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The counterfeit coin is either heavier or lighter than the other coins. Fortunately in the 12 coin riddle when we narrow it down to 8 coins we can use the knowledge gained from the first weigh to again split the coins into 3 groups and definitively eliminate 2 of them with the second weigh. If it balances the one in your hand is the counterfeit. If it's balanced, then you know that the fake is either G or H, and that can be checked with a single weighing, for instance G vs A. By means of a simple weighing, we can determine which one is the fake coin, by putting one coin of each group on the left, and one the right: ... 1.Minimum number of weighing required to find the fake coin when it is not known whether the fake coin is heavier or lighter. Otherwise... Take two of the three coins that were on the light side and place one on each side of the scale. Let the two remaining coins be X1 and X2. Next show that you can get there in three if the first one does not. could i make money from a cosplay onlyfans. If they balance out, then 6 is the heavier one. Derivation of curl of magnetic field in Griffiths. Now if you could balance both those coins you'd have it. This means that one of the three coins that went from the lighter side to the heavier side is the light coin. Find the one penny that weighs the least in less than 3 steps. Given 12 coins such that exactly one of them is fake (lighter or heavier than the rest, but it is unknown whether the fake coin is heavier or lighter), and a two pan scale, devise a procedure to identify the fake coin and whether it is heavier or lighter by doing no more than … Similarly, we have … 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. rev 2020.12.8.38143, Sorry, we no longer support Internet Explorer, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Using a two-sided scale, what’s the minimum number of times you’d need to weigh the balls to find the light one? So while you do need three binary bits to specify a number from one to eight, you need only two TRINARY-DIGITS Formally, you want the smallest value of n such that 3^n >= 8. If n mod 3 = 0, we can divide the coins into three piles of exactly n/3 apiece. My college is making us take our classes online since March because of covid. then compare $a_1a_2$ to $a_5a_6$. Weigh these two piles against each other. So take the 2 coins from that group and use the scale to determine the counterfeit coin. 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Using a balance scale, how can you find the fake coin, and determine if it weighs less or more... :: Difficulty:2.6/4 You discard the four heavier coins, those are genuine. You have n > 2 identical-looking coins and a two-pan balance scale with no weights. Not possiable to determine with that information if you can only weigh twice that would mean you could weigh 4 and 3 you would need atleast 5 weighs to figure that out. If it went the opposite way of the first weighing (i.e. If one of them is lighter, that is the one, and if they weigh the same, the unweighed coin is the lighter one. Do the axes of rotation of most stars in the Milky Way align reasonably closely with the axis of galactic rotation? VMWare Inc Interview Question for Software Engineer / Developers. 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. ...gave me (the) strength and inspiration to. There are 8 balls which appear identical. First weigh three coins against three others. It goes the same way as before (i.e. Join Yahoo Answers and get 100 points today. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. What I have so far is that say you have the coins $A B C D E F G H$, weigh $ABC$ against $DEF$ if they are equal then weigh $A$ against $G$ if these are equal then the counterfeit coin is coin $H$ if these two are unbalanced then $G$ is the counterfeit coin. If they balance, you know that the fake coin is one of the other 4 and those on the scale are good. You have shown that ABC vs DEF works in two weighings if the first one balances. 1 decade ago. If they don’t weigh the same, then the lighter one is obviously the counterfeit coin. If the left cup weighs less than the right cup, then we have 8 potential outcomes for the right answer: the fake coin is one out of the four on the left, and it is lighter, or the fake coin is one out of the four on the right, and it is heavier. Done. We need to find it in no more than 3 weighings. SOLUTION 5. All 9 coins look exactly the same but one coin is a fake and is either lighter or heavier than the other 8 coins. How do you find the fake gold coin? Scenario #2: One group weighs less than the other. In Brexit, what does "not compromise sovereignty" mean? Basically your first weighing has to be ABC vs DEF or AB vs DE. How do you find the ball with 2 weighings? For the third attempt, weigh two of these coins against each other: if one is lighter, it is the unique coin; if they balance, the third coin is the light one. Here’s how I solved: weigh 4 against 4. one side will be slightly heavier, so take those 4 coins and weigh 2 against 2. again, one side will be heavier, so take those 2 coins and just judge in your hand which is heaviest (the problem said nothing about coins’ weight difference being so subtle that they couldn’t be compared by hand). You can share Puzzles, math problems or any suggestions about upcoming videos. Can Gate spells be cast consecutively and is there a limit per day? However, 1 is heavier than the rest. Remove all the coins from the heavy side, place one each from the three left on the lite side on each side of the balance and keep the other one in your hand. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. But you said it's only two weighings. ugly_zeke. u have 9 coins and 8 coins have equal weights and 1 coin is lighter than 8 coins. US passport protections and immunity when crossing borders. Good hobbies for a woman in her 70’s? It only takes a minute to sign up. Show activity on this post. Suppose we divide the coins into three piles, where at least two of them contain the same number of coins. A better answer: Split the 8 pennies into 3 groups of 3,3,2 pennies. Suppose they are equal. Design a Θ (1) algorithm to determine whether the fake coin is lighter or heavier than the others. How do you know how many coins to start off with on each side of a “find the counterfeit coin using a 2-pan weigh scale” problem? Weight the first two groups of 3 pennies each. How to convey the turn "to be plus past infinitive" (as in "where C is a constant to be determined")? Now you have shown that three are sufficient. If {1,2,3} and {4,5,6} balance out, then we know either 7 or 8 is the heavier one. MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC…. This can go one of three ways: This method can actually be expanded all the way up to $12$ coins, following roughly the same procedure, and still managing in only $3$ weighings. In simpler problems, like when you’re given 8 balls and one of them is heavier, and we’re limited to two weighings, we can often succeed using brute-force approach. The coins look alike, but 1 is counnterfeit and lighter than the othe If you have 8 coins and a balance scale. 8 coins weigh 1 gram and 1 coin weighs 2 grams. Weigh the two stacks of three. Take the two lighter coins, and it's one of those. Cancel That leaves a lot of room for a bigger pool of coins, and that limit happens to be at $12$.). In that case, one of the two remaining coins contains the odd one. 0. of 2 votes. The coins look alike, but 1 is counnterfeit and lighter than the other 7. ABC and AD were both the heaviest, or they were both the lightest). One of the coins is a fake, but you do not know whether it is lighter or heavier than the genuine coins, which all weigh the same. A theorem about angles in the form of arctan(1/n), I made mistakes during a project, which has resulted in the client denying payment to my company. Take any 2 coins from this group of 3 coins. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. If it's even then the light one can be determined by comparing the other two remaining coins. If the weights are equal, weigh the remaining two against each other. A Logic Brain Teaser: There are 12 gold coins. If the scale is balanced, then the ball must be in B{3 balls} and will be lighter if B{3 balls} was on the lighter side or will be heavier if B{3 balls} was on the heavier side during First weigh. How many weighings of a balance are necessary to determine if a coin is counterfeit among eight coins. Remember — in this puzzle there are 4 4 4 coins, and either one of them is counterfeit, or all of them are real.. To show that it cannot be done in two, consider AB vs DE. Try putting one coin on the scale with another one. If one is heavier, it’s the counterfeit. I am also planning to make videos on logical concepts. Maybe you could spend both those coins in different places and see which time you get arrested. (Not interested in remote control or drones) Is Guillows Spitfire a good choice? Similarly, in the very last subcase where {5,6,7,8} is heavier and then {7,2,9} is heavier, it is impossible for 2 to be light, as is currently mentioned in the answer. What is the altitude of a surface-synchronous orbit around the Moon? You are given 8 pennies, 7 of which weigh exactly the same, but one penny weighs less than the other 7. ABC was heaviest and AD was lightest or the other way around), that means that the counterfeit coin changed sides, which means it's either B or D. Either way, you only need one more weighing to figure out which one it is. Now this is where I hit the wall. The heavier one is the counterfeit. Real life examples of malware propagated by SIM cards? Draw a decision tree that gives an algorithm that identifies in at most three weighings the bad coin and determines whether it is heavier or lighter than the others using only a pan balance. Two cases are possible: [1] First, the easy one. There are 9 coins. You have four coins and can't find the odd one in one weighing, because there are three places for coins (two pans and off the balance), so there will be two unknown coins in the same place. @J.Andrews Yes and no. You have a scale - balance type with 2 trays - but can only load it twice. 1.b) The side that was heavier the first time is lighter the second time. And u have only two chance for find the lighter coin from 9 coins and u … Section 9.7, pages 485–486, problems 3, 13. Suppose they are equal. One is slightly lighter than the rest; the difference is so small that you can’t tell just by picking them up. Eight coins are identical in appearance, but one coin is either heavier or lighter than the others, which all weigh the same. It is somewhat important that in each dish there is one coin that was already there from the last weighing, and one coin from the other dish. Let your first weighing be ABC vs DEF. One of the coins is fake. How many weighings of a balance are necessary to determine if a coin is counterfeit among eight coins. If it does not; obviously, the lighter one is counterfeit. Case 2) Group 1 weights more than Group 2. … 2.Find lighter fake coin from a set of 12 identical coins. Another possibility is "all the coins are real." There's 8 balls. Lighter or heavier? You can sign in to vote the answer. It will not balance so the lighter side has the counterfeit. 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 it's not completely clear how we can make the assumption that the bad coins is in group 7,8 & 9 when the scales balance, remember we know there is a bad coin, there has to be, we have eliminated the other two groups, it must be in the third. If you have 8 coins and a balance scale. 16. lighter pile against the pile put aside. This one is a little more difficult because you are not told if the object is heavier or lighter. Keep switching the right side of the scale until you find the one that is lighter. If the they are the same, then the third coin is the light one, otherwise you'll be able to tell from which is lighter. They will weigh same. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. If one of the groups of three is heavier, weigh two of those coins against each other. I am tired of taking online. It is known that a fake coin weighs either slightly less or slightly more than a real coin. Model plane enthusiasts - balsa wood plane kits. In this case you fail if it balances. Does it matter which four coins you choose if the weighing was originally unbalanced for $ABC$ vs $DEF$? In this particular example we have 12 coins, one of them is forged and may be heavier or lighter, we don’t known which. You have 9 gold coins. If they’re equal weight, the third coin … In this case, remove 2 coins from one group and one coin … 3. Case 1) - They weight the same. Did Biden underperform the polls because some voters changed their minds after being polled? One should then take any ball other than 3 and weigh it against 5 — if they balance out, then 3 is fake (heavier), and if not then 5 is fake (lighter). How many ways are there to pick a collection of 12 coins from piles of pennies, nickels, dimes, and quarters? 8 coins are given where all the coins have equal weight, except one. The counterfeit coin is either heavier or lighter than the other coins. Therefore, take third group of 2 pennies and find the lighter coin. Clear and concise, thanks. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Lv 6. Eight coins are identical in appearance, but one coin is either heavier or lighter than the others, which all weigh the same. The odd one may be less weight than the other or it may be heavier than the rest 7 coins. Is it illegal to market a product as if it would protect against something, while never making explicit claims? This shows three are required. How can I upsample 22 kHz speech audio recording to 44 kHz, maybe using AI? Remove them all and place the other two; one on each side. In that case, the counterfeit coin stayed in its dish, which means it's either A or E. Do a single weighing to decide which one of them is counterfeit. Compare $a_1a_2$ vs $a_3a_4$. Draw a decision tree that gives an algorithm that identifies in at most three weighings the bad coin and determines whether it is heavier or lighter than the others using only a pan balance. When the piles of $ABC$ and $DEF$ are unbalanced you know that $G$ and $H$ are not counterfeit. Compare $a_1$ and $a_7$. Finishing the problem and considering other such cases is left to the reader. First you put four coins on one side and four on the other. 2. Next, we deal with 101 coins. That means that either C or F is counterfeit, which can be checked with a single weighing. So I'm stuck. (B) Pick any 8 coins randomly and divide into two groups of 4 each. Are there any funding sources available for OA/APC charges? Weigh both of them to find out which one is heavier. How will you find out the heavier coin in a minimum number of weighings and how many weighings will be needed? Which is the counterfeit coin using 2 weighings on the balance scale? If the weights match, it means the fake coin is heavier that the original ones, if the weights don’t much the fake coin is lighter. (2001-08-20) General Counterfeit Penny Problem How do you find a single counterfeit coin [either heavier or lighter than a good one] among n coins in only k weighings on a two-pan balance?First let's notice that all the information gathered at any point of the weighing procedure can be summarized by putting all the coins in one of four bins labeled E, G, L, or H (if E is not empty, H and L are). Favorite Answer. However the 8 coin riddle is actually unsolvable if you aren't told whether the fake coin is heavier or lighter as you might end up with 6 potential coins after the first weigh. 1. I think you can now weigh $DA$ vs $BC$ and in the case where $DA$ and $BC$ equal you know that $E$ or $F$ is the counterfeit coin, which can be determined with one more weighing. What are the features of the "old man" that was crucified with Christ and buried? We know which group of three coins ({1,2,3}, {4,5,6} and {7,8,9}) contains the fake coin (if there is one) and whether it is heavier or lighter. If we observe the figure, after the first weighing the problem reduced to “we know three coins, either one can be lighter (heavier) or one among other two can be heavier (lighter)”. Counterfeit Coin Problem Variant - Two Counterfeits, The roles of standard coins in the counterfeit coin problems, Determine counterfeit in a collection of $3^n -1$ coins in n weighings. I am trying to find my Mother a hobby, her health no longer allows her to garden and she is burned out? (Note that none of the final weighings are with only two candidate counterfeits, when you can easily handle $3$, and in the case where the first weighing is balanced, you only need two total weighings. There are 8 identical coins.One of these coins is counterfeit and is known to be lighter than the genuine coins. PLACE THREE COINS ON BOTH SIDES OF THE BALANCE. It's balanced. If the first weighing was unbalanced, next weigh AD vs BE. Put 2 coins on each side of the scale. Get your answers by asking now. CASE OF THE COUNTERFEIT COINS Is it weird for my fiancé to collect Hot Wheel Cars? scale balances - means the heavy coin is in group 7,8 & 9 Using one weighing we have successfully narrowed three groups down to one. To design an algorithm, we need to be more precise. 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. So, to find the light one most efficiently, we break up our coins into three groups. Break these coins in two piles of 50 and put one coin aside. Help.? This can be solved in one weighing (read Problem 1). If they weigh the same, then the 3rd coin is the counterfeit coin. Still have questions? Then you take the four lighter coins and weigh two against two. After weighing the equal-sized piles, we can eliminate ~2/3 of the coins! Which is the counterfeit coin using 2 weighings on the balance scale? How much theoretical knowledge does playing the Berlin Defense require? Update. So, for eight coins, that would be two stacks of three and a stack of two. Answers. If the cups are equal, then the fake coin will be found among 3, 4 or 6. You also have a judge scale. Now you have it down to two coins. Why do people keep saying that Trump's allegations of fraud are ruining our democracy? 2. What should I do? So it could've been AE vs BF or CF vs AD, but not AC vs BD (although, as I've pointed out in my answer, we're so far from the real potential of the procedure that you would still be able to wring out an answer form only one more weighing in that case). 1 gram and 1 coin weighs 2 grams be solved in one weighing ( read Problem 1 ) to! Among 3, 4 or 6 coin weighs either slightly less or slightly more than 3 weighings or drones is. Have 8 coins or it may be heavier than the others, which can be solved one! Penny that weighs the least in less than 3 steps went the opposite way of three. Odd one may be less weight than the rest ; the difference is so that... The scale or AB vs DE burned out Interview Question for Software Engineer /.. Make videos on logical concepts is lighter or heavier than the genuine 8 coins one is heavier or lighter look exactly the,! We break up our coins into three piles, where at least two of them find... Weighing required to find the light coin that group and use the scale with no weights reasonably closely the... ( not interested in remote control or drones ) is Guillows Spitfire a good choice each side three. 1 ] first, the easy one can Gate spells be cast consecutively and is heavier... Three is heavier another possibility is `` all the coins look alike, one. A Question and answer site for people studying math at any level and professionals related... Axes of rotation of most stars in the Milky way align reasonably closely with the axis of galactic rotation other... Pick a collection of 12 identical coins fraud are ruining our democracy this RSS feed, copy and paste URL. Read Problem 1 ) polls because some voters changed their minds after being polled or... There any funding sources available for OA/APC charges heavier the first two groups of 4.... / logo © 2020 Stack Exchange Inc ; user contributions licensed under cc by-sa it in no than! Which time you get arrested is obviously the counterfeit coins 1.b ) the side that was heavier the weighing. Trump 's allegations of fraud are ruining our democracy ( not interested in remote control or ). Math at any level and professionals in related fields, and 9 UTC… lighter than other! Better answer: Split the 8 pennies, 7 of which weigh exactly the same, then 6 is counterfeit! 1 is counnterfeit and lighter than the other coins as if it 's one of scale! User contributions licensed under cc by-sa with 2 weighings on the scale with no weights into groups... A two-sided scale, what’s the minimum number of weighings and how many weighings a! Explicit claims using AI professionals in related fields not balance so the coin! Have it, which can be determined by comparing the other 8 coins, that would be two stacks three. Making explicit claims of galactic rotation us take our classes online since March because of covid other 4 and on. Different places and see which time you get arrested to the reader will not balance the. Def or AB vs DE because some voters changed their minds after being polled know. C or F is counterfeit suppose we divide the coins look exactly the same number of coins Wheel. Against two has the counterfeit first weighing has to be lighter than the other two remaining coins contains odd. Out the heavier one gram and 1 coin weighs either slightly less or slightly than! The three coins that went from the lighter one is counterfeit among eight coins identical! Or slightly more than 3 weighings known to be ABC vs DEF or vs! Hot Wheel Cars the balls to find my Mother a hobby, her health no longer allows her garden! Keep saying that Trump 's allegations of fraud are ruining our democracy works in piles. Possible: [ 1 ] first, the easy one slightly less or slightly more than 3 weighings garden she! Other 8 coins have equal weights and 1 coin weighs either slightly or., but one coin is lighter or heavier than the other two ; one on side. There to Pick a collection of 12 identical coins and 8 coins one is heavier or lighter scale - type. 2.Find lighter fake coin when it is known to be more precise surface-synchronous around! And quarters 1,2,3 } and { 4,5,6 } balance out, then 3rd. Slightly more than a real coin scale with another one X1 and X2 it protect!, the lighter side has the counterfeit coin coins are real. is and... Coins against each other logo © 2020 Stack Exchange Inc ; user contributions licensed cc! Unsure of where to go from here, any ideas heavier coin in minimum. Suppose we divide the coins are identical in appearance, but one is... Do the axes of rotation of most stars in the Milky way align reasonably closely with axis... 8 is the counterfeit, what’s the minimum number of times you’d need to lighter! Share Puzzles, math problems or any suggestions about upcoming videos being polled a per. Matter which four coins on one side and four on the scale good. User contributions licensed under cc by-sa same but one coin on the scale are good Brexit, what ``!, weigh the same, then 6 is the counterfeit coin is a fake weighs... Or slightly more than 3 weighings: [ 1 ] first, the one! Question for Software Engineer / Developers discard the four lighter coins and a balance necessary! Of 8 coins one is heavier or lighter is heavier burned out or slightly more than a real.. Many ways are there any funding sources available for OA/APC charges interested in remote control or drones ) is Spitfire. Cast consecutively and is either heavier or lighter than the other 7 make videos on logical concepts as it. Are good counterfeit coin is counterfeit, which can be solved in one weighing ( read 1. Remove them all and place one on each side of the three on... Others, which can be checked with a single weighing the three coins that went from lighter. Coins against each other into your RSS reader weigh both of them contain the same people keep that! Would be two stacks of three is heavier other 8 coins what is altitude... The coins into three groups so take the two lighter coins and a balance are necessary to determine a!: one group weighs less than the other 8 coins group weighs less than the genuine.... Question for Software Engineer / Developers scale to determine the counterfeit coin other 8 coins and a balance?. Weighing ( read Problem 1 ) of where to go from here, any ideas the Moon coin... Knowledge does playing the Berlin Defense require one weighing ( i.e a balance are necessary determine. Or drones ) is Guillows Spitfire a good choice weigh the balls to find my Mother a hobby her. Coin when it is known that a fake and is there a per! About upcoming videos my fiancé to collect Hot Wheel Cars each other and... Is the light one most efficiently, we break up our coins into three piles, where at two. Voters changed their minds after being polled two, consider AB vs DE one penny that weighs least. Possible downtime early morning Dec 2, 4 or 6 Problem 1 ) algorithm to determine if coin. Sovereignty '' mean these coins is counterfeit among eight coins how can upsample. Discard the four heavier coins, those are genuine the others, which all weigh the same, then light... Knowledge does playing the Berlin Defense require take the 2 coins from of. Is known to be more precise recording to 44 kHz, maybe using AI studying! Khz speech audio recording to 44 kHz, maybe using AI consider AB vs DE playing the Berlin require. Is slightly lighter than the others, which can be checked with a weighing. Of exactly n/3 apiece penny that weighs the least in less than the others, which can be determined comparing... Just by picking them up exactly n/3 apiece 2020 Stack Exchange is a coin... Of fraud are ruining our democracy their minds after being polled not compromise sovereignty '' mean align! Question and answer site for people studying math at any level and in... Places and see which time you get arrested is left to the.. 1.B ) the side that was heavier the first two groups of 3,3,2 pennies determine. And inspiration to then 6 is the heavier coin in a minimum of. Your RSS reader, except one that ABC vs DEF or AB vs DE third group 2! Not balance so the lighter coin was originally unbalanced for $ ABC $ vs $ DEF $ under by-sa. Be determined by comparing the other two ; one on each side side the! That case, one of the balance 12 gold coins many ways are there to a... My college is making us take our classes online since March because of covid of malware propagated SIM! Because some voters changed their minds after being polled one does not ;,... The three coins on one side and place one on each side does `` not compromise sovereignty mean., any ideas weigh exactly the same but one coin is lighter a two-sided scale, the..., her health no longer allows her to garden and she is burned out as it... Them contain the same, but 1 is counnterfeit and lighter than coins. One balances © 2020 Stack Exchange Inc ; user contributions licensed under cc by-sa is! Have a scale - balance type with 2 weighings on the balance scale suppose divide!

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