[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:06.88,0:00:10.35,Default,,0000,0000,0000,,Your favorite band is great\Nat playing music,
Dialogue: 0,0:00:10.35,0:00:13.47,Default,,0000,0000,0000,,but not so great at being organized.
Dialogue: 0,0:00:13.47,0:00:16.26,Default,,0000,0000,0000,,They keep misplacing \Ntheir instruments on tour,
Dialogue: 0,0:00:16.26,0:00:18.76,Default,,0000,0000,0000,,and it's driving their manager mad.
Dialogue: 0,0:00:18.76,0:00:20.57,Default,,0000,0000,0000,,On the day of the big concert,
Dialogue: 0,0:00:20.57,0:00:22.88,Default,,0000,0000,0000,,the band wakes up \Nto find themselves tied up
Dialogue: 0,0:00:22.88,0:00:26.56,Default,,0000,0000,0000,,in a windowless, \Nsoundproof practice room.
Dialogue: 0,0:00:26.56,0:00:28.80,Default,,0000,0000,0000,,Their manager explains what's happening.
Dialogue: 0,0:00:28.80,0:00:32.03,Default,,0000,0000,0000,,Outside, there are ten large boxes.
Dialogue: 0,0:00:32.03,0:00:34.39,Default,,0000,0000,0000,,Each contains one of your instruments,
Dialogue: 0,0:00:34.39,0:00:39.21,Default,,0000,0000,0000,,but don't be fooled by the pictures -\Nthey've been randomly placed.
Dialogue: 0,0:00:39.21,0:00:42.75,Default,,0000,0000,0000,,I'm going to let you out one at a time.
Dialogue: 0,0:00:42.75,0:00:46.24,Default,,0000,0000,0000,,While you're outside, you can look\Ninside any five boxes
Dialogue: 0,0:00:46.24,0:00:50.23,Default,,0000,0000,0000,,before security\Ntakes you back to the tour bus.
Dialogue: 0,0:00:50.23,0:00:51.90,Default,,0000,0000,0000,,You can't touch the instruments
Dialogue: 0,0:00:51.90,0:00:55.80,Default,,0000,0000,0000,,or in any way communicate what you find\Nto the others.
Dialogue: 0,0:00:55.80,0:00:59.57,Default,,0000,0000,0000,,No marking the boxes, shouting, nothing.
Dialogue: 0,0:00:59.57,0:01:02.14,Default,,0000,0000,0000,,If each one of you \Ncan find your own instrument,
Dialogue: 0,0:01:02.14,0:01:04.00,Default,,0000,0000,0000,,then you can play tonight.
Dialogue: 0,0:01:04.00,0:01:07.00,Default,,0000,0000,0000,,Otherwise, the label is dropping you.
Dialogue: 0,0:01:07.00,0:01:10.79,Default,,0000,0000,0000,,You have three minutes to think about it\Nbefore we start.
Dialogue: 0,0:01:10.79,0:01:12.33,Default,,0000,0000,0000,,The band is in despair.
Dialogue: 0,0:01:12.33,0:01:16.85,Default,,0000,0000,0000,,After all, each musician only has a 50%\Nchance of finding their instrument
Dialogue: 0,0:01:16.85,0:01:19.33,Default,,0000,0000,0000,,by picking five random boxes.
Dialogue: 0,0:01:19.33,0:01:22.65,Default,,0000,0000,0000,,And the chances that all ten will succeed\Nare even lower -
Dialogue: 0,0:01:22.65,0:01:25.77,Default,,0000,0000,0000,,just 1 in 1024.
Dialogue: 0,0:01:25.77,0:01:28.46,Default,,0000,0000,0000,,But suddenly, the drummer comes up\Nwith a valid strategy
Dialogue: 0,0:01:28.46,0:01:33.04,Default,,0000,0000,0000,,that has a better than 35% chance\Nof working.
Dialogue: 0,0:01:33.04,0:01:35.74,Default,,0000,0000,0000,,Can you figure out what it was?
Dialogue: 0,0:01:35.74,0:01:43.76,Default,,0000,0000,0000,,Pause the video on the next screen\Nif you want to figure it out for yourself!
Dialogue: 0,0:01:43.76,0:01:45.21,Default,,0000,0000,0000,,Answer in: 3
Dialogue: 0,0:01:45.21,0:01:46.22,Default,,0000,0000,0000,,Answer in: 2
Dialogue: 0,0:01:46.22,0:01:47.64,Default,,0000,0000,0000,,Answer in: 1
Dialogue: 0,0:01:47.64,0:01:49.28,Default,,0000,0000,0000,,Here's what the drummer said:
Dialogue: 0,0:01:49.28,0:01:53.06,Default,,0000,0000,0000,,Everyone first open the box\Nwith the picture of your instrument.
Dialogue: 0,0:01:53.06,0:01:55.38,Default,,0000,0000,0000,,If your instrument is inside, you're done.
Dialogue: 0,0:01:55.38,0:01:57.56,Default,,0000,0000,0000,,Otherwise, look at whatever's in there,
Dialogue: 0,0:01:57.56,0:02:00.44,Default,,0000,0000,0000,,and then open the box \Nwith that picture on it.
Dialogue: 0,0:02:00.44,0:02:03.50,Default,,0000,0000,0000,,Keep going that way until you find\Nyour instrument.
Dialogue: 0,0:02:03.50,0:02:05.16,Default,,0000,0000,0000,,The bandmates are skeptical,
Dialogue: 0,0:02:05.16,0:02:08.22,Default,,0000,0000,0000,,but amazingly enough,\Nthey all find what they need.
Dialogue: 0,0:02:08.22,0:02:12.10,Default,,0000,0000,0000,,And a few hours later, they're playing\Nto thousands of adoring fans.
Dialogue: 0,0:02:12.10,0:02:14.66,Default,,0000,0000,0000,,So why did the drummer's strategy work?
Dialogue: 0,0:02:14.66,0:02:17.28,Default,,0000,0000,0000,,Each musician follows a linked sequence
Dialogue: 0,0:02:17.28,0:02:20.94,Default,,0000,0000,0000,,that starts with the box whose outside\Nmatches their instrument
Dialogue: 0,0:02:20.94,0:02:25.00,Default,,0000,0000,0000,,and ends with the box \Nactually containing it.
Dialogue: 0,0:02:25.00,0:02:28.50,Default,,0000,0000,0000,,Note that if they kept going,\Nthat would lead them back to the start,
Dialogue: 0,0:02:28.50,0:02:30.11,Default,,0000,0000,0000,,so this is a loop.
Dialogue: 0,0:02:30.11,0:02:32.92,Default,,0000,0000,0000,,For example, if the boxes \Nare arranged like so,
Dialogue: 0,0:02:32.92,0:02:36.03,Default,,0000,0000,0000,,the singer would open the first box\Nto find the drums,
Dialogue: 0,0:02:36.03,0:02:38.26,Default,,0000,0000,0000,,go to the eighth box to find the bass,
Dialogue: 0,0:02:38.26,0:02:41.06,Default,,0000,0000,0000,,and find her microphone in the third box,
Dialogue: 0,0:02:41.06,0:02:43.89,Default,,0000,0000,0000,,which would point back to the first.
Dialogue: 0,0:02:43.89,0:02:46.48,Default,,0000,0000,0000,,This works much better \Nthan random guessing
Dialogue: 0,0:02:46.48,0:02:49.82,Default,,0000,0000,0000,,because by starting with the box \Nwith the picture of their instrument,
Dialogue: 0,0:02:49.82,0:02:54.44,Default,,0000,0000,0000,,each musician restricts their search to\Nthe loop that contains their instrument,
Dialogue: 0,0:02:54.44,0:02:57.84,Default,,0000,0000,0000,,and there are decent odds, about 35%,\N
Dialogue: 0,0:02:57.84,0:03:02.28,Default,,0000,0000,0000,,that all of the loops \Nwill be of length five or less.
Dialogue: 0,0:03:02.28,0:03:03.92,Default,,0000,0000,0000,,How do we calculate those odds?
Dialogue: 0,0:03:03.92,0:03:07.73,Default,,0000,0000,0000,,For the sake of simplicity, \Nwe'll demonstrate with a simplified case,
Dialogue: 0,0:03:07.73,0:03:13.49,Default,,0000,0000,0000,,four instruments and no more than \Ntwo guesses allowed for each musician.
Dialogue: 0,0:03:13.49,0:03:16.33,Default,,0000,0000,0000,,Let's start by finding \Nthe odds of failure,
Dialogue: 0,0:03:16.33,0:03:19.68,Default,,0000,0000,0000,,the chance that someone will need\Nto open three or four boxes
Dialogue: 0,0:03:19.68,0:03:21.72,Default,,0000,0000,0000,,before they find their instrument.
Dialogue: 0,0:03:21.72,0:03:24.32,Default,,0000,0000,0000,,There are six distinct four-box loops.
Dialogue: 0,0:03:24.32,0:03:27.46,Default,,0000,0000,0000,,One fun way to count them \Nis to make a square,
Dialogue: 0,0:03:27.46,0:03:29.63,Default,,0000,0000,0000,,put an instrument at each corner,
Dialogue: 0,0:03:29.63,0:03:31.79,Default,,0000,0000,0000,,and draw the diagonals.
Dialogue: 0,0:03:31.79,0:03:34.62,Default,,0000,0000,0000,,See how many unique loops you can find,
Dialogue: 0,0:03:34.62,0:03:38.11,Default,,0000,0000,0000,,and keep in mind that these two \Nare considered the same,
Dialogue: 0,0:03:38.11,0:03:40.14,Default,,0000,0000,0000,,they just start at different points.
Dialogue: 0,0:03:40.14,0:03:42.40,Default,,0000,0000,0000,,These two, however, are different.
Dialogue: 0,0:03:42.40,0:03:47.10,Default,,0000,0000,0000,,We can visualize the eight distinct\Nthree-box loops using triangles.
Dialogue: 0,0:03:47.10,0:03:49.45,Default,,0000,0000,0000,,You'll find four possible triangles\N
Dialogue: 0,0:03:49.45,0:03:52.02,Default,,0000,0000,0000,,depending on which instrument \Nyou leave out,
Dialogue: 0,0:03:52.02,0:03:54.95,Default,,0000,0000,0000,,and two distinct paths on each.
Dialogue: 0,0:03:54.95,0:03:58.33,Default,,0000,0000,0000,,So of the 24 possible \Ncombinations of boxes,
Dialogue: 0,0:03:58.33,0:04:00.80,Default,,0000,0000,0000,,there are 14 that lead to faliure,
Dialogue: 0,0:04:00.80,0:04:03.40,Default,,0000,0000,0000,,and ten that result in success.
Dialogue: 0,0:04:03.40,0:04:08.47,Default,,0000,0000,0000,,That computational strategy works for any\Neven number of musicians,
Dialogue: 0,0:04:08.47,0:04:09.85,Default,,0000,0000,0000,,but if you want a shortcut,
Dialogue: 0,0:04:09.85,0:04:13.81,Default,,0000,0000,0000,,it generalizes to a handy equation.
Dialogue: 0,0:04:13.81,0:04:18.11,Default,,0000,0000,0000,,Plug in ten musicians, \Nand we get odds of about 35%.
Dialogue: 0,0:04:18.11,0:04:20.43,Default,,0000,0000,0000,,What if there were 1,000 musicians?
Dialogue: 0,0:04:20.43,0:04:21.80,Default,,0000,0000,0000,,1,000,000?
Dialogue: 0,0:04:21.80,0:04:26.20,Default,,0000,0000,0000,,As n increases, \Nthe odds approach about 30%.
Dialogue: 0,0:04:26.20,0:04:32.23,Default,,0000,0000,0000,,Not a guarantee, but with a bit of\Nmusician's luck, it's far from hopeless.
Dialogue: 0,0:04:32.23,0:04:35.92,Default,,0000,0000,0000,,Hi everybody, if you liked this riddle,\Ntry solving these two.