Math Magic

hey Vsauce Michael here if you rearrange the letters in William Shakespeare you can spell here was I like a song in the King James Bible in psalm 46 the 46th word is shake and the 46th word from the bottom is sphere William Shakespeare was 46 years old when the King James Bible was completed is this just a coincidence yes it is given enough searching enough words enough data you can eventually find and should indeed expect all kinds of neat coincidences it's just probability here's another good example think of a card any card value and suit okay got it mentally focus on that card deliver it to my brain using ESP whoa I got it here it is amazing right now think of another card are you ready okay it's some well it's this one right one more time think of a card really think about it and there it is impressed probably not but approximately seven of you might be easy there are only 52 cards if each card is equally likely to be thought of I had a one in 52 chance of guessing your card and about a 1 in every 140,000 chance of doing so three times in a row if say a million people watched this video and play along after three tries you could expect about seven people to be left who had their imagined card pulled by mean each time except not more than seven actually you see there's even more magic happening here and by magic I mean math when asked to arbitrarily think of a card studies have shown that people tend to think of certain cards more often than others these are the results from a study of a few hundred people hearts are named a bit more often than the other suits the three cards I drew are the top three most often thought of but would it be more amazing if it actually was magic yes and no obviously magic as a performance is important it reminds us that being stupefied and curious and humbled is fun without a visceral reaction to the unknown would we care to learn or study or investigate but yet explanations don't dull mystery they empower the mystified explanations aren't ends they're in trances the start of an ability to use reassemble and reimagine new things out of what you now know here's a cool fact if you divide a deck of cards exactly in half there will always be the same number of red cards in one half as there are black cards in the other no matter how mixed up the cards were before you began it's not magic though it's just that think of it this way half of the cards in a deck are red so all the red cards could fit in one deck half this means that the number of cards in a half that aren't red that is our black cards are necessarily equal in number to the Reds that must be in the other half knowing this you can devise an even neater trick like the really cool trick Matt Parker just showed off on his channel which you should subscribe to or like this one take an easier to count number of cards like say 10 and then place another 10 cards on top but face up have a friend shuffle the whole mess however they wish cuts messy cuts riffle smooshing they just can't cause any card to flip over then ask them to hand you the pack and behind your back even while blindfolded show them that you can separate the deck into two halves with the same number of face-up cards every time over and over again all you have to do is count out ten cards the number of face-up cards in one hand will equal the number of facedown cards in the other flip one of the packs over and all of its facedown cards will turn into face-up ones tada the face-up counts are now equal here's another trick that shows off some neat mathematical properties to help me with this trick I've brought in Vanessa from the YouTube channel braincraft it's an awesome Channel please go check it out but first check out this trick are you ready I'm ready we've got ten cards on the table five diamonds and five clubs ace to five of each go ahead and put the clubs on top of the diamonds face up face up yep alright now I am going to mix these cards up alright and you were gonna use your ESP abilities to get the cards paired up at the end you ready okay I want to mix these up take some off the top moving to the bottom taking parts from the bottom and just like I don't okay I'm gonna I'm gonna cut these as well let me know where you want to cut them okay yeah there okay we'll cut the cards there and I'm gonna take half of what's here five cards and deal them down one two three four five I'm gonna set the other five right there now we have two piles and it is your turn to start swapping we got poor swaps you can make and a swap involves taking the top card and moving it to the bottom okay alright now you can divvy up these four swaps however you want you can do four here you can do 2 & 2 1 & 3 whatever you want okay I'll do three here yeah interesting yeah right you do it I'll do it one two three and then one swap here good yes now we're left with these two cards on top I'm gonna take them off and set them right here now three swaps all three okay it's your choice one two three and the top two cards I will peel off and we'll set aside all right now two swaps two here mm-hmm all right one two once again the cards that are on top get peeled off and set aside one swap one here okay so I'll take these away and we're left with two cards will they be the same will they be a match let's find out okay whoa don't say okay yet because things aren't okay if something magic just happened we've got all of these cards we set aside and yet they're all matches both aces right there what are these both fives you also matched up the threes and of course the fours you're welcome thank you very much but not you Vanessa you mathematics every time no matter how your friend decided to swap matching pairs will be together how well it's all about cycles if you have some cards in a particular order and you cut them somewhere doesn't matter waiter you will only change the positions of the individual cards but not the sequence the next card down will always be the next card in order with the whole thing wrapping back around it helps to think of the order not as a tower but as a clock now when the trick we just did each cards match is always just five cards away in the sequence because five happens to also be the number of cards in half the pack dividing the pack in half leaves pairs lined up no matter how we've cut the deck in the trick however we added a bit of extra confusion by dealing the top half down onto the table one two three four five this has the effect of reversing its order now instead of having the same position in each package pairs have mirrored positions the number of cards below any given card in the bottom pack is the number above it in the top pack this balance means that swapping all the cards above a target in the bottom pack leaves it on top and swapping the number of cards below it from the other pack leaves its match on the other top of course the number of cards above and below a target card in the bottom pack is just the number of all the other cards in that half which is the total amount in each pack minus one that's why with five cards in each pack we started with four swaps then after each pack had shrunk to four we did three swaps as long as the total number of swaps adds up to the pack total minus 1 we'll have matching pairs on top you can extend these facts about cyclical sequences to other tricks of your own imagining this is one of my favorite tricks it comes from Peter Magowan on the stem maths magic channel trick time once again I'm joined by Vanessa from brain craft and ace through five of diamonds and ace through five of clubs Vanessa could you please take the ACE through five of diamonds and put them on top of the clubs but flip them over so they're facedown okay so we now have a kind of messy little deck here where all the black cards are face up and the Reds are face down we're gonna mix these up a lot we can even riffle shuffle this riffle shuffle right kind of divide it in half like this and then well it's hard to do with a few cards and they do a little you know yeah thank you where would you like me to cut these right there okay cool now there's another way you can kind of mix up a deal cards it's called the down over deal goes like this one card goes down the next goes over one goes down one goes over down over down over down over until you're done okay but now it's your turn all right I'm gonna go two at a time and you're gonna tell me whether these first two cards should be dealt down or should be turned over okay down down what about these down down over down over all right now let's go four at a time okay one two three four should these go down or over over what about these four down death and the final two now we can do this as long as as often as you want I can keep doing this we can do - we can do four at a time how do you feel let's do it once more and do two at a time two time okay down down down down you're getting faster buff down okay all right now I'm going to make two little piles like a little book all right one okay now Vanessa should I close this book like this or like this all right there we go now you had completely free choice here mm-hmm you made a lot of different decisions somewhat arbitrarily but yet like oil and water the black and the red cards have separated and only the Reds have face-up I knew this would happen but it's not because of magic it's because of math here's how it works the red cards all start out facedown the black cards face-up no amount of cutting or rippling will change the fact that Reds are facedown and blacks are face-up but when you down over Deal you cause every other card to have its original facing direction reversed down over down over down over down over down over this alternating sequence is maintained through any number of cuts because it's cyclical this is the key to the trick because when we alternatively deal cards into book halves one pile will contain cards reversed by the down over deal the other cards not reversed depending on how you close the book either all reversed carts are unrehearsed or all unrehearsed cards are reversed either way the original division by color comes back the swaps missed things up by introducing an illusion of control but in reality the swaps are what are known as a hummer deal a swap flips the facing direction of the cards you flip but simultaneously moves them to the correct category preserving the sequence as long as you swap an even number of cards the facing direction of all of them is reversed but the positions they wind up in are still correct according to the sequence finally let's end with something popular that also sounds cool on the surface until you go deeper understand it better and it winds up being well even cooler the number of different ways 52 cards can be arranged there's obviously a lot the top card could be any card so there's 52 different possibilities any remaining cards can follow so there are 51 possibilities for the next card 54 the third spot 49 for the fourth and so on multiplying all of these numbers together gives us how many unique ways fifty-two cards can be arranged 52 factorial is a gigantic amount so large in fact every time you shuffle a deck of cards well smoosh it for a few minutes or riffle at seven or more times chances are you have put those playing cards into an order that they have never been in in the entire history of cards or humans or the universe seriously this is because 52 factorial is eight point zero six five eight times ten to the 67th in comparison the observable universe is only about 10 to the 18th seconds old even if you had been properly shuffling a deck every single second since the universe began 13.7 billion years ago you still to this day wouldn't have even come close to assembling every arrangement possible but even that doesn't paint the whole picture of just how big 52 factorial is scotch appeal wrote what in my opinion are some of the most mind-boggling visualizations of the size of 52 factorial imagine setting a timer to count down 52 factorial seconds while the timer runs stand on the equator and wait one billion years after a billion years have passed take a single step forward and then wait another billion years before taking a second step and so on once you have walked all the way around the earth take a single drop of water out of the Pacific Ocean that's five hundredths of a milliliter and set it aside now continue walking at a rate of one step every billion years removing one drop after every journey around the entire Earth and by the time the Pacific Ocean is completely empty put a single sheet of paper on the ground refill the ocean and keep going until the stack of paper reaches the Sun at that point how many seconds will be left on the timer will it be zero a few hundred a few billion no there will still be eight times ten to the sixty seven seconds left you have barely made a dent if you start all over again and do that whole thing a thousand more times you will only be 1/3 of the way done luckily Scott has a great idea for how to pass the rest of the time if you're bored of paper and water and walking instead he says deal yourself five cards every billion years when you finally deal yourself a royal flush buy a lottery ticket if the ticket wins the lottery throw a single grain of sand into the Grand Canyon as soon as the Grand Canyon is completely full of sand remove one ounce of rock that's about 28 grams from Mount Everest by the time Mount Everest is leveled take a look at the clock this is what will be left do the whole Royal Flush lottery ticket Grand Canyon Mount Everest thing 256 more times and then and only then will your timer have reached zero that is how big 52 factorial is it's pretty big but now think about this the number of possible people the number of different humans there could be is well even larger what that means is that even though you will probably die most people including possibly the smartest or funniest or most annoying possible person won't even get to die like you do they won't even get to be born so I'm glad you were and as always thanks for watching

Channel: Vsauce

Views: 8,726,134

Rating: 4.9516459 out of 5

Keywords: math, maths, magic, card trick, learn, vsauce, michael stevens, michael, stevens, education, counting, william shakespeare, psalm 46, bible, king james bible, sequence, science, psychology, research, study

Id: ObiqJzfyACM

Length: 19min 31sec (1171 seconds)

Published: Fri Feb 05 2016

Reddit Comments

[u/TheClouse]

plus a few bonus self-working tricks if you watch the whole thing.

This is a great way to explain to your audience how random a shuffle is.

Obviously doing a 7min explanation isn't practical, but you can gain a lot of knowledge and paraphrase it... "If every human that ever existed could shuffle a deck of cards once a second... and had been doing so since the beginning of time... we still wouldn't have two shuffles that matched exactly..."

Then maybe go into a routine where you guys match cards.

[u/resorcinarene]

I'm having an existential crisis after watching this.

[u/betteraskpat]

Mike Powers delves into 52! in his new book “Tesseract”. He also explains what the tesseract is by cross-referencing Flatworld and Sphereworld. And if that doesn’t beat all, he includes A LOT of dynamite close-up material (not just cards!) that’s well within reach of any serious magician.

[u/aghawa]

Always the perfect explanation for ACAAN, OOTW and other “coincidence” effects!