Can you solve the counterfeit coin riddle? - Jennifer Lu — Transcript

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1. 0:07You’re the realm’s greatest mathematician,
2. 0:09but ever since you criticized the emperor's tax laws,
3. 0:13you’ve been locked in the dungeon with only a marker to count the days.
4. 0:17But one day you're suddenly brought before the emperor,
5. 0:21who looks even angrier than usual.
6. 0:23One of his 12 governors has been convicted of paying his taxes
7. 0:28with a counterfeit coin,
8. 0:29which has already made its way into the Treasury.
9. 0:32As the kingdom's greatest mathematician,
10. 0:35you’ve been granted a chance to earn your freedom by identifying the fake.
11. 0:40Before you are the 12 identical looking coins and a balance scale.
12. 0:45You know that the false coin will be very slightly lighter or heavier than the rest.
13. 0:51But the emperor’s not a patient man.
14. 0:54You may only use the scale three times
15. 0:57before you’ll be thrown back into the dungeon.
16. 1:00You look around for anything else you can use,
17. 1:02but there's nothing in the room, just the coins,
18. 1:05the scale, and your trusty marker.
19. 1:08How do you identify the counterfeit?
20. 1:11Pause now to figure it out yourself!
21. 1:14Answer in 3
22. 1:15Answer in 2
23. 1:17Answer in 1
24. 1:18Obviously, you can't weigh each coin against all of the others,
25. 1:22so you’ll have to weigh several coins at the same time
26. 1:25by splitting the stack into multiple piles,
27. 1:28then narrowing down where the false coin is.
28. 1:32Start by dividing the 12 coins into three equal piles of four.
29. 1:37Placing two of these on the scale gives us two possible outcomes.
30. 1:42If the two sides balance, all eight coins on the scale are real
31. 1:46and the fake must be among the remaining four.
32. 1:50So how do you keep track of these results? That’s where the marker comes in.
33. 1:54Mark the eight authentic coins with a zero.
34. 1:57Now take three of them and weigh them against three unmarked coins.
35. 2:02If they balance, the remaining unmarked coin must be the fake.
36. 2:06If they don’t, draw a plus on the three unmarked coins
37. 2:11if they’re heavier, or a minus if they’re lighter.
38. 2:14Now take two of the newly marked coins and weigh them against each other.
39. 2:19If they balance, the third coin is fake.
40. 2:22Otherwise, look at their marks.
41. 2:24If they are plus coins, the heavier one is the impostor.
42. 2:28If they are marked with minus, it's the lighter one.
43. 2:31But what if the first two piles you way don't balance?
44. 2:35Mark the coins on the heavier side with a plus
45. 2:39and those on the lighter side with a minus.
46. 2:42You can also mark the remaining four coins with zeros,
47. 2:46since you know the fake one is already somewhere on the scale.
48. 2:51Now you'll need to think strategically
49. 2:53so you can remove all remaining ambiguity in just two more ways.
50. 2:58To do this, you’ll need to reassemble the piles.
51. 3:01One method is to replace three of the plus coins
52. 3:04with three of the minus coins
53. 3:06and replace those with three of the zero coins.
54. 3:10From here, you have three possibilities.
55. 3:13If the previously heavier side of the scale is still heavier,
56. 3:17that means either the remaining plus coin on that side
57. 3:20is actually the heavier one,
58. 3:22or the remaining minus coin on the lighter side is actually the lighter one.
59. 3:27Choose either one of them and weigh it against one of the regular coins
60. 3:31to see which is true.
61. 3:33If the previously heavier side became lighter,
62. 3:36that means one of the three minus coins you moved is actually the lighter one.
63. 3:41Weigh two of them against each other.
64. 3:43If they balance, the third is counterfeit.
65. 3:46If not, the lighter one is.
66. 3:49Similarly, if the two sides balanced after your substitution,
67. 3:53then one of the three plus coins you removed must be the heavier one.
68. 3:58Weigh two of them against each other.
69. 4:00If they balance, the third one is fake.
70. 4:03If not, then it's the heavier one.
71. 4:06The emperor nods approvingly at your finding,
72. 4:09and the counterfeiting lord takes your place in the dungeon.