The Dot Game That Breaks Your Brain

published on July 9, 2020

Vsauce Kevin here with a game you can'tpossibly comprehend really it's too hardfor you your brain can't take it lookI'll show you that's it are you sweatingyet you should be real quick huge thanksto expressvpn for sponsoring this videoand supporting Vsauce – if your device

Is unsecured you're gonna want to getexpressvpn to take care of that I'llexplain more later but first let'sexplain our dots all right as you stareinto these dots your brain starts toshort-circuit doesn't it no why would it

I mean it's just two dots I could drawout all the possible moves for a gamethis simple look I'll show youokay my award-winning handwriting asidethis was a lot more complicated than Ithought it was gonna be and the thing is

As its scales analyzing what appears tobe the simplest game in the worlddoesn't just break your brain Computerscan't even crunch the possibilitieshere's how it works the game of sproutstarts with any number of dots placed

Anywhere the boundaries of the gameboard are limitless so put the dotswherever you want we'll play with twodots but you can't just play withyourself you need an opponent yes yes aworthy adversary you need let's go over

The three rules of sprouts first aplayer draws a line from one dot toanother or from one dot back to itselflines can be curved or they can bestraight they just can't cross anotherline or themselves when you draw a line

You get to place a new dot anywhere onthat new line and in sprouts no dot canhave more than three lines coming fromit or going to it once a dot has threelinesit's an unplayable dead dot the winner

Of sprouts is the last person to draw aline or to put it another way the playerwho can't draw another line loses okaynow my friend and I will play a two dotgame of sprouts go first I will allright Yoda dude okay hang on all right

Finejust go right all right great job yougot to make sure you draw a new dot onthe line yes yes yes invented this gameI did spouts trained many Jedi mindshydrants of years hundreds of years no

No no no sprouts was created in 1967 byCambridge mathematicians John Conway andMichael Martin airsdartz lead two lines lines lead two dotssprout just a path to the light side ofthe end I just won huh alive this dot

Still is yeah yeah but you can't connectit to anything look dead dead dead deadand you can't draw a line to get to thisone explain why I lost he'll must allright the first player can always lose ato dot game against a perfect opponent

Because even though it's complex yourbrain can analyze two dots sprouts Imean you could literally just memorizethis whole game tree chart to makeexactly the right move as player torendering player one helpless player to

Can engineer the two games so that itends on a fourth move win for them butConway and Patterson figured out whenthe game has to endcheck it out they discovered that a gameof sprouts must be completed by three n

Minus one moves where n equals thenumber of starting dots so that means atwo dot game is concluded in no morethan five moves because three times twominus one equals five so problem solvesright no why because the game can play

Out in so many different ways what'sinteresting is that player one actuallyhas 11 ways of winningcompared to player two having only sixit's just that if player 2 knows exactlywhat they're doing they can always

Facilitate one of theirsix winning outcomes what's amazing tome about sprouts is this is all justwith two dots as soon as we add a thirddot to the game become more difficult toanalyze than tic-tac-toe it does adding

A third dot at the beginning means thatwe could have up to eight moves todetermine a winner since three timesthree minus one equals eight but we havemore possible moves to start it isn'thard to figure out how many

Possibilities we begin with it's just ntimes n plus 1 over 2 so here we haveour number of dots at start and numberof initial possible moves n times n plus1 over 2 and number of moves todetermine a winner that's 3n minus 1 so

If you have two dots to start the gamethe initial possible moves would bethree with three dots to start thatjumps to six for four it's 10 for 5 it's15 and so on now that we know thiswhat's the guaranteed strategy for

Winning every time there isn't onebecause since the game can develop in somany different ways especially once youstart playing with four or five dotsplayers will have to constantlyreanalyze and adapt their moves to force

Their opponent into a loss you need tofactor in which dots are still live andwhich ones are dead you need to forceyour opponent into bad moves andeventually no moves and allthere's just no formula for this adapt

And overcomeyou must what we do know kind of is whocan win the first real glimpse intodominant sprout ology came from DennisMollison a professor of appliedprobability at heriot-watt University

Conway bet Mollison 10 shillings beforethe 1971 decimalisation of the britishmonetary system and equivalent to alittle under 10 bucks today that hecouldn't complete a full analysis of asix sprout game within a month well he

Did and it only took 47pages not looking forward to pickingthose up mollison's analysis led to theconclusion that sprouts games with zeroone or two dots could always be won bythe second player games with three four

And five dots could always be won by thefirst player the second player canalways win with six dots but that'swhere the computational power of thehuman mind started to strain under theweight of the sprout they were just too

Many scenarios to compute wait how canyou have a game with zero dots well ifthere are zero dots the first playerwouldn't be able to draw a line so thesecond player wins one thing that'sreally weird about sprouts is you'd

Think that playing the game wouldvisually result in nothing but nearrandom lines and patterns but Conway andMollison unearthed something augs theycall this Fatah 'some the fundamentaltheorem of zeroeth order more abundant e

Which states that any sprouts game of ndots must last at least two and movesand if it lasts exactly two and movesthe final board will consist of one offive insect patterns louse beetlecockroach earwig and scorpion surrounded

By any number of lice scorpions arearachnids not insects but these guysdon't have no time for no biology andthat's Fattah Zim for you but this wasall fifty years ago how has sprout ologyprogressed since will it lay dormant for

Decades until carnegie mellon universityfired up its computers in 1990 usingsome of the most advanced processors ofthe error computer scientist DavidApplegate died Jacobson and DanielSlater were able to map sprouts

Conclusively up to 11 dots they foundthe same pattern 6 7 & 8 favored thesecond player 9 10 and 11 favored thefirst player there appears to be anendless three lawthree wind pattern with a cycle length

Of six dots in 2001 Bacardi and Luciopublished a new analysis technique forthe sprouts game that showed a simplerproof of sprouts to seven dots by handnow we're up to eleven so we're makingprogress on the pencil and paper front

But what about one thousand two hundredand seventy two dots or a billion dotswere not even close like really notclose julienne LeBlanc and Simone Enocreated a computer program called blobthat could calculate sprouts results

More efficiently and in 2011 they wereonly able to process up to forty fourdots consecutively their results were inline with Carnegie Mellon's cycle of sixbut the computational power and timerequired to get us to proving results

With say a million dots is way beyondour reachit's been over half a century sinceConway and Patterson were drinking teain the Cambridge math departments commonroom and playing around with inventing a

Simple pencil and paper based game theynoticed that the game was spreadingthroughout the department and then thecampus seeing students hunched overtables and spotting the discardedremnants of epic sprouts battles they

Stumbled on something so big and socomplex that the human mind can't fullyfathom it beyond a very limited pointand it all started by just connecting acouple of dots and as always thanks forwatching

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