Can you solve the prisoner boxes riddle? - Yossi Elran
-
0:07 - 0:10Your favorite band is great
at playing music, -
0:10 - 0:13but not so great at being organized.
-
0:13 - 0:16They keep misplacing
their instruments on tour, -
0:16 - 0:19and it's driving their manager mad.
-
0:19 - 0:21On the day of the big concert,
-
0:21 - 0:23the band wakes up
to find themselves tied up -
0:23 - 0:27in a windowless,
soundproof practice room. -
0:27 - 0:29Their manager explains what's happening.
-
0:29 - 0:32Outside, there are ten large boxes.
-
0:32 - 0:34Each contains one of your instruments,
-
0:34 - 0:39but don't be fooled by the pictures -
they've been randomly placed. -
0:39 - 0:43I'm going to let you out one at a time.
-
0:43 - 0:46While you're outside, you can look
inside any five boxes -
0:46 - 0:50before security
takes you back to the tour bus. -
0:50 - 0:52You can't touch the instruments
-
0:52 - 0:56or in any way communicate what you find
to the others. -
0:56 - 1:00No marking the boxes, shouting, nothing.
-
1:00 - 1:02If each one of you
can find your own instrument, -
1:02 - 1:04then you can play tonight.
-
1:04 - 1:07Otherwise, the label is dropping you.
-
1:07 - 1:11You have three minutes to think about it
before we start. -
1:11 - 1:12The band is in despair.
-
1:12 - 1:17After all, each musician only has a 50%
chance of finding their instrument -
1:17 - 1:19by picking five random boxes.
-
1:19 - 1:23And the chances that all ten will succeed
are even lower - -
1:23 - 1:26just 1 in 1024.
-
1:26 - 1:28But suddenly, the drummer comes up
with a valid strategy -
1:28 - 1:33that has a better than 35% chance
of working. -
1:33 - 1:36Can you figure out what it was?
-
1:36 - 1:44Pause the video on the next screen
if you want to figure it out for yourself! -
1:44 - 1:45Answer in: 3
-
1:45 - 1:46Answer in: 2
-
1:46 - 1:48Answer in: 1
-
1:48 - 1:49Here's what the drummer said:
-
1:49 - 1:53Everyone first open the box
with the picture of your instrument. -
1:53 - 1:55If your instrument is inside, you're done.
-
1:55 - 1:58Otherwise, look at whatever's in there,
-
1:58 - 2:00and then open the box
with that picture on it. -
2:00 - 2:04Keep going that way until you find
your instrument. -
2:04 - 2:05The bandmates are skeptical,
-
2:05 - 2:08but amazingly enough,
they all find what they need. -
2:08 - 2:12And a few hours later, they're playing
to thousands of adoring fans. -
2:12 - 2:15So why did the drummer's strategy work?
-
2:15 - 2:17Each musician follows a linked sequence
-
2:17 - 2:21that starts with the box whose outside
matches their instrument -
2:21 - 2:25and ends with the box
actually containing it. -
2:25 - 2:29Note that if they kept going,
that would lead them back to the start, -
2:29 - 2:30so this is a loop.
-
2:30 - 2:33For example, if the boxes
are arranged like so, -
2:33 - 2:36the singer would open the first box
to find the drums, -
2:36 - 2:38go to the eighth box to find the bass,
-
2:38 - 2:41and find her microphone in the third box,
-
2:41 - 2:44which would point back to the first.
-
2:44 - 2:46This works much better
than random guessing -
2:46 - 2:50because by starting with the box
with the picture of their instrument, -
2:50 - 2:54each musician restricts their search to
the loop that contains their instrument, -
2:54 - 2:58and there are decent odds, about 35%,
-
2:58 - 3:02that all of the loops
will be of length five or less. -
3:02 - 3:04How do we calculate those odds?
-
3:04 - 3:08For the sake of simplicity,
we'll demonstrate with a simplified case, -
3:08 - 3:13four instruments and no more than
two guesses allowed for each musician. -
3:13 - 3:16Let's start by finding
the odds of failure, -
3:16 - 3:20the chance that someone will need
to open three or four boxes -
3:20 - 3:22before they find their instrument.
-
3:22 - 3:24There are six distinct four-box loops.
-
3:24 - 3:27One fun way to count them
is to make a square, -
3:27 - 3:30put an instrument at each corner,
-
3:30 - 3:32and draw the diagonals.
-
3:32 - 3:35See how many unique loops you can find,
-
3:35 - 3:38and keep in mind that these two
are considered the same, -
3:38 - 3:40they just start at different points.
-
3:40 - 3:42These two, however, are different.
-
3:42 - 3:47We can visualize the eight distinct
three-box loops using triangles. -
3:47 - 3:49You'll find four possible triangles
-
3:49 - 3:52depending on which instrument
you leave out, -
3:52 - 3:55and two distinct paths on each.
-
3:55 - 3:58So of the 24 possible
combinations of boxes, -
3:58 - 4:01there are 14 that lead to faliure,
-
4:01 - 4:03and ten that result in success.
-
4:03 - 4:08That computational strategy works for any
even number of musicians, -
4:08 - 4:10but if you want a shortcut,
-
4:10 - 4:14it generalizes to a handy equation.
-
4:14 - 4:18Plug in ten musicians,
and we get odds of about 35%. -
4:18 - 4:20What if there were 1,000 musicians?
-
4:20 - 4:221,000,000?
-
4:22 - 4:26As n increases,
the odds approach about 30%. -
4:26 - 4:32Not a guarantee, but with a bit of
musician's luck, it's far from hopeless. -
4:32 - 4:36Hi everybody, if you liked this riddle,
try solving these two.
- Title:
- Can you solve the prisoner boxes riddle? - Yossi Elran
- Speaker:
- Yossi Elran
- Description:
-
View full lesson: http://ed.ted.com/lessons/can-you-solve-the-prisoner-boxes-riddle-yossi-elran
Your favorite band is great at playing music...but not so great at being organized. They keep misplacing their instruments on tour, and it’s driving their manager mad. Can you solve the brain-numbing riddle their manager assigns them and make sure the band stays on their label? Yossi Elran shows how.
Lesson by Yossi Elran, animation by Artrake Studio.
- Video Language:
- English
- Team:
- closed TED
- Project:
- TED-Ed
- Duration:
- 04:52
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