1
00:00:06,589 --> 00:00:08,249
You heard the traveler's tales,

2
00:00:08,249 --> 00:00:10,029
you followed the crumbling maps,

3
00:00:10,029 --> 00:00:12,899
and now, after a long and dangerous quest,

4
00:00:12,899 --> 00:00:16,024
you have some good news and some bad news.

5
00:00:16,024 --> 00:00:19,469
The good news is you've managed to locate
the legendary dungeon

6
00:00:19,469 --> 00:00:22,661
containing the stash 
of ancient Stygian coins

7
00:00:22,661 --> 00:00:25,595
and the eccentric wizard 
who owns the castle

8
00:00:25,595 --> 00:00:29,244
has even generously
agreed to let you have them.

9
00:00:29,244 --> 00:00:31,964
The bad news is that he's not
quite as generous

10
00:00:31,964 --> 00:00:37,703
about letting you leave the dungeon,
unless you solve his puzzle.

11
00:00:37,703 --> 00:00:39,683
The task sounds simple enough.

12
00:00:39,683 --> 00:00:44,284
Both faces of each coin bear
the fearsome scorpion crest,

13
00:00:44,284 --> 00:00:45,373
one in silver,

14
00:00:45,373 --> 00:00:47,113
one in gold.

15
00:00:47,113 --> 00:00:50,844
And all you have to do is separate them
into two piles

16
00:00:50,844 --> 00:00:56,294
so that each has the same number
of coins facing silver side up.

17
00:00:56,294 --> 00:01:00,305
You're about to begin when all
of the torches suddenly blow out

18
00:01:00,305 --> 00:01:02,943
and you're left in total darkness.

19
00:01:02,943 --> 00:01:05,022
There are hundreds 
of coins in front of you

20
00:01:05,022 --> 00:01:08,756
and each one feels the same on both sides.

21
00:01:08,756 --> 00:01:11,602
You try to remember 
where the silver-facing coins were,

22
00:01:11,602 --> 00:01:13,273
but it's hopeless.

23
00:01:13,273 --> 00:01:14,771
You've lost track.

24
00:01:14,771 --> 00:01:17,141
But you do know one thing for certain.

25
00:01:17,141 --> 00:01:18,612
When there was still light,

26
00:01:18,612 --> 00:01:23,382
you counted exactly 
20 silver-side-up coins in the pile.

27
00:01:23,382 --> 00:01:24,842
What can you do?

28
00:01:24,842 --> 00:01:29,437
Are you doomed to remain in the dungeon
with your newfound treasure forever?

29
00:01:29,437 --> 00:01:31,529
You're tempted to kick the pile of coins

30
00:01:31,529 --> 00:01:34,892
and curse the curiosity 
that brought you here.

31
00:01:34,892 --> 00:01:37,302
But at the last moment, you stop yourself.

32
00:01:37,302 --> 00:01:41,815
You just realized there's 
a surprisingly easy solution.

33
00:01:41,815 --> 00:01:43,463
What is it?

34
00:01:43,463 --> 00:01:46,763
Pause here if you want to figure
it out for yourself.

35
00:01:46,763 --> 00:01:47,793
Answer in: 3

36
00:01:47,793 --> 00:01:49,163
Answer in: 2

37
00:01:49,163 --> 00:01:50,893
Answer in: 1

38
00:01:50,893 --> 00:01:54,185
You carefully move aside 20 coins
one by one.

39
00:01:54,185 --> 00:01:57,775
It doesn't matter which ones:
any coins will do,

40
00:01:57,775 --> 00:02:00,323
and then flip each one of them over.

41
00:02:00,323 --> 00:02:02,063
That's all there is to it.

42
00:02:02,063 --> 00:02:04,505
Why does such a simple solution work?

43
00:02:04,505 --> 00:02:08,044
Well, it doesn't matter how many
coins there are to start with.

44
00:02:08,044 --> 00:02:12,924
What matters is that only 20
of the total are facing silver side up.

45
00:02:12,924 --> 00:02:15,503
When you take 20 coins in the darkness,

46
00:02:15,503 --> 00:02:19,088
you have no way of knowing how many
of these silver-facing coins

47
00:02:19,088 --> 00:02:21,947
have ended up in your new pile.

48
00:02:21,947 --> 00:02:24,274
But let's suppose you got 7 of them.

49
00:02:24,274 --> 00:02:27,715
This means that there are 13
silver-facing coins left

50
00:02:27,715 --> 00:02:29,606
in the original pile.

51
00:02:29,606 --> 00:02:33,215
It also means that the other
13 coins in your new pile

52
00:02:33,215 --> 00:02:35,796
are facing gold side up.

53
00:02:35,796 --> 00:02:40,500
So what happens when you flip
all of the coins in the new pile over?

54
00:02:40,500 --> 00:02:44,771
Seven gold-facing coins and
13 silver-facing coins

55
00:02:44,771 --> 00:02:47,995
to match the ones in the original pile.

56
00:02:47,995 --> 00:02:52,930
It turns out this works no matter how
many of the silver-facing coins you grab,

57
00:02:52,930 --> 00:02:55,930
whether it's all of them, 
a few, or none at all.

58
00:02:56,521 --> 00:03:00,515
That's because of what's known
as complementary events.

59
00:03:00,515 --> 00:03:04,227
We know that each coin only has
two possible options.

60
00:03:04,227 --> 00:03:08,476
If it's not facing silver side up,
it must be gold side up,

61
00:03:08,476 --> 00:03:09,847
and vice versa,

62
00:03:09,847 --> 00:03:12,117
and in any combination of 20 coins,

63
00:03:12,117 --> 00:03:15,289
the number of gold-facing
and silver-facing coins

64
00:03:15,289 --> 00:03:17,777
must add up to 20.

65
00:03:17,777 --> 00:03:20,927
We can prove this mathematically
using algebra.

66
00:03:20,927 --> 00:03:24,498
The number of silver-facing coins
remaining in the original pile

67
00:03:24,498 --> 00:03:29,167
will always be 20 minus
however many you moved to the new pile.

68
00:03:29,167 --> 00:03:32,517
And since your new pile also
has a total of 20 coins,

69
00:03:32,517 --> 00:03:35,257
its number of gold-facing coins will be

70
00:03:35,257 --> 00:03:39,487
20 minus the amount of 
silver-facing coins you moved.

71
00:03:39,487 --> 00:03:42,318
When all the coins in the new pile
are flipped,

72
00:03:42,318 --> 00:03:45,966
these gold-facing coins become
silver-facing coins,

73
00:03:45,966 --> 00:03:51,538
so now the number of silver-facing
coins in both piles is the same.

74
00:03:51,538 --> 00:03:55,159
The gate swings open
and you hurry away with your treasure

75
00:03:55,159 --> 00:03:57,268
before the wizard changes his mind.

76
00:03:57,268 --> 00:04:00,877
At the next crossroads, you flip
one of your hard-earned coins

77
00:04:00,880 --> 00:04:03,800
to determine the way 
to your next adventure.

78
00:04:03,800 --> 00:04:07,350
But before you go, we have another 
quick coin riddle for you –

79
00:04:07,350 --> 00:04:11,549
one that comes from this video 
sponsor’s excellent website.

80
00:04:11,549 --> 00:04:14,010
Here we have 8 arrangements of coins.

81
00:04:14,010 --> 00:04:18,790
You can flip over adjacent pairs of coins 
as many times as you like.

82
00:04:18,790 --> 00:04:22,919
A flip always changes gold to silver, 
and silver to gold.

83
00:04:22,919 --> 00:04:26,176
Can you figure out how to tell, 
at a glance,

84
00:04:26,176 --> 00:04:29,636
which arrangements can be made all gold?

85
00:04:29,636 --> 00:04:33,916
You can try an interactive version of 
this puzzle and confirm your solution

86
00:04:33,916 --> 00:04:36,296
on Brilliant’s website.

87
00:04:36,296 --> 00:04:39,386
We love Brilliant.org because the site 
gives you tools

88
00:04:39,386 --> 00:04:42,916
to approach problem-solving in 
one of our favorite ways—

89
00:04:42,916 --> 00:04:47,228
by breaking puzzles into smaller pieces 
or limited cases,

90
00:04:47,228 --> 00:04:49,648
and working your way up from there.

91
00:04:49,648 --> 00:04:52,928
This way, you're building up a 
framework for problem solving,

92
00:04:52,928 --> 00:04:55,728
instead of just memorizing formulas.

93
00:04:55,728 --> 00:04:59,308
You can sign up for Brilliant for free, 
and if you like riddles

94
00:04:59,308 --> 00:05:02,783
a Brilliant.org premium membership 
will get you access

95
00:05:02,783 --> 00:05:05,794
to countless more interactive puzzles.

96
00:05:05,794 --> 00:05:09,636
Try it out today by visiting 
brilliant.org/TedEd

97
00:05:09,636 --> 00:05:12,528
and use that link so they know 
we sent you.

98
00:05:12,528 --> 00:05:15,887
The first 833 of you to visit that link

99
00:05:15,887 --> 00:05:20,457
will receive 20% off the annual premium 
subscription fee.