-to-node=\"16\">In every round, the deal alternates (Left, Right, Left, Right). Because of the original 15-card offset, the face-up King and the face-down King will <b data-path-to-node=\"16\" data-index-in-node=\"149\">always have the same parity<\/b>.<\/p>\n<ul data-path-to-node=\"17\">\n<li>\n<p data-path-to-node=\"17,0,0\">If the face-up King is dealt into the &#8220;Right&#8221; pile (even positions), the face-down King is mathematically guaranteed to land in the &#8220;Right&#8221; pile as well.<\/p>\n<\/li>\n<li>\n<p data-path-to-node=\"17,1,0\">They are &#8220;locked&#8221; into the same half of the binary split.<\/p>\n<\/li>\n<\/ul>\n<h3 data-path-to-node=\"18\">5. The Reveal<\/h3>\n<p data-path-to-node=\"19\">The reason the remaining card is the other King is that the &#8220;discard&#8221; process eliminates every card <i data-path-to-node=\"19\" data-index-in-node=\"100\">except<\/i> the one that shares the exact same mathematical index as the face-up King. Since you placed them in the stack at a relative distance that matches the &#8220;power of two&#8221; cycle, the face-up King acts as a <b data-path-to-node=\"19\" data-index-in-node=\"306\">homing beacon<\/b> for the other King.<\/p>\n<p data-path-to-node=\"20\">As long as the initial count is exactly 15 and 15, the math is inescapable. It isn&#8217;t a &#8220;trick&#8221; of the hand\u2014it&#8217;s an <b data-path-to-node=\"20\" data-index-in-node=\"115\">algorithm<\/b>.<\/p>\n<p><\/div><input type='hidden' bg_collapse_expand='6a20283e33dfc3051305873' value='6a20283e33dfc3051305873'><input type='hidden' id='bg-show-more-text-6a20283e33dfc3051305873' value='Claude&#039;s Analysis'><input type='hidden' id='bg-show-less-text-6a20283e33dfc3051305873' value='Show Less'><button id='bg-showmore-action-6a20283e33dfc3051305873' class='bg-showmore-plg-button bg-orange-button  '   style=\" color:#4a4949;\">Claude&#039;s Analysis<\/button><div id='bg-showmore-hidden-6a20283e33dfc3051305873' ><\/p>\n<div>\n<div class=\"standard-markdown grid-cols-1 grid [&amp;_&gt;_*]:min-w-0 gap-3 standard-markdown\">\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">The Core Mechanics<\/h2>\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Why 32 Cards?<\/h3>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The setup produces exactly <strong>32 cards<\/strong> in the final stack:<\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Upper pile: 15 cards<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Lower pile: 15 cards<\/li>\n<li class=\"whitespace-normal break-words pl-2\">2 Kings<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Total: <strong>32 = 2\u2075<\/strong><\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">This enables exactly <strong>5 halvings<\/strong> (32\u219216\u21928\u21924\u21922), landing perfectly on 2 remaining cards.<\/p>\n<hr class=\"border-border-200 border-t-0.5 my-3 mx-1.5\" \/>\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">The Kings Are Always 16 Apart<\/h3>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">This is the central insight. Let&#8217;s say the upper pile cut produces <strong>x<\/strong> cards on top (UA) and <strong>15\u2212x<\/strong> on the bottom (UB). Similarly <strong>y<\/strong> and <strong>15\u2212y<\/strong> for the lower pile.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">After all the assembly steps, the final stack is:<\/p>\n<div class=\"overflow-x-auto w-full px-2 mb-6\">\n<table class=\"min-w-full border-collapse text-sm leading-[1.7] whitespace-normal\">\n<thead class=\"text-left\">\n<tr>\n<th class=\"text-text-100 border-b-0.5 border-border-300\/60 py-2 pr-4 align-top font-bold\" scope=\"col\">Position<\/th>\n<th class=\"text-text-100 border-b-0.5 border-border-300\/60 py-2 pr-4 align-top font-bold\" scope=\"col\">Contents<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">1 to (15\u2212x)<\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">UB<\/td>\n<\/tr>\n<tr>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\"><strong>16\u2212x<\/strong><\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\"><strong>King 2 (face up)<\/strong><\/td>\n<\/tr>\n<tr>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">&#8230;<\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">LB, LA<\/td>\n<\/tr>\n<tr>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\"><strong>32\u2212x<\/strong><\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\"><strong>King 1<\/strong><\/td>\n<\/tr>\n<tr>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">32<\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">UA<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">King 2 is always at position <strong>16\u2212x<\/strong>. King 1 is always at position <strong>32\u2212x<\/strong>. Their separation is always exactly <strong>16<\/strong>, regardless of where you cut.<\/p>\n<hr class=\"border-border-200 border-t-0.5 my-3 mx-1.5\" \/>\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Why 16 Apart Guarantees They Stay Together<\/h3>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">When dealing left-right, odd positions go left and even positions go right. Since <strong>(32\u2212x) \u2212 (16\u2212x) = 16<\/strong>, and 16 is even, both kings <strong>always have the same parity<\/strong> \u2014 so they always land in the same pile at every single round.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">After each halving, their separation simply halves too:<\/p>\n<div class=\"overflow-x-auto w-full px-2 mb-6\">\n<table class=\"min-w-full border-collapse text-sm leading-[1.7] whitespace-normal\">\n<thead class=\"text-left\">\n<tr>\n<th class=\"text-text-100 border-b-0.5 border-border-300\/60 py-2 pr-4 align-top font-bold\" scope=\"col\">Round<\/th>\n<th class=\"text-text-100 border-b-0.5 border-border-300\/60 py-2 pr-4 align-top font-bold\" scope=\"col\">Total Cards<\/th>\n<th class=\"text-text-100 border-b-0.5 border-border-300\/60 py-2 pr-4 align-top font-bold\" scope=\"col\">King Separation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">Start<\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">32<\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">16<\/td>\n<\/tr>\n<tr>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">1<\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">16<\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">8<\/td>\n<\/tr>\n<tr>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">2<\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">8<\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">4<\/td>\n<\/tr>\n<tr>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">3<\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">4<\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">2<\/td>\n<\/tr>\n<tr>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">4<\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">2<\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">1 (adjacent)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Why the Cut Doesn&#8217;t Matter<\/h3>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Cutting at any point shifts <strong>both<\/strong> kings by the same amount \u2014 so their separation stays locked at exactly 16. The mark&#8217;s &#8220;free choice&#8221; of cut position is completely inconsequential to the outcome.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The face-up King 2 is simply a <strong>tracking beacon<\/strong> \u2014 it tells you which pile to keep at each step, and since King 1 is always exactly 16 positions away, it&#8217;s always in the same pile.<\/p>\n<\/div>\n<\/div>\n<p><\/div><\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Kings Aside RETURN TO CARD TRICK MENU Remove the kings from a deck of cards Deal out two piles of 15 cards face down &#8211; an upper and lower pile Set remaining cards aside.\u00a0 We don&#8217;t need them. Cut the cards anywhere you like, setting the selected cards next to [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21],"tags":[],"class_list":["post-3514","post","type-post","status-publish","format-standard","hentry","category-entertainment"],"_links":{"self":[{"href":"https:\/\/www.robertandrews.org\/LIFE\/wp-json\/wp\/v2\/posts\/3514","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.robertandrews.org\/LIFE\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.robertandrews.org\/LIFE\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.robertandrews.org\/LIFE\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.robertandrews.org\/LIFE\/wp-json\/wp\/v2\/comments?post=3514"}],"version-history":[{"count":2,"href":"https:\/\/www.robertandrews.org\/LIFE\/wp-json\/wp\/v2\/posts\/3514\/revisions"}],"predecessor-version":[{"id":3522,"href":"https:\/\/www.robertandrews.org\/LIFE\/wp-json\/wp\/v2\/posts\/3514\/revisions\/3522"}],"wp:attachment":[{"href":"https:\/\/www.robertandrews.org\/LIFE\/wp-json\/wp\/v2\/media?parent=3514"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.robertandrews.org\/LIFE\/wp-json\/wp\/v2\/categories?post=3514"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.robertandrews.org\/LIFE\/wp-json\/wp\/v2\/tags?post=3514"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}