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.
Hey, the brand new Curiosity Box which includes the matchstick game and a bunch of other awesome hand-picked, designed and developed science toys is available right now. Michael, Jake and I created this subscription box to bring physical Vsauce to your doorstep. So, check out the link below, it’s CuriosityBox.com, to secure yours right now. I’m going to stay here and, uh, try and figure out some of these puzzles.
Alright. Move one match to make a square. Uhh. Um. SQUARE.