Question

Medal to the best answer.

You can place weights on both side of weighing balance and you need to measure all weights between 1 and 1000. For example if you have weights 1 and 3,now you can measure 1,3 and 4 like earlier case, and also you can measure 2,by placing 3 on one side and 1 on the side which contain the substance to be weighed. So question again is how many minimum weights and of what denominations you need to measure all weights from 1kg to 1000kg.

Answer 1

i didnt get it

Answer 2

read it again u ll understand it

Answer 3

1, 3, 9, 27, 81, 243, 729

Answer 4

i dont know how to solve it

Answer 5

lets rephrase the problem like this , i want to get all numbers from 1 to 1000 as a linear combination of selected numbers ;

See , any number N=3q+r ,where r={0,1,-1}
now for any number N repeating the same thing we get , N=3(3q'+r')3 =3^3q' +3^2r' (if q>3)
Similarly continuing till numbers q,r are {0,1,-1}
then we will have
\[3^{n}q +3^{n-1}q'+ ......+ 3^{1}q'' +3^{0}q''' \] where q's are {0,1,-1 }

there fore we can get any number .

Answer 6

I was inspired by the answer of @rational to think about it , and when i was confident , i was thinking a way to prove it and got that .

now you can also see , if we take the numbers in the set {1,4,16..} we cannot get 2 . so any other powers greater than 3 will not work .

now if we write in powers of 2 , N=2q+1 , ie any number can be written as N= \[2^{n}+2^{n-1}......+2^{1}+2^{0}\] but takes a lot more numbers , (note this fact is used in binqary number system) .

now why cant any other set of numbers that is more optimum than the powers of 3 , remains to be proved , i am still thinking about that

Answer 7

Now our optimum answer till now is 7 weights . let assume that there is a set that has only 6 weights and still able to measure 1000 weights ,
say {a,b,c,d,e,f} we can measure {a-b+c+d-e-f ,...etc} then # of possible measurements is 3^6=(can be +.- or 0 in between) so only 729 different measurements possible .

So , our answer must have atleast 7 weights .

Answer 8

@LichKing ??? which is the best answer here ??

Answer 9

ah sorry didnt notice