The solution: Binary numbers

So what is the way to convert a binary number, say, 11011 to its decimal equivalent? A basic simple question for anyone who has gone through high school mathematics. Since I am one of them let me solve it here.

1*16 + 1*8 + 0*4 + 1*2 + 1*1 = 27

In words, number 27 is the sum of 1, 2, 8, and 16. More importantly any decimal number can be seen as a sum of numbers that are powers of two. And the binary equivalent of the number decides which of the power of two numbers take part in the summation. Like in the above case it was 1, 2, 8 and 16. Basic binary number concept right?

Lets take a fresh look at the puzzle I shared at this link. And the solution would become obvious to you. So all numbers less than 128 (binary equivalent of 128 is 1111111) can be seen as summation of numbers 1,2,4,8,16,32 and 64.

Off-course I didn't come to the solution through this route. I picked 1 and 2 and then kept including all the numbers that I could not sum up by using the numbers I already had. So 3 I got from 1+2. Because I could not get 4 from 1 and 2, I included 4 and marched on. By the time I reached 8, I noticed the pattern and  tried 16 to verify. And then the above concept of binary numbers came to my mind. And so I was sure I had the answer.

Also I would like to share one interesting solution Rajesh gave. I will quote his response directly here. You can find his answer at this link in the comment section.

The answer is:
ONE each weight of 1,3,9,27 and 81 kg.
...
I want to weigh 2 as 3-1 and 4 kg as 9-(3+1).
I think this sequence can be extended up to 1000. If fact 1,3,9,27 and 81 can weigh up to 121 kgs.

One puzzle that starts a story

One fine day I happened to mistakenly enter my father's boss's office room. Once I introduced myself and apologized he welcomed me but after knowing I had just written my standard X board exams, he started to bombard me  with a series of puzzles and mathematical questions that afternoon as if it was punishment I had to endure for being in the wrong room at the wrong time. Also I sensed he was using me as a guinea pig  to test the  current schooling standards. I managed to tackle, all his question, but for one. I eventually cracked it that evening and the complete process made me move a notch up in my understanding of mathematics and made me realize how a puzzle can be one of the best teaching methodology. Since then I have asked many this puzzle and have in most cases got the answer in quick time. Much quicker than I managed.

What I will do now is state the puzzle here without further delay and wait for some weeks to get answers from whoever cares to read this post. Most likely some of you have come across it and and I am expecting some participation. In the process of solving the question, each of you, I am sure, will understand the wider implication of this puzzle and would agree to me when I say it could be a good starting point when one introduces a particular of branch of mathematics to students.

The question:

I have been asked to weight all weights up to 100. That is I might be asked to weight, lets say 49 kg or any integer weight under 100 (including). What would be the least number of weights I should have and which ones.

It's possible I was not able to frame the question in best possible way. Let me know if you have any queries.