There is a 1 gm weight on each side of a balance. If Harry Potter casts a spell with his magic wand on any of the weights, then the mass of that particular weight doubles, but the other weight remains unchanged. How many masses between 1 and 1000 gm inclusive can be measured using this balance?
Please give some hints, not solution to solve this question . Thank you.
On
You can weigh amounts $2^a-2^b$ for $a>b\ge 0,$ and no other amounts. If $a\ge 11$ and $b\le a-1$ then $2^a-2^b\ge 2^{10}>1000.$ So the answer may seem to be the number of ordered pairs $(a,b)$ with $1\le a\le 10$ and $0\le b\le a-1,$ minus the $5$ pairs $(10, b)$ with $0\le b\le 4$ that give $2^a-2^b>1000$. BUT you must not assume without proof that there are no duplicates. If $(a,b)\ne (a',b')$ is it possible that $2^a-2^b=2^{a'}-2^{b'}?$
The most obvious hint is to reduce the maximum number from 1,000 to, say, 10, or even 5, and then enumerate the possible cases. Double one side and you can weigh 2-1=1g, then double twice and you can weigh 4-1=3g, then double the other side and you can weigh 4-2=2g, double the other side again and you can weigh 8-2=6g, and so on. If you just enumerate all the possibilities for a lower number, the way to solve the problem for the general case should become fairly obvious.