0:00:07.507,0:00:09.944 You're the realm's greatest mathematician, 0:00:09.944,0:00:13.348 but ever since you criticized [br]the Emperor's tax laws, 0:00:13.348,0:00:15.249 you've been locked in the dungeon 0:00:15.249,0:00:18.198 with only a marker to count the days. 0:00:18.198,0:00:21.358 But one day, you're suddenly brought[br]before the Emperor 0:00:21.358,0:00:24.149 who looks even angrier than usual. 0:00:24.149,0:00:27.938 One of his twelve governors has been[br]convicted of paying his taxes 0:00:27.938,0:00:29.945 with a counterfeit coin 0:00:29.945,0:00:33.239 which has already made its way[br]into the treasury. 0:00:33.239,0:00:35.160 As the kingdom's greatest mathematician, 0:00:35.160,0:00:40.930 you've been granted a chance to earn[br]your freedom by identifying the fake. 0:00:40.930,0:00:46.099 Before you are the twelve identical[br]looking coins and a balance scale. 0:00:46.099,0:00:50.300 You know that the false coin[br]will be very slightly lighter or heavier 0:00:50.300,0:00:51.868 than the rest. 0:00:51.868,0:00:54.489 But the Emperor's not a patient man. 0:00:54.489,0:00:56.979 You may only use the scale three times 0:00:56.979,0:01:00.640 before you'll be thrown back [br]into the dungeon. 0:01:00.640,0:01:02.829 You look around for anything else[br]you can use, 0:01:02.829,0:01:04.650 but there's nothing in the room - 0:01:04.650,0:01:05.681 just the coins, 0:01:05.681,0:01:06.921 the scale, 0:01:06.921,0:01:08.952 and your trusty marker. 0:01:08.952,0:01:11.317 How do you identify the counterfeit? 0:01:11.317,0:01:14.379 Pause here if you want [br]to figure it out for yourself! 0:01:14.379,0:01:15.849 Answer in: 3 0:01:15.849,0:01:17.350 Answer in: 2 0:01:17.350,0:01:19.107 Answer in: 1 0:01:19.107,0:01:23.309 Obviously you can't weigh each coin[br]against all of the others, 0:01:23.309,0:01:25.941 so you'll have to weigh several coins[br]at the same time 0:01:25.941,0:01:29.070 by splitting the stack [br]into multiple piles 0:01:29.070,0:01:32.648 then narrowing down [br]where the false coin is. 0:01:32.648,0:01:37.361 Start by dividing the twelve coins[br]into three equal piles of four. 0:01:37.361,0:01:42.201 Placing two of these on the scale[br]gives us two possible outcomes. 0:01:42.201,0:01:47.212 If the two sides balance,[br]all eight coins on the scale are real, 0:01:47.212,0:01:50.392 and the fake must be among [br]the remaining four. 0:01:50.392,0:01:52.672 So how do you keep track of these results? 0:01:52.672,0:01:54.660 That's where the marker comes in. 0:01:54.660,0:01:58.170 Mark the eight authentic coins[br]with a zero. 0:01:58.170,0:02:02.822 Now, take three of them and weigh them[br]against three unmarked coins. 0:02:02.822,0:02:07.333 If they balance, the remaining [br]unmarked coin must be the fake. 0:02:07.333,0:02:12.181 If they don't, draw a plus on the three[br]unmarked coins if they're heavier 0:02:12.181,0:02:15.152 or a minus if they're lighter. 0:02:15.152,0:02:19.662 Now, take two of the newly marked coins[br]and weigh them against each other. 0:02:19.662,0:02:22.701 If they balance, the third coin is fake. 0:02:22.701,0:02:24.612 Otherwise, look at their marks. 0:02:24.612,0:02:28.553 If they are plus coins, [br]the heavier one is the imposter. 0:02:28.553,0:02:31.753 If they are marked with minus,[br]it's the lighter one. 0:02:31.753,0:02:36.232 But what if the first two piles you weigh[br]don't balance? 0:02:36.232,0:02:39.082 Mark the coins on the heavier side[br]with a plus 0:02:39.082,0:02:42.974 and those on the lighter side [br]with a minus. 0:02:42.974,0:02:46.173 You can also mark the remaining four coins[br]with zeros 0:02:46.173,0:02:51.524 since you know the fake one[br]is already somewhere on the scale. 0:02:51.524,0:02:53.414 Now, you'll need to think strategically 0:02:53.414,0:02:58.475 so you can remove all remaining ambiguity[br]in just two more weighings. 0:02:58.475,0:03:01.944 To do this, you'll need [br]to reassemble the piles. 0:03:01.944,0:03:04.786 One method is to replace [br]three of the plus coins 0:03:04.786,0:03:07.113 with three of the minus coins, 0:03:07.113,0:03:10.953 and replace those [br]with three of the zero coins. 0:03:10.953,0:03:13.423 From here, you have three possibilities. 0:03:13.423,0:03:17.475 If the previously heavier side of[br]the scale is still heavier, 0:03:17.475,0:03:20.725 that means either the remaining[br]plus coin on that side 0:03:20.725,0:03:22.826 is actually the heavier one, 0:03:22.826,0:03:25.756 or the remaining[br]minus coin on the lighter side 0:03:25.756,0:03:28.235 is actually the lighter one. 0:03:28.235,0:03:31.915 Choose either one of them, and weigh[br]it against one of the regular coins 0:03:31.915,0:03:33.735 to see which is true. 0:03:33.735,0:03:36.466 If the previously heavier side[br]became lighter, 0:03:36.466,0:03:39.236 that means one of the three minus[br]coins you moved 0:03:39.236,0:03:41.507 is actually the lighter one. 0:03:41.507,0:03:43.706 Weigh two of them against each other. 0:03:43.706,0:03:46.666 If they balance, the third is counterfeit. 0:03:46.666,0:03:49.765 If not, the lighter one is. 0:03:49.765,0:03:53.596 Similarly, if the two sides balanced[br]after your substitution, 0:03:53.596,0:03:56.546 then one of the three plus coins[br]you removed 0:03:56.546,0:03:58.526 must be the heavier one. 0:03:58.526,0:04:00.527 Weigh two of them against each other. 0:04:00.527,0:04:03.588 If they balance, the third one is fake. 0:04:03.588,0:04:07.068 If not, then it's the heavier one. 0:04:07.068,0:04:10.076 The Emperor nods approvingly[br]at your finding, 0:04:10.076,0:04:13.476 and the counterfeiting Lord[br]takes your place in the dungeon.