| operations |
| comment | | "parent_author":"",<br>"parent_permlink":"steemstem",<br>"author":"mcfarhat",<br>"permlink":"how-many-balls-are-in-the-vase-at-noon-or-the-ross-littlewood-paradox",<br>"title":"How Many Balls Are In The Vase At Noon? (or The Ross-Littlewood Paradox)",<br>"body":"So if you enjoyed our latest infinity related paradox post - [Hilbert's Grand Hotel Paradox (https:\/\/steemit.com\/steemstem\/@mcfarhat\/hilbert-s-grand-hotel-paradox),<br> today we got an even wayyyy cooler infinity related paradox,<br> which i believe to be one of the most mind-blowing ones i've recently written about,<br> and it's called:\n<center>**The Ross-Littlewood Paradox**<\/center>\n_You can see an additional list of my other cool paradox posts at the end of the article,<br> don't scroll down yet :D_\n***\n### What are we talking about?\n###\nSo,<br> the paradox essentially aims at answering the question of the post's title,<br> \"how many balls are in the vase at noon?\"\nYou might think,<br> tell me how big it is,<br> i'll just calculate the volume,<br> divide by the volume of each ball,<br> and there you have it - mathematically speaking.\nWell,<br> there is a slight problem,<br> actually few slight problems,<br> since the paradox goes as follows:\n***\n### Assumptions\n###\n* We have a giant vase,<br> which is infinitely big,<br> that can hold an infinite amount of balls.\n* We also have an infinite amount of balls that we can insert into the vase.\n* The balls are numbered in order 1,<br> 2,<br> 3,<br> ...\n* We are adding balls at a rate of 10 balls per each task.\n\n<center>![ (https:\/\/steemitimages.com\/DQmaLbqMm6jMpG7gB4NUFQ2LnWuZDhMfys4raJ5cKJ5xQF9\/image.png)\n*Our imaginary vase and balls*<\/center>\n* Tasks are performed at decreasing timing,<br> with first task starting at 11:59:00 AM:\nThe first task would is completed in half a min (1\/2 min).\nThe second task,<br> would be completed in half the amount,<br> so a quarter of a min (1\/4 min).\nThe third one,<br> in half the last one's amount,<br> so one eigth of a min (1\/8 min).\nand so on and so forth...until,<br> the last task to be completed by noon.\n\n<center>![ (https:\/\/steemitimages.com\/DQmccSpmuGH3xrioWbEaiXuQrUZx37u1o7EJjFCMyc2Z6rT\/image.png)\n*Time lapse of the different tasks*\n<\/center>\n\nSo basically,<br> it drills down to performing an **infinite amount of tasks,<br> in an finite amount of time**,<br> which is the definition of what we refer as a **supertask**.\n***\n_Cool Fact: If we stay in the realm of possible,<br> supertasks simply cannot exist,<br> due to both space and time's properties of NOT being infinitely divisible. \nIn the real world,<br> or at least the known universe,<br> the smallest distance is actually the **Planck length** = 1.616 299(38)x10<sup>-35<\/sup>meters,<br> which practically translates to 100 quintillionth of proton's diameter.\nSimilarly,<br> the **Planck time**,<br> is the fastest time to travel through the Planck length at the speed of light,<br> which is equal to 5.391 16(13)x10<sup>-44<\/sup>seconds\nYet,<br> Math simply doesn't care about limitations,<br> nor does the human mind,<br> so on with our quest :)_\n\nNow,<br> let us consider several scenarios of how we are feeding the balls into the vase,<br> and then get an answer to our question,<br> how many balls are in the vase at noon..\n***\n### Scenario one: Adding 10 balls and removing the first\n###\n<center>![ (https:\/\/steemitimages.com\/DQmc1SUUmv4nxzpDos5p1LLfnzgkyadsbZXDJnsHBjb6xHQ\/image.png)\n*10 balls to be added at each task*<\/center>\n\nSo in this scenario,<br> we start with task 1,<br> where we add balls 1 to 10,<br> and remove the first,<br> which is ball 1\nAt task 2,<br> we add balls 11-20,<br> and remove the remaining first,<br> which is ball 2\nAt task 3,<br> we add balls 21-30,<br> and remove the remaining first,<br> ball 3,<br>\nAnd so on and so forth...\n\n<center>![ (https:\/\/steemitimages.com\/DQmYtnB5Lhq8xjmVJ56tJc2MZ5zACzxr8dprqWBQewUnhpZ\/image.png)\n*Balls in the vase after 3 tasks*<\/center>\n\nNow the question is .. how many balls would be in the vase at noon?\nMy personal guess was,<br> well,<br> as probably many of you would suggest,<br> infinity. \nAnd that is actually **INCORRECT**\n\nBut how,<br> we just practically added 9,<br> and then another 9,<br> and yet another 9,<br>... so we should have .. infinity?\n\nLet us look into what happened on the actual the steps. At step 1,<br> we removed ball 1,<br> and then step 2,<br> removed ball 2,<br> step 3,<br> removed ball 3,<br>...so at step n,<br> we would remove ball n,<br> and hence all the balls will be removed from the vase by the end of the tasks\n\n<center>![ (https:\/\/steemitimages.com\/DQmRm2oHD5ZXyZ4p5rD1n7t8HD9SSWjMAV64cji2VAME3ej\/image.png)\n*Progress of removing balls at every step*\n<\/center>\n\n**But wait,<br> there's more...**\n\nApparently,<br> the order in which we add balls,<br> and remove them,<br> affects how many balls are in the vase at noon. Let us look at an alternative scenario..\n***\n### Scenario two: Adding 10 balls and removing the last\n###\nSo in this alternative scenario,<br> let us proceed as follows:\nAt task 1,<br> we add balls 1 to 10,<br> and remove the last,<br> which is ball 10\nAt task 2,<br> we add balls 11-20,<br> remove the last,<br> which is ball 20\nAt task 3,<br> we add balls 21-30,<br> remove the last,<br> ball 30,<br>\nAnd so on and so forth...\n\n<center>![ (https:\/\/steemitimages.com\/DQmNr64SsYVmgH7BV3CkJYX6L8tCxTxYDrfgz5QTMnb7zvo\/image.png)\n*Balls in vase at end of task 4*\n<\/center>\n\nSo again,<br> how many balls would be in the vase at noon?\nThis time,<br> the answer is different,<br> and it is **Infinity**. \n\nSeriously? yes seriously. \n\nThis is actually due to the fact that only balls which are multiples of ten are the ones being removed,<br> so the vase will not contain balls 10,<br> 20,<br> 30,<br>... yet all the other balls will still be there,<br> hence the infinite amount.\n***\n### Other answers?\n###\nOther philosophers had different opinions about this problem statement. \nFor instance,<br> philosopher mathematician Paul Benacerraf claimed that the problem statement is not well specified,<br> since it explains all the steps that happen prior to reaching noon,<br> yet no specification is clear about noon. The vase might as well burst into dust,<br> blow up...\nAnother philosopher,<br> Jean Paul Van Bendegem,<br> claimed that the problem is not properly formed,<br> due to the fact that noon is a moment that can never be reached because of the infinite amount of tasks that need to be done. And the question in itself is assuming that noon will be reached,<br> which is contradictory.\n***\n### Not to forget proper historical acknowledgment\n###\nThe Ross-Littlewood paradox was actually initially formulated in 1953 by mathematician John E. Littlewood,<br> and then expanded in 1988 by Sheldon Ross.\n***\n### What to take from this?\n###\nSo at the end,<br> while this might seem like a crazy concept to dwel upon,<br> or something that would never happen in our universe,<br> yet it is such a challenging thought,<br> such an out of the box thinking that grows our knowledge,<br> helps us explore uncharted territories,<br> and that made us who we are at the moment.\n\nHope you found this as fun and educational as I did !\n\n@mcfarhat\n\n_Other cool paradoxes i've recently written about:_\n* [Hilbert's Grand Hotel Paradox (https:\/\/steemit.com\/steemstem\/@mcfarhat\/hilbert-s-grand-hotel-paradox)\n* [The Napkin Ring Paradox (https:\/\/steemit.com\/steemstem\/@mcfarhat\/the-napkin-ring-paradox) \n* [The Zipf's Law (https:\/\/steemit.com\/steemstem\/@mcfarhat\/century-old-mystery-of-the-zipf-s-law-and-a-cool-experiment)_\n\n***\n**References:**\n* [Wikipedia (https:\/\/en.wikipedia.org\/wiki\/Ross%E2%80%93Littlewood_paradox)\n* [Wikipedia (https:\/\/en.wikipedia.org\/wiki\/Planck_units)\n* [Youtube PBS Infinite Series (https:\/\/www.youtube.com\/watch?v=Sdp_V0L99sw)\n* [Youtube Vsauce (https:\/\/www.youtube.com\/watch?v=ffUnNaQTfZE)\n\n**Photo Credits:**\n* [Images 1,<br> 2,<br> 3 (https:\/\/www.youtube.com\/watch?v=Sdp_V0L99sw)\n\n***\n**_Founder of Arab Steem_**\nArab Steem is a community project to expand Steemit to the Arab world,<br> by supporting the existing Arab steemians and promoting others to join.\nYou can connect with us on @arabsteem or via discord channel https:\/\/discord.gg\/g98z2Ya\nYour support is well appreciated!\n\n***\n**_Proud Member Of_**\n* **steemSTEM**: SteemSTEM is a project that aims to increase both the quality as well as visibility of Science,<br> Technology,<br> Engineering and Mathematics (and Health). You can check out some great scientific articles via visiting the project tag #steemSTEM ,<br> project page @steemstem,<br> or connecting with us on chat https:\/\/steemit.chat\/channel\/steemSTEM\n* **MAP(Minnows Accelerator Project)**: MAP is a growing community helping talented minnows accelerate their growth on Steemit.\nTo join,<br> check out the link at the home page of @accelerator account",<br>"json_metadata":" \"community\":\"busy\",<br>\"app\":\"busy\/2.0.0\",<br>\"format\":\"markdown\",<br>\"users\":[\"mcfarhat\",<br>\"arabsteem\",<br>\"steemstem\",<br>\"accelerator\" ,<br>\"links\":[\"https:\/\/steemit.com\/steemstem\/@mcfarhat\/hilbert-s-grand-hotel-paradox\",<br>\"https:\/\/steemit.com\/steemstem\/@mcfarhat\/hilbert-s-grand-hotel-paradox\",<br>\"https:\/\/steemit.com\/steemstem\/@mcfarhat\/the-napkin-ring-paradox\",<br>\"https:\/\/steemit.com\/steemstem\/@mcfarhat\/century-old-mystery-of-the-zipf-s-law-and-a-cool-experiment\",<br>\"https:\/\/en.wikipedia.org\/wiki\/Ross%E2%80%93Littlewood_paradox\",<br>\"https:\/\/en.wikipedia.org\/wiki\/Planck_units\",<br>\"https:\/\/www.youtube.com\/watch?v=Sdp_V0L99sw\",<br>\"https:\/\/www.youtube.com\/watch?v=ffUnNaQTfZE\",<br>\"https:\/\/www.youtube.com\/watch?v=Sdp_V0L99sw\",<br>\"https:\/\/discord.gg\/g98z2Ya\",<br>\"https:\/\/steemit.chat\/channel\/steemSTEM\" ,<br>\"image\":[\"https:\/\/steemitimages.com\/DQmaLbqMm6jMpG7gB4NUFQ2LnWuZDhMfys4raJ5cKJ5xQF9\/image.png\",<br>\"https:\/\/steemitimages.com\/DQmccSpmuGH3xrioWbEaiXuQrUZx37u1o7EJjFCMyc2Z6rT\/image.png\",<br>\"https:\/\/steemitimages.com\/DQmc1SUUmv4nxzpDos5p1LLfnzgkyadsbZXDJnsHBjb6xHQ\/image.png\",<br>\"https:\/\/steemitimages.com\/DQmYtnB5Lhq8xjmVJ56tJc2MZ5zACzxr8dprqWBQewUnhpZ\/image.png\",<br>\"https:\/\/steemitimages.com\/DQmRm2oHD5ZXyZ4p5rD1n7t8HD9SSWjMAV64cji2VAME3ej\/image.png\",<br>\"https:\/\/steemitimages.com\/DQmNr64SsYVmgH7BV3CkJYX6L8tCxTxYDrfgz5QTMnb7zvo\/image.png\" ,<br>\"tags\":[\"steemstem\",<br>\"math\",<br>\"education\",<br>\"busy\",<br>\"arab\" " | | vote | "voter":"mcfarhat",
"author":"mcfarhat",
"permlink":"how-many-balls-are-in-the-vase-at-noon-or-the-ross-littlewood-paradox",
"weight":10000 |
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