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I’ll eventually write a long series of posts on Minecraft, but for the moment, I’m going to start at the end, because, well, we got to the end game a little earlier than one normally would.

The reasons that we got to the End Game are that, first, ever since we’ve been doing Minecraft, which isn’t actually very long (although it’s a long story, for another time), Leo has been obsessively studying, and obsessively telling me all about, the End Game. In the Minecraft end game you: Discover a Stronghold, Find an End Portal, Get into the Ender World (or whatever it’s called), Kill the Ender Dragon, and Solve a slightly complex mechanical puzzle, all while trying not to die in various stupid ways that you can die in Minecraft. Once you’ve done all this you’ve basically won Minecraft, and are reworded with an actually quite moving and lovely (albeit long) poem. I was hoping that by going through this with Leo I would be rewarded by his no longer bothering me about all this…. (We’ll see!)

So, anyway, in keeping with my theory of math education, where everything fun is mathematical, and everything mathematical is fun, we turned the search for the Stronghold into a math problem, which is described by the scribbling on this whiteboard:

To make a long story short, the Strongholds are randomly located slightly underground along three axes 120 degrees apart at between 640 and 1550 (or so) blocks from the world center. In order to find them efficiently, you can thrown an “Eye of the Ender” up in the air, and it travels for a bit in the direction of one of the Strongholds.

Finding the stronghold along a given line is analogous to the children’s game where someone thinks of a number in a range, say 1-100, and then the magician asks if it’s say, 50, and if not, whether it’s higher or lower, at which point you go to 1/2 way between (25, for lower, for example).

In mathematics, this is approximately Newton’s Method — or at least I always think of Newton’s Method when I think of this game. In algorithms design it’s called binary search, and it’s easy to prove that it’s big-O log-base-2 in the size of the ordered list, so, in the case of 100, it’s O(Log2(100)) or about 6-7 tries, max.

Leo and I calculated all this, and figured out where the middle of the line connecting the center (where we started out) between 640 and 1550 was. (That’s what all the trig is in the picture above — the line wasn’t exactly along the X or Y axis, so there’s some complexity.)

And then we started to search for the Stronghold, and lo-and-behold, it worked! In about 5 tosses of the Eye of The Ender we hit pay-dirt (literally — well, digitally, anyway), and upon drilling into the ground where we ended up, found a stronghold, played the end game, and won!

Now that he’s won, let’s see if Leo will come down from his Minecraft addiction! I have low hopes; He’s already made a list of the numerous other Minecraft challenges he wants to solve.