Vsauce -- Kevin here, with a game you can always win.
Grab a friend and take turns counting using numbers from 1 to 10.
The first person to get to 100 wins.
Which can always be you.
To demonstrate, let’s have a friendly game between me and uhhh…
I guess..
I’ll just play with myself.
Hey me!
It’s good to see me.
Am I ready to play?
Yes.
Yes I am.
Okay, add numbers between 1 and 10 and the person who gets to 100 first wins.
I’ll write the sum in between mes so you can follow along.
I’ll start the game.
1
7
4
9
2
10
1
You’re just picking numbers randomly like me, right?
Uhh… yeahhh...
Okay cool, 3
8
3
5
6
Seriously, though.
We both have 50/50 odds of getting to 100.
Um.
Sure!
Alright.
4
7
9
2
Uh. Wait.
89?
If I play 1, that’s 90 and you can play 10 and win and… if I play 10 that’s 99
and you can play 1 and win.
Oh, weird.
I, uh, must’ve gotten lucky.
Yeah, right.
5.
[94]
6! 100, I WIN!
Every.
Time.
Here’s how.
Uhh..you can go now, me.
Uh, fine.
Well, I want a rematch later.
I guess I’ll just uhhh...see ya in the mirror?
Yeahhhhh.
Okay, now let’s play a match between me and you and I’ll explain along the way how
it works.
The easiest way for me to know I’m going to win every time is for me to go first and
start with the number 1.
No matter what number you add next, 5, 3, 8, whatever it is, I just subtract your number
from 11 to get my next play.
So if I start with 1 and if you say 4, then our sum is 5.
To get my next number, I do 11 minus 4, which equals 7, so 7 is my next play: 7 plus 5 gives
our game a new total of 12. Then you say...
8. Cool.
8 plus 12 gives our game a new total of 20.
I do 11 minus your 8, which is 3.
So, I play 3 and our new total is 23.
As long as I stick to this strategy I’m guaranteed to be the first to 100.
This is why it works.
Regardless of your number, mine will always result in a sum that’s part of an arithmetic
series separated by elevens: 1, 12, 23, 34, 45, 56, 67, 78, 89, and… then finally 100.
Since the highest number you can play in the game is 10 -- if I separate your moves by
eleven...
I rule the world!
Or at least… this simple number game.
Gotta start somewhere.
When you play this with your friend can they ever win?
Yes.
If you mess up the series.
While going first, playing 1, and subtracting every opponent’s number by 11 is the easiest
way to stick with the series, after a few rounds you might want to switch things up
to prevent them from figuring out your strategy.
You could do that by starting with a number other than 1 and choosing wrong numbers until
later in the game when you land on a number in the arithmetic series like 67 or 78.
So if you’re playing with a friend and you pick random numbers until the sum is 65, then
you could play 2 to bring your total to 67 and then you’re right on track.
From there just subtract your friend’s numbers every single time by 11 and you will be the
first to 100.
To players who don’t realize that the optimal strategy depends on that series, this game
seems like a game of chance -- but once you know the series, you have the secret number
knowledge.
But if adding up to 100 isn’t your cup of tea you could try a similar game with matchsticks.
These are Inq’s Durable Match-Like Puzzle Sticks that come in the new Curiosity Box
that is out right now.
It’s packed with a booklet that features 22 different puzzles that you can try to solve
yourself when you get these matchsticks.
But for right now we’ll use the matchsticks to visualize another game you can always win.
Alright.
You have eleven matchsticks spread out on a table.
You and a friend take turns removing either 1, 2 or 3 matchsticks and the player who picks
up the last match loses.
If you go first, you can always win this game.
Since you don’t want to pick up that last match, let’s work backwards to uncover the
secret.
We’ll play a game between Mr. T and Skeletor because… why not?.
Alright, let’s skip to the end of the game to workout the winning strategy.
If it’s Skeletor’s turn and Mr T. leaves Skeletor with 2, 3, or 4 matches, Skeletor
can leave T. with the final losing match.
So, T. will want to make sure he leaves Skeletor with 5, to guarantee that Mr. T keeps his
jewelry.
Here’s why.
If there are 2 matches left, then Skeletor just takes 1 and leaves Mr. T. with the final
losing match.
If there are 3 matches left, then Skeletor takes 2, Mr. T. loses.
And if there are 4, then Skeletor takes 3, then Mr. T loses again.
But if T leaves 5, no matter what Skeletor plays next -- 3, 2, or 1 -- T can make a winning
move and pity the fool accordingly.
The arithmetic series that will rig the game for T is separated by 4’s…
1, then 5, and then…
9.
Mr. T. will want to leave Skeletor with 9 matches to make sure that ALL his plays match
up with the series.
If Skeletor and T start with 11 matches and T goes first, that means his initial play
will be to remove 2.
Since each player can only remove a maximum of 3 matches per turn, you can dominate the
game every time by going first and making moves that stick to this series.
… which is why the game works with 20 matches, too.
To guarantee a win with 20 matches on the table, your first play should be to remove
3 to land you at 17.
So let’s say T. starts and removes 3 to get to 17.
No matter what Skeletor does, T just needs to get to 13 to stay on course to win.
So, if Skeletor removes 2 to get to 15, then Mr. T. just needs to remove 2 to get to 13.
Now T. needs to get down to 9.
So, if Skeletor removes 3 to get to 10, then Mr. T. just has to remove 1 match to get to
9.
The next milestone is 5, so if Skeletor removes 1 to get to 8, then Mr. T can remove 3 to
get to 5.
And now it’s officially over.
Skeleton can’t do anything -- 1, 2, 3, it doesn’t matter.
Mr. T. is leaving Skeletor with the final losing match.
Let’s see if it’ll fit in his hand.
What about this one?
Good enough.
Both the counting game and this matchstick one are referred to as “Nim-like,” because
they’re conceptually similar to a game from
ancient times that evolved into what mathematicians
now call Nim.
Players take turns removing objects from heaps or stacks, and the player to remove the last
object loses… but it gets a lot more complex than sticking to a basic arithmetic series.
We like complex.
Humans have been inventing brain-teasing games to pass the time, sharpen their minds, and
extend collective knowledge for as long as recorded history shows.
One of the oldest games we know about is Senet, with board pieces from Ancient Egypt dating
back over 5,000 years.
About the same time humans were inventing the precursors to our modern written language
systems, we were developing number games to
occupy ourselves and tease out a better understanding
of the quantifiable world.
Creating artificial challenges -- and then out-thinking them -- is a way we exercise
our minds.
By discovering the hidden patterns that govern reality, whether we’re just practicing with
a matchstick game or unwinding the great scientific
challenges of our times, we’re engaging
in an integral part of what makes us -- us.
Even those of us who use Skeletor to explain math games.
And as always - thanks for watching.