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