Keeping you Sharp
Something a little different today. Finish that cup of coffee, and let’s do a few mental push ups to get the blood flowing around the grey matter.
Imagine you have 10 stacks of 10 gold coins. All the coins in one stack are counterfeit, all the others are genuine. A genuine coin weighs 10 grams, and a counterfeit weighs 11 grams. Happily, you have a modern digital scale that can accurately weigh the coins.
Being a fan of efficiency, you don’t want to weigh every stack to find the counterfeits. You want to short-circuit the process to identify the dodgy pile in the minimum possible number of weighings. How will you approach it, and how many weighings will you need to be sure you have identified the fake stack?
Neatest correct answer gets bragging rights.
Tim joined Montgomery in July 2012 and is a senior member of the investment team. Prior to this, Tim was an Executive Director in the corporate advisory division of Gresham Partners, where he worked for 17 years. Tim focuses on quant investing and market-neutral strategies.
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Tim Kelley
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Shortly we’ll publish comments from those who had the correct answer. Once again, nice work if you figured this one out for yourself!
Mike Fox
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Hi Tim
You don’t need to add the tenth stack, because if weight so far is 450, 10th stack is counterfeit, otherwise reason as before.
Best Regards
Mike
Dave B
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Take 1 from pile 1, 2 from pile 2 etc and weigh together (keep seperated on scales). Delta over 550g tells you which pile is fake, e.g. 3g over means pile 3 is fake.
Boris
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Take one coin from pile #1, two coins from pile #2, three coins from pile #3 and so on until we have 10 coins from pile #10.
This gives us 55 coins which if they were all pure would give us 550g. However, some of them will be fake. Lets say we place them all on the scale, for our one and only weighing, and it reads 555g. The only possible way this could have happened is if we placed 5 counterfeit coins on the balance, and that means that pile #5 is counterfeit.
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Questions for you, try to say 50 words in 1 minute that dont contain letter A.
Tim Kelley
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And now the solution: Number your stacks from one to ten. Take one coin from stack one, two coins from stack two, three coins from stack three and so on. The weight of this combination would be 550g if all the coins were genuine. If stack one were counterfiet it would weigh 551g, if stack two were fake it would weigh 552g, and so on. The last digit on our scale tells us which is the fake stack.
If you solved it without having seen this type of problem before, give youself a pat on the back – that’s good insight.
As an aside, this type of insight has some useful practical applications in error detection and correction in digital information storage and transmission.
Austin Gan
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Name those stacks from 1-10. Take one coin from stack 1, two coins from stack 2, three coins from stack 3 and so on…
Then the result should be 550g as you have 55 coins, the additional weight equal the stack number (eg. If result is 557g, stack 7 is counterfeit)
colin
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Keeping You Sharp: Answer = 1 !
We take one coin from stack #1, two coins from stack #2, three coins from stack #3 and so on until we have 10 coins from stack #10.
This gives us 55 coins which if they were all pure would give us 550g. However, some of them will be fake. Lets say we place them all on the scale, for our one and only weighing, and it reads 553g. The only possible way this could have happened is if we placed 3 counterfeit coins on the balance, and that means that stack #3 is counterfeit.
Jack Ross
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1- Label the stacks one to ten.
2- Create a pile of coins using as many coins as the label of the pack. (one from pack one, two from pack two and so on)
3- Weigh the resulting pack.
4- The result would be 10+9+8+7+6+5+4+3+2+1 + X
5- X is the counterfeit pack
Tim Kelley
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A lot of answers in already, all thoughtful, some spot on. Before we reveal the solution, a clue: there is a way to identify the counterfeit stack with just a single weighing, and without needing any luck.
Andrew Podgornik
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It would take 1 weigh
I would take 1 coin from the first stack, 2 coins from the second stack, 3 coins from the third stack and so on up to 10 coins from tenth stack then weigh them together on the scale and subtract the normal weight of the coins from the measured weight ie. measured weight subtract (1+2+3+4+5+6+7+8+9+10) x 10 grams. The amount over would tell me which stack is counterfeit. eg if the measured weight was 3 grams over normal weight I would know third stack was counterfeit as 3 x 11 gram coins included.
Lucas
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Take one coin from the first stack, two from the second and so on leaving out the last stack (i.e. you weigh 45 coins in total). The fake stack is the one with the same last number of the reading on the scale unless it reads 450gm in which case the last stack is fake. i.e. 456g means you are 6g over and they must come from stack 6.
You can also do it by weighing the last stack with the rest but leaving it out is more efficient.
Anthony Doe
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One weighing only.
Order the stacks carefully in a row.
Number them, in order, one to ten.
Take one coin from the first stack, two coins from the second, three coins from the third, four from the fourth and so on until all from the tenth.
Weigh the coins you have taken. Record this.
Calculate the weight of the coins assuming they are all non counterfeit.
Subtract this from the weight you have recorded.
The difference in grams is the column number containing the counterfeits