0:00:06.589,0:00:08.249 You heard the traveler's tales, 0:00:08.249,0:00:10.029 you followed the crumbling maps, 0:00:10.029,0:00:12.899 and now, after a long and dangerous quest, 0:00:12.899,0:00:16.024 you have some good news and some bad news. 0:00:16.024,0:00:19.469 The good news is you've managed to locate[br]the legendary dungeon 0:00:19.469,0:00:22.661 containing the stash [br]of ancient Stygian coins 0:00:22.661,0:00:25.595 and the eccentric wizard [br]who owns the castle 0:00:25.595,0:00:29.244 has even generously[br]agreed to let you have them. 0:00:29.244,0:00:31.964 The bad news is that he's not[br]quite as generous 0:00:31.964,0:00:37.703 about letting you leave the dungeon,[br]unless you solve his puzzle. 0:00:37.703,0:00:39.683 The task sounds simple enough. 0:00:39.683,0:00:44.284 Both faces of each coin bear[br]the fearsome scorpion crest, 0:00:44.284,0:00:45.373 one in silver, 0:00:45.373,0:00:47.113 one in gold. 0:00:47.113,0:00:50.844 And all you have to do is separate them[br]into two piles 0:00:50.844,0:00:56.294 so that each has the same number[br]of coins facing silver side up. 0:00:56.294,0:01:00.305 You're about to begin when all[br]of the torches suddenly blow out 0:01:00.305,0:01:02.943 and you're left in total darkness. 0:01:02.943,0:01:05.022 There are hundreds [br]of coins in front of you 0:01:05.022,0:01:08.756 and each one feels the same on both sides. 0:01:08.756,0:01:11.602 You try to remember [br]where the silver-facing coins were, 0:01:11.602,0:01:13.273 but it's hopeless. 0:01:13.273,0:01:14.771 You've lost track. 0:01:14.771,0:01:17.141 But you do know one thing for certain. 0:01:17.141,0:01:18.612 When there was still light, 0:01:18.612,0:01:23.382 you counted exactly [br]20 silver-side-up coins in the pile. 0:01:23.382,0:01:24.842 What can you do? 0:01:24.842,0:01:29.437 Are you doomed to remain in the dungeon[br]with your newfound treasure forever? 0:01:29.437,0:01:31.529 You're tempted to kick the pile of coins 0:01:31.529,0:01:34.892 and curse the curiosity [br]that brought you here. 0:01:34.892,0:01:37.302 But at the last moment, you stop yourself. 0:01:37.302,0:01:41.815 You just realized there's [br]a surprisingly easy solution. 0:01:41.815,0:01:43.463 What is it? 0:01:43.463,0:01:46.763 Pause here if you want to figure[br]it out for yourself. 0:01:46.763,0:01:47.793 Answer in: 3 0:01:47.793,0:01:49.163 Answer in: 2 0:01:49.163,0:01:50.893 Answer in: 1 0:01:50.893,0:01:54.185 You carefully move aside 20 coins[br]one by one. 0:01:54.185,0:01:57.775 It doesn't matter which ones:[br]any coins will do, 0:01:57.775,0:02:00.323 and then flip each one of them over. 0:02:00.323,0:02:02.063 That's all there is to it. 0:02:02.063,0:02:04.505 Why does such a simple solution work? 0:02:04.505,0:02:08.044 Well, it doesn't matter how many[br]coins there are to start with. 0:02:08.044,0:02:12.924 What matters is that only 20[br]of the total are facing silver side up. 0:02:12.924,0:02:15.503 When you take 20 coins in the darkness, 0:02:15.503,0:02:19.088 you have no way of knowing how many[br]of these silver-facing coins 0:02:19.088,0:02:21.947 have ended up in your new pile. 0:02:21.947,0:02:24.274 But let's suppose you got seven of them. 0:02:24.274,0:02:27.715 This means that there are thirteen[br]silver-facing coins left 0:02:27.715,0:02:29.606 in the original pile. 0:02:29.606,0:02:33.215 It also means that the other[br]thirteen coins in your new pile 0:02:33.215,0:02:35.796 are facing gold side up. 0:02:35.796,0:02:40.500 So what happens when you flip[br]all of the coins in the new pile over? 0:02:40.500,0:02:44.771 Seven gold-facing coins and[br]thirteen silver-facing coins 0:02:44.771,0:02:47.995 to match the ones in the original pile. 0:02:47.995,0:02:52.930 It turns out this works no matter how[br]many of the silver-facing coins you grab, 0:02:52.930,0:02:54.356 whether it's all of them, 0:02:54.356,0:02:54.924 a few, 0:02:54.924,0:02:56.521 or none at all. 0:02:56.521,0:03:00.515 That's because of what's known[br]as complementary events. 0:03:00.515,0:03:04.227 We know that each coin only has[br]two possible options. 0:03:04.227,0:03:08.476 If it's not facing silver side up,[br]it must be gold side up, 0:03:08.476,0:03:09.847 and vice versa, 0:03:09.847,0:03:12.117 and in any combination of 20 coins, 0:03:12.117,0:03:15.289 the number of gold-facing[br]and silver-facing coins 0:03:15.289,0:03:17.777 must add up to 20. 0:03:17.777,0:03:20.927 We can prove this mathematically[br]using algebra. 0:03:20.927,0:03:24.498 The number of silver-facing coins[br]remaining in the original pile 0:03:24.498,0:03:29.167 will always be 20 minus[br]however many you moved to the new pile. 0:03:29.167,0:03:32.517 And since your new pile also[br]has a total of 20 coins, 0:03:32.517,0:03:35.257 its number of gold-facing coins will be 0:03:35.257,0:03:39.487 20 minus the amount of [br]silver-facing coins you moved. 0:03:39.487,0:03:42.318 When all the coins in the new pile[br]are flipped, 0:03:42.318,0:03:45.966 these gold-facing coins become[br]silver-facing coins, 0:03:45.966,0:03:51.538 so now the number of silver-facing[br]coins in both piles is the same. 0:03:51.538,0:03:55.159 The gate swings open[br]and you hurry away with your treasure 0:03:55.159,0:03:57.268 before the wizard changes his mind. 0:03:57.268,0:04:00.877 At the next crossroads, you flip[br]one of your hard-earned coins 0:04:00.877,0:04:03.450 to determine the way [br]to your next adventure.