The weaver was visiting his friend Adam, a black sheep. They had sat down for a cup of tea and a game of chess.<\/p>\n\n\n\n
\u201dYes, I have three bags of wool left, although I\u2019ve already promised two of them for others.\u201d said Adam, responding to the weavers inquiry.<\/p>\n\n\n\n
\u201dA single bag won\u2019t get me far.\u201d replied the weaver, and made an initial move. Adam was, as always, playing as black. \u201dI am at wits end. I have just signed a contract to deliver handkerchiefs to the National Association of Hay Fever Victims. We\u2019ll start small, but in ten years I\u2019ll need a thousand bags to meet the terms. So far I only have a ready supply of 5 bags a year.\u201d<\/p>\n\n\n\n
Adam looked thoughtful and made a move. He lay back down on his chair and sipped his tea.<\/p>\n\n\n\n
\u201dI do enjoy the grains you bring me.\u201d he said, and glanced over to a white bucket filled to the brim with groats. \u201dI can get you 1000 bags a year by then. Right now, it\u2019s just me and the wife, but we are expecting twins, and the family should be big enough by then.\u201d<\/p>\n\n\n\n
The weaver picked up a bishop and held it thoughtfully over the board. \u201dI don\u2019t believe that. This can only be solved through legwork. I\u2019ll walk down the road and knock on the door of every sheep around here until I have what I need.\u201d<\/p>\n\n\n\n
Adam stood up and walked over to the white bucket. \u201cNot every sheep has as splendid wool as me and Ewe. It will take you time to find enough suppliers. Realistically, perhaps ten each year?\u201d He leaned down and nibbled on some groats.<\/p>\n\n\n\n
The weaver finally made up his mind and made his move. \u201cYes, that sounds about right.\u201d<\/p>\n\n\n\n
Adam was by now halfway down the bucket. He pulled his head out and went back to the table.\u201cSo, in ten years you\u2019ll have a hundred suppliers.\u201d<\/p>\n\n\n\n
\u201cYeah, that won\u2019t suffice.\u201d said the Weaver with a troubled frown.<\/p>\n\n\n\n
\u201cIn the meantime, my family will be four this year. Since sheep has between one and six lambs per term, we can expect that our children will have another two lambs each, on average, so in two years there\u2019ll be four new lambs and eight of us in total. In ten years, our grandlambs grandlambs grandlambs grandlambs will have around 1024 lambs and our family will be about 2048 strong.\u201d<\/p>\n\n\n\n
The weaver made a startled jerk and spilled his tea. He produced a black handkerchief and started wiping the table. \u201cThat may just solve my problem. Just for the sake of argument, though, I\u2019d like to consider some additional strategies. Humor me for a moment, will you?\u201d<\/p>\n\n\n\n
Adam had already completed his draw and was on his way back to the bucket. \u201cBy all means.\u201d<\/p>\n\n\n\n
\u201cPosit that I proceed as planned for this year, except that I hire an apprentice that I will train in the craft of recognising fine wool. Next year, my new lieutenant will be ready to work independently, but to speed things up further, both of us will have an assistant. In that way, we\u2019ll be able to proceed four times as fast.The year after, I will promote another lieutenant, and each of us will have two assistants. At that point, we have speed things up by a factor of nine, and acquired ninety new wool suppliers, compared to your meekly eight!\u201d<\/p>\n\n\n\n
\u201cThen by the tenth year, your lead is surely insurmountable?\u201d replied Adam, his voice muted as he didn\u2019t bother to remove his head from the bucket.<\/p>\n\n\n\n
The weaver looked up from the chessboard. \u201cYes, I would think so. By that year, I\u2019ll have a staff of one hundred, good for finding a thousand sheep! That is vastly more than your\u2026\u201d<\/p>\n\n\n\n
\u201c1024 lambs.\u201d Said Adam and raised his head. The bucket was by now empty but had gotten stuck around his neck. \u201cWould you please?\u201d<\/p>\n\n\n\n
His friend laughed and got up. \u201cI find myself outsmarted by a sheep in a bucket. How is it possible that two sheep can outdo a hundred workers?\u201d He put his foot on Adam\u2019s chuck and pulled the bucket back. The bucket came loose. The weaver stumbled backwards.<\/p>\n\n\n\n
\u201cThink about the annual increase. In your initial plan, your supply increased by the same amount every year. In your revised plan, you accumulate suppliers more quickly for every year, but let\u2019s consider how quickly it gets quicker. In the first year, you have 10 bags of wool. By the second, you have 40, and by the third 90. By the fourth you have 160.\u201d<\/p>\n\n\n\n
\u201cSo the increase is 30 between the first and second year, then 50, and then 70.\u201d<\/p>\n\n\n\n
\u201cRight, so as you may have noticed, the increase<\/em> increases by the same amount, that is 20 bags.\u201d<\/p>\n\n\n\n \u201cYou\u2019ve lost me. I was under the impression that this is what you said of my first idea?\u201d<\/p>\n\n\n\n \u201cIn that case, the number of sheep increased by the same amount annually. In this case, the number increases by an ever-larger amount every year. However, the annual change itself grows by a constant amount.\u201d<\/p>\n\n\n\n The weaver just stood silent, with a bemused look on his face.<\/p>\n\n\n\n \u201cIt is possible to take it one step further: What if you and your lieutenants had sergeants who in turn has ensigns? In that case, the rate of change is at the same pace as your second example. To be specific, you would have 80 suppliers by the second year, and 270 by the second year.\u201d Explained Adam. <\/p>\n\n\n\n \u201cThen surely you will no longer have a chance to catch up?\u201d<\/p>\n\n\n\n \u201cIt will take us a bit longer than 10 years, but even then we will eventually overtake you. Your three strategies are respectively called linear, quadratic and cubic, but you can of course have arbitrarily many levels, where the rate of change of any particular level corresponds to the level below. Eventually, I will always prevail, because my strategy is called exponential, and the rate of increase of the exponential is also exponential.\u201d<\/p>\n\n\n\n The weaver and Adam went back to the chess table, and settled down again. \u201cClearly I must accept such an erudite argument. I trust that we have an agreement then, that you will supply 1000 bags of wool annually within the next 10 years? Only the matter of price remains to be settled, then?\u201d<\/p>\n\n\n\n Adam smirked. \u201cI do not require much. For the first square of this chess board, I would like a single groat. For the second square, I ask for one multiplied by two groats, and for the third I would request one multiplied by two multiplied by three groats \u2013 you get the idea. For every year that we are in business, we add another square, according to the aforementioned pattern.\u201d<\/p>\n","protected":false},"excerpt":{"rendered":" The weaver was visiting his friend Adam, a black sheep. They had sat down for a cup of tea and a game of chess. \u201dYes, I have three bags of wool left, although I\u2019ve already promised two of them for others.\u201d said Adam, responding to the weavers inquiry. \u201dA single bag won\u2019t get me far.\u201d […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/extrapolerat.se\/index.php\/wp-json\/wp\/v2\/posts\/5"}],"collection":[{"href":"https:\/\/extrapolerat.se\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/extrapolerat.se\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/extrapolerat.se\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/extrapolerat.se\/index.php\/wp-json\/wp\/v2\/comments?post=5"}],"version-history":[{"count":14,"href":"https:\/\/extrapolerat.se\/index.php\/wp-json\/wp\/v2\/posts\/5\/revisions"}],"predecessor-version":[{"id":22,"href":"https:\/\/extrapolerat.se\/index.php\/wp-json\/wp\/v2\/posts\/5\/revisions\/22"}],"wp:attachment":[{"href":"https:\/\/extrapolerat.se\/index.php\/wp-json\/wp\/v2\/media?parent=5"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/extrapolerat.se\/index.php\/wp-json\/wp\/v2\/categories?post=5"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/extrapolerat.se\/index.php\/wp-json\/wp\/v2\/tags?post=5"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}