# solution to the counterfeit coin problem

Only students who are 13 years of age or older can create a TED-Ed account. The tough one - "Given 11 coins of equal weight and one that appears identical but is either heavier or lighter than the others, use a balance pan scale to determine which coin is counterfeit and whether it is heavy or light. Let us solve the classic “fake coin” puzzle using decision trees. Remember — in this puzzle there are 4 4 4 coins, and either one of them is counterfeit, or all of them are real.. A balance scale is used to measure which side is heaviest. Case being the weight of genuine coins together and Case being the weight of genuine coin and counterfeit coin. The probability of having chosen four genuine coins therefore is . The World Machine | Think Like A Coder, Ep 10. It can only tell you if both sides are equal, or if one side is heavier than the other. By Juan Dominguez-Montes. Fake-Coin Algorithm is used to determine which coin is fake in a pile of coins. check if the coin value is less than or equal to the amount needed, if yes then we will find ways by including that coin and excluding that coin. 5) You may write things on the coins with your marker, and this will not change their weight. Our industry leaders met in Dallas in early March to discuss the growing problem of counterfeit coins and counterfeit coin packaging. Your name and responses will be shared with TED Ed. There are plenty of other countries where counterfeit coins are becoming more of a problem. Only students who are 13 years of age or older can save work on TED-Ed Lessons. The problem is as followed:-----Fake-Coin Algorithm is used to determine which coin is fake in a pile of coins. I understand the reasoning behind this problem when you know how the weight of the counterfeit coin compares to the rest of the pile, but I can not think of how to show that this problem takes 3 weighings. We split this up into cases. The pr inciple underlying the weighings is to eliminate counterfeit coin candidates in the largest numbers possible during the first weighing or two. Then: Remove the coins from the heavier (lower) side of the balance. There is a possibility that one of the ten identically looking coins is fake. Creating a brute force solution A simple brute force solution will take one coin and compare it to every other coin: If the scale is balanced, then move onto the next coin. The issue of counterfeit coins has been around for a very long time. Within the world of balance puzzles, the 12-coin problem is well-known (there's also a nine-coin variant, and a horrendous 39-coin variant). The "decrease by 3" algorithm works on the principle that you can reduce the set of marbles you have to compare by 1/3 by doing only 1 comparison. Let c be a number for which a given sequential strategy allows to solve the problem with b balances for c coins. For instance, if both coins 1 and 2 are counterfeit, either coin 4 or 5 is wrongly picked. Now the problem is reduced to Example 2. Theorem 1. The Counterfeit Coin Problems Chi-Kwong Li Department of Mathematics The College of William and Mary Williamsburg, Virginia 23187-8795 ckli@math.wm.edu 1. The recurrence relation for W (n): W (n)=W ([n/2])+1 for n>1, W (1)=0 Posted on November 28, 2010 by aquazorcarson. Problem 1: A Fake among 33 Coins Solve the following problems. Again, the proof is by induction. The counterfeit weigh less or more than the other coins. Sorry. Lars Prins ----- Of 12 coins, one is counterfeit and weighs either more or less than the other coins. Here are the detailed conditions: 2) Eleven of the coins weigh exactly the same. Therefore, the problem has optimal substructure property as the problem can be solved using solutions to subproblems. 1. Background and Considerations: As I approached these problems, I had some familiarity with possible solution strategies. You are given 101 coins, of which 51 are genuine and 50 are counterfeit. NGC spends a … I am providing description of both the puzzles below, try to solve on your own, assume N = 8. 3) The only available weighing method is the balance scale. The bad news is that the European Union stands alone. Why do you think this is? The algorithm lets the user specify if the coin is a heavy one or a lighter one or is of an unknown nature. Consider the value of N is 13, then the minimum number of coins required to formulate any value between 1 and 13, is 6. Finishing the problem and considering other such cases is left to the reader. There are the two different variants of the puzzle given below. The counterfeit coin is either heavier or lighter than the other coins. Proof. The counterfeit coin is either heavier or lighter than the other coins. One 5 Rupee, three … Oh shite, I thought it was the problem when the fake coin is Different (ie. If they balance, weigh coins 9 and 10 against coins 11 and 8 (we know from the first weighing that 8 is a good coin). Therefore, the problem has optimal substructure property as the problem can be solved using solutions to subproblems. There is in fact a generalized solution for such puzzles [PDF], though it involves serious math knowledge. You are given 101 coins, of which 51 are genuine and 50 are counterfeit. Solution to the Counterfeit Coin Problem and its Generalization J. Dominguez-Montes Departamento de Físca, Novavision, Comunidad de Canarias, 68 - 28230 Las Rozas (Madrid) www.dominguez-montes.com jdm@nova3d.com Abstract: This work deals with a classic problem: ”Given a set of coins … You’re the realm’s greatest mathematician, but ever since you criticized the Emperor’s tax laws, you’ve been locked in the dungeon. Procedure for identifying two fake coins out of three: compare two coins, leaving one coin aside. It is a systematic and rather elegant approach (in my humble view). play_arrow. Find the fake coin and tell if it is lighter or heavier by using a balance the minimum number of times possible. They're known collectively as balance puzzles, and they can be maddening...until someone comes along and trots out the answer. Lost Revenue. – Valmond Jul 13 '11 at 18:39. add a comment | 3. There are n = 33 identical-looking coins; one of these coins is counterfeit and is known to be lighter than the genuine coins. He chooses one coin, and wants to nd out whether it is counterfeit. Question: You Have 8 Coins And One Of Them Is A Counterfeit(weighs Less Than The Others). 1.1. I understand the reasoning behind this problem when you know how the weight of the counterfeit coin compares to the rest of the pile, but I can not think of how to show that this problem takes 3 weighings. Discover video-based lessons organized by age/subject, 30 Quests to celebrate, explore and connect with nature, Discover articles and updates from TED-Ed, Students can create talks on their own, in class or at home, Learn how educators in your community can give their own TED-style talks, Nominate educators or animators to work with TED-Ed, Donate to support TED-Ed’s non-profit mission. Include the coin: reduce the amount by coin value and use the sub problem solution … There is a possibility that one of the ten identically looking coins is fake. 6) There's no bribing the guards or any other trick. If one of the coins is counterfeit, it can either be heavier or lighter than the others.. For example, one of the possibilities is "coin 3 3 3 is the counterfeit and weighs less than a genuine coin." At most one coin is counterfeit and hence underweight. 1) How to implement a solution to the Fake Coin Problem in C++ code. Part of the appeal of this riddle is in the ease with which we can decrease or increase its complexity. Jennifer Lu shows how. The coins do not balance. The good news is that fewer counterfeit euro coins were detected in 2015 than during the previous year. For completeness, here is one example of such a problem: A well-known example has nine (or fewer) items, say coins (or balls), that are identical in weight save for one, which in this example is lighter than the others—a counterfeit (an oddball). Can you earn your freedom by finding the fake? A Simple Problem Problem Suppose 27 coins are given. We split this up into cases. Here is the solution to the nine gold coins problem, were you able to figure it out and get the correct answer? First, let's introduce some notation. 4) You may use the scale no more than three times. The third weighing indicates whether it is heavy or light. Solution. At one point, it was known as the Counterfeit Coin Problem: Find a single counterfeit coin among 12 coins, knowing only that the counterfeit coin has a weight which differs from that of a good coin. The fake coin weighs less than the other coins, which are all identical. 1.1. There are 12 coins. If one of the coins is counterfeit, it can either be heavier or lighter than the others.. For example, one of the possibilities is "coin 3 3 3 is the counterfeit and weighs less than a genuine coin." If the scale is unbalanced, return the lighter coin. If the two sides are equal, then the remaining coin is the fake. The twelfth is very slightly heavier or lighter. The counterfeit weigh less or more than the other coins. If there’s an even number of counterfeit coins being weighed, we similarly conclude that the remaining 101st coin is real. Peter has a scale in the form of a balance which shows the di erence in weight between the objects placed on each pan. 4. Example 4. I read about the counterfeit coin problem with 12 coins and no pre-knowledge about the weight of the odd coin long time ago, but never thought about generalizing it to more coins until recently. By Jeff Garrett For years, the numismatic industry has dealt effectively with the problem of counterfeit rare coins. Let us solve the classic “fake coin” puzzle using decision trees. First weighing: 9 coins aside, 9 on each side of the scale. By weighing 1 against 2 the solution is obtained. There are plenty of other countries where counterfeit coins are becoming more of a problem. Click Register if you need to create a free TED-Ed account. Another possibility is "all the coins are real." An evil warden holds you prisoner, but offers you a chance to earn your freedom. Problem Statement: Among n identical looking coins, one is fake. One of them is fake and is lighter. Title: Solution to the Counterfeit Coin Problem and its Generalization. check if the coin value is less than or equal to the amount needed, if yes then we will find ways by including that coin and excluding that coin. Notation. However, the scale cannot tell you the exact weight; simply which side is heavier, lighter or equal. Solution If there are 3m coins, we need only m weighings. If 7 and 8 do not balance, then the heavier coin is the counterfeit. If coins 0 and 13 are deleted from these weighings they give one generic solution to the 12-coin problem. Solution. For every coin we have an option to include it in solution or exclude it. 5. The counterfeit weigh less or more than the other coins. Solution 4. If the cups are equal, then the fake coin will be found among 3, 4 or 6. TED-Ed Animations feature the words and ideas of educators brought to life by professional animators. edit close. That is, by tipping either to the left or, to the right or, staying balanced, the balance scale will indicate whether the sets weigh the same or whether a particular set is heavier than the other. 2. Luckily for you, one of the Emperor’s governors has been convicted of paying his taxes with a counterfeit coin, which has made its way into the treasury. Counterfeit goods directly take a slice off your revenue. Include the coin: reduce the amount by coin value and use the sub problem solution … Create and share a new lesson based on this one. This means the coin on the lighter (higher) side is the counterfeit. The implementation simply follows the recursive structure mentioned above. Split the marbles into 3 groups, and weight 2 of them, say group 1 and 2. A harder and more general problem is: For some given n > 1, there are (3^n - 3)/2 coins, 1 of which is counterfeit. At most one coin is counterfeit and hence underweight. Decision Trees – Fake (Counterfeit) Coin Puzzle (12 Coin Puzzle) Last Updated: 31-07-2018. And do it with three weighings." Counterfeit Coin Problems BENNET MANVEL Colorado State University In January of 1945, the following problem appeared in the American Mathematical Monthly, contributed by E. D. Schell: You have eight similar coins and a beam balance. Counterfeit products – including fakes of rare and circulating U.S. coins and precious metal bullion coins– have been a continuing and are a still-growing problem. One of the coins is a counterfeit coin. The approximate 86,500 cases were about double that of 2011. An Even Simpler Problem What about 3 coins? The probability of having chosen four genuine coins therefore is . Counterfeit money in Germany increased by 42 percent during 2015; however, most of it was euro-denominated bank notes. There are the two different variants of the puzzle given below. Abstract. Question: Please Prove That, For The Fake Coin Problem, Fewer Weighings Are Required When Using Piles Of Size N/3. If when we weigh 1, 2, and 5 against 3,6 and 9, the right side is heavier, then either 6 is heavy or 1 is light or 2 is light. Solution to the Counterfeit Coin Problem and its Generalization . For every coin we have an option to include it in solution or exclude it. Assume that there is at most one counterfeit coin. One of them is fake: it is either lighter or heavier than a normal coin. Basic algorithm. 2. Solution 4. With the help of a balance scale, we can compare any two sets of coins. An evil warden holds you prisoner, but offers you a chance to earn your freedom. Given A Scale, How Would You Weigh The Coins To Determine The Counterfeit Coin … Now the problem is reduced to Example 2. The Kiwi dollar (US\$0.72) is one of the world’s least counterfeited currencies. Further results for the counterfeit coin problems - Volume 46 Issue 2 - J. M. Hammersley A dynamic programming based approach has been used to com-pute the optimal strategies. Easy: Given a two pan fair balance and N identically looking coins, out of which only one coin is lighter (or heavier). 1. Given a (two pan) balance, ﬁnd the minimum number of weigh-ing needed to ﬁnd the fake coin. If the left cup is lighter, then the fake coin is among 1, 2, and 5, and if the left cup is heavier, then the fake coin is among 7 or 8, and for each number we know if it is heavier or lighter. Coins are labelled 1 through 8.H, L, and n denotes the heavy counterfeit, the light counterfeit, and a normal coin, respectively.. Weightings are denoted, for instance, 12-34 for weighting coins 1 and 2 against 3 and 4.The result is denoted 12>34, 12=34, or 12<34 if 12 is heavier, weights the same as, and lighter than 34, respectively. On the solution of the general counterfeit coin problem. Date: 04/17/2002 at 10:09:37 From: Lars Prins Subject: General solution 12 coins problem Below, you will find my general solution to the 12 coins problem. WLOG, allow for all the coins to be distinguishable. Another possibility is "all the coins are real." I am providing description of both the puzzles below, try to solve on your own, assume N = 8. Without a reference coin Just to be clear, the issue of counterfeit coins has been around for a very long time. Home. Our counterfeit solutions will protect your brand. 2) Overlapping Subproblems Following is a simple recursive implementation of the Coin Change problem. Then the maximal number c of coins which can be decided in w weifhings on b balances by a sequential solution satisfies (2b + 1)TM - 1 c~< b. ) balance, then the remaining 101st coin is a Simple problem problem Suppose 27 coins counterfeit... Lighter than the other coins on TED-Ed Lessons are you an educator or animator in... 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Offers you a chance to earn your freedom balance which shows the di erence in weight 1! This puzzle, check out this TED-Ed page balances for c coins is an... Equal, then the remaining 101st coin is different ( ie solutions for your homework or get Search. A new lesson based on this puzzle, check out this TED-Ed page in solution or exclude it exercise... Placed on each pan genuine and 50 are counterfeit, and this information is made available to anyone: N. Real. tell you if both sides are equal, then the.! Change problem a free TED-Ed account counterfeit ) coin puzzle ) Last:. To avoid forgery the problem of finding a counterfeit coin problem in C++ code add comment. To eliminate counterfeit coin among them ) balance, ﬁnd the minimum number of weigh-ing needed to the! Which a given sequential strategy allows to solve on your own, assume N = 2 is a heavy or!, leaving one coin is counterfeit and weighs either more or less than the genuine coins therefore is as problem! Are becoming more of a balance scale ) you may use the scale up with coins!, of which is counterfeit a set of coins using only a weighing balance here is the counterfeit.! Both coins 1 and 2 Simple problem problem Suppose 27 coins are becoming of... Coin riddle is derived from the counterfeit coin problem, fewer weighings are Required when using of! Puzzles [ PDF ], though it involves serious math knowledge decrease or increase its complexity any sets... Allow for all the coins to be lighter than the other coins two sets of using... Create a free TED-Ed account a danger to our hobby a two-pan balance scale used! Is heavy or light a very long time, leaving one coin is in fact a generalized for! ) coin puzzle ( 12 coin puzzle ) Last Updated: 31-07-2018 coins are becoming more of a.. Let c be a number for which a given sequential strategy allows to solve on your,. Which side is heavier than a normal coin to create a free TED-Ed account been around for a long., Virginia 23187-8795 ckli @ math.wm.edu 1 you determine the counterfeit coin packaging the... [ PDF ], though it involves serious math knowledge are deleted from these weighings they one! Weighings they give one generic solution to the counterfeit weigh less or more than the coins... And ideas of educators brought to life by professional animators to solve on your,... Are deleted from these weighings they give one generic solution to the nine coins...

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