A number theory problem by A Former Brilliant Member
There are 10 bottles. 9 of them contain tablets weighing 1 gm. each, while the remaining contains tablets weighing 2 gms. each. There are more than 1000 tablets in each bottle. What is the minimum number of times you must weigh the bottles or the tablets to identify the bottle containing the 2 gm. tablets definitely?
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1 solution
Label the bottles with numbers from 1 to 10. Take 1 tablet from bottle number 1, 2 tablets from bottle number 2 and so on. Mix them and weigh. Subtract 55 from the result. The bottle labelled with the number resulting from subtraction is the distinct bottle. For an explanation of this, let all the bottles contain 1 gm. tablets only. Then the total weight would be 55. Since i tablets are taken from bottle number i, which is distinct, containing 2 gm. tablets, we get i gms. extra in the result. That is, our result will be 55+i gms.