An evil warden holds you prisoner, but offers you a chance to earn your freedom. You cannot find the fake among more than 121 coins, as there are simply too many possibilities. One stack will be heavier than the other. Greg Charles wrote:Drat. ... To find the fake coin out of 3 coins in 2 weightings... 1 <--> 2 1 <--> 3 ... From there, you can develop the algorithm for 12 coins in 3 weightings using the method on the site mentioned above. Given N coins with a fake among them which has a weight slightly more than the real one, what is the minimum number of times you need to use the balance to correctly identify the fake one in the worst case? Solution. Analysis: In general, if we know that the coin is heavy or light, we can trace the coin in log3(N) trials (rounded to next integer). The counterfeit coin problem With n coins all the same weight except for one which could weigh more or less, determine the minimum number of weighings x which must be performed in balance scales to identify whether this coin exists and whether it is heavier or lighter than the rest. of ECE, Arasu Engineering College, Kumbakonam, Tamilnadu ,India #5Assistant professor,Dept. If one of the sides of the scale is heavier, the fake coin is on that side. (1234) = (5678), both groups are equal. Creativity is allowing yourself to make mistakes; art is knowing which ones to keep. Similar way we can also solve the right subtree (third outcome where (1234) > (5678)) in two more weighing. Observe the above figure that not all the branches are generating leaves, i.e. c. Solution The problem solved is a general n coins problem. From the above analysis we may think that k = 2 trials are sufficient, since a two level 3-ary tree yields 9 leaves which is greater than N = 4 (read the problem once again). The fake coin problem can be solved recursively using the decrease-by-constant factor strategy. Yes, I meant 3 groups - though if you go bottom-up rather than top-down it's similar (I was thinking of the 9-coin version, where it's the same thing!). First note that for 121 coins, there are 242 possibilities. The left side of tree corresponds to the case (G1) < (23). You are given 101 coins, of which 51 are genuine and 50 are counterfeit. Writing code in comment? e.g. In both the cases, we know that (ABCD) is genuine. Let us solve the classic “fake coin” puzzle using decision trees. For each coin, x, in Sn-1, replace it with 3x-2, 3x-1 and 3x. Consider the second outcome where (1234) < (5678). These two outcomes can be declared at the root of tree itself (prior to first weighing), can you figure these two out comes? There can't be \$2\$ coins that go through the same weighing course (because we won't be able to know which one is fake). Set up a recurrence relation for the number of weighing in the divide-into three algorithm for the fake-coin problem and solve it for n = 3k. For instance, if both coins 1 and 2 are counterfeit, either coin 4 or 5 is wrongly picked. Analysis: Given (N + 1) coins, one is genuine and the rest N can be genuine or only one coin is defective. To figure out the odd coin, how many minimum number of weighing are required in the worst case? Set up a recurrence relation for the number of weighings in the divide-into-three algorithm for the fake-coin problem and solve it for n = 3k. – – – by Venki. You are given n gold coins, and one of them is fake.Assume that all the coins are identical, except that the fake coin is lighter.Given a balance scale, where you can put a bunch of coins on the left and the right and determine which is heavier, design the fastest algorithm for determining the fake coin.. If you're already bored with the thread up to this point, this post will only make matters worse. Set j=i and B[l] 2. We can get (3) > (2) in which 2 is lighter, or (3) = (2) in which 1 is lighter. In this video, the Fake Coin problem is discussed This video has no prerequisites. Keep this tiny ad: current ranch time (not your local time) is, https://coderanch.com/t/730886/filler-advertising. Specifically read section 5.5 and section 11.2 including exercises. We can group the coins in two different ways, [(12, 34)] or [(1, 2) and (3, 4)]. We still have to worry about dealing with an odd number during a weighing, but Put the remaining n -2⌊ n /2⌋ coins into a separate pile, C. Note that it is impossible to solve above 4 coins problem in two weighing. An algorithm for reaching Ivo's answer: You can find the fake out of up to 121 coins, but you cannot necessarily distinguish if the fake is heavy or light. of ECE, Arasu Engineering College,Kumbakonam,Tamilnadu, India Abstract-This paper presents a method for detection of fake Pick one genuine coin from any of weighed groups, and proceed with (ABCD) as explained in Problem 3. 4. Let us shuffle coins as (1235) and (4BCD) as new groups (there are different shuffles possible, they also lead to minimum weighing). There are the two different variants of the puzzle given below. And also, we know a set of coins being lighter and a set of coins being heavier. One weighing. Apache Tomcat, SSL: how to add new algorithm. Defective coin may be in (ABCD) group. We are able to solve the 12 coin puzzle in 3 weighing in the worst case. Let us consider the combination (12, 34), the corresponding decision tree is given below. Otherwise, divide the coins into 2 equal piles, A and B, of size ⌊ n /2⌋ coins. Infact, we can get 27 leaves of 3 level full 3-ary tree, but only we got 11 leaves including impossible cases. If two coins are counterfeit, this procedure, in general, does not pick either of these, but rather some authentic coin. we go on to left subtree or (12) > (34) i.e. From the above two examples, we can ensure that the decision tree can be used in optimal way if we can reveal atleaset one genuine coin. The outcome of second trail can be three ways. Devise a brute-force algorithm to identify the stack with the fake coins and determine its worst-case e ﬃ ciency class. This is possible in two ways, either 1 is heavier or either of (2, 3) should be lighter. An algorithm is a procedure or formula for solving a problem, based on conducting a sequence of specified actions. Label the coins as 1, 2, 3, 4 and G (genuine). https://www.geeksforgeeks.org/decision-trees-fake-coin-puzzle How can we trace which coin, if any, is odd one and also determine whether it is lighter or heavier in minimum number of trials in the worst case? fake coin, and in exactly 10balance weighings, we determine the coin. Remember to group coins such that the first weighing reveals atleast one genuine coin. We need two weighing in worst case. Also, the tree is not full tree, middle branch terminated after first weighing. Without a reference coin The outcome can be (12) < (34) i.e. Both the combinations need two weighing in case of 5 coins with prior information of one coin is lighter. The algorithm lets the user specify if the coin is a heavy one or a lighter one or is of an unknown nature. Here are the detailed conditions: 1) All 12 coins look identical. Note that it impossible to get (3) < (2), it contradicts our assumption leaned to left side. Among them are f fake coins. The solution to this problem for a given set of coin … 8. Let us start with relatively simple examples. //Output:- The position of fake coin or appropriate message. The problem is, we're only allowed the use of a marker (to make notes on the coins) and three uses of a balance scale. Here we need one more weighing to check a genuine coin against 1 or 2. Count inversions in an array | Set 3 (Using BIT), Segment Tree | Set 2 (Range Minimum Query), XOR Linked List – A Memory Efficient Doubly Linked List | Set 2, proto van Emde Boas Trees | Set 1 (Background and Introduction), Self-Balancing-Binary-Search-Trees (Comparisons), Remove edges connected to a node such that the three given nodes are in different trees, Proto Van Emde Boas Trees | Set 4 | Deletion, Dynamic Segment Trees : Online Queries for Range Sum with Point Updates, LCA for general or n-ary trees (Sparse Matrix DP approach ), Segment Trees | (Product of given Range Modulo m), Difference between Backtracking and Branch-N-Bound technique, K Dimensional Tree | Set 1 (Search and Insert), Write Interview In other words, we need atleast k > log3(2N + 1) weighing to find the defective one. fake coin problem 1. C) (3) = (4), yielding ambiguity. The left subtree is possible in two ways, Further on the left subtree, as second trial, we weigh (1, 2) or (3, 4). we are missing valid outputs under some branches that leading to more number of trials. Example 2. If we proceed as in Problem 1, we will not generate best decision tree. This is another problem in which i will show you the advantage of Dynamic programming over recursion. Easy: Given a two pan fair balance and N identically looking coins, out of which only one coin is lighter (or heavier). Since the total possible outcomes are (2(N + 1) + 1), number of weighing (trials) are given by the height of ternary tree, k >= log3[2(N + 1) + 1]. In this video, the Fake Coin problem is discussed This video has no prerequisites. When possible, we should group the coins in such a way that every branch is going to yield valid output (in simple terms generate full 3-ary tree). With just c-3 (i.e. By using our site, you Sort the list in alphabetical order by selection sort. 1. 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Write pseudocode for the divide-into-three algorithm for the fake-coin problem i will show the... And proceed with ( ABCD ) things in computer science: cache invalidation, naming things, and off-by-one.! Faster that yours to solve on your own, assume N = 2k, this procedure, in Sn-1 replace! All coins are counterfeit i took ( 3 ) = ( 23 ) is similar be applied the... Coin left, then it is the minimum number of trials ) weighing to the... Are impossible cases are unable to utilize two outcomes of 3-ary trees to get ( 3 should! It more efficiently than that N ) weighings coins 1 and 2 are counterfeit either! That there are at most n-1 fake coins and determine its worst-case ﬃ... In both the combinations need two weighing and return and N = 8 assumption leaned to left side tree. Groups in such a way that we end up with smaller height tree. Or a lighter one or a lighter one or a lighter one a! We also need to get ( 3 ) should be `` 3 groups '' 's: c = 23.
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