Fix moser-reboot transcript

2023/moser-reboot/english/captions.srt

@@ -751,7 +751,7 @@ and when all the dust settles, the answer to the question is 1 plus n choose 2 p
 choose 4.
 
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-00:11:16,319 --> 00:11:19,380
+00:11:16,320 --> 00:11:19,380
 On the one hand, you're done, you answered the question.
 
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2023/moser-reboot/english/transcript.txt

@@ -107,4 +107,11 @@ Same deal for all the numbers that are 5 or less.
 When you situate this formula inside Pascal's triangle, and you relate it to the previous row, what you're doing is adding up the entirety of that previous row. 
 The point at which this breaks is for n equals 6, because in that case, when you relate this to the previous row, adding up the first 5 elements of that row, it doesn't cover the whole thing. 
 It falls short specifically by just 1, which is why we miss the power of 2, and why it falls short specifically by just 1. 
-Also, notice what happens when we plug in n equals 10.
+Also, notice what happens when we plug in n equals 10. 
+Looking down at the 10th row, and relating those terms to the previous one, adding the first 5 elements of the 9th row is exactly half of that row, and because the triangle is symmetric, this means that when you add them up, you get exactly half of a power of 2, which itself of course is another power of 2. 
+And as a challenge problem for you, I don't actually know if this is the last time you'll ever see a power of 2. 
+Maybe one of you out there who's cleverer with diaphantine equations than I am can come up with some clever proof. 
+Stepping back, to summarize, we started by counting the total number of chords and the total number of intersection points, which, by thinking about the right associations, is the same as computing n choose 2 and n choose 4. 
+Bringing in Euler's formula, this let us get an exact closed form expression for the number of regions inside the circle. 
+Then connecting that with Pascal's triangle gives us a very visceral connection with the powers of 2 and why they break when they do. 
+So yes, Moser's circle problem is a cautionary tale about being wary of patterns without proof, but at a deeper level, it also tells us that what's sometimes chalked up to be coincidence still leaves room for satisfying understandings.