How can it be done with least number of weighing?

u weich each box, so u would only weigh 10 times. if the amount is over 1000, then thats the box.

split the 10 boxes into 2 groups of 5, group A and group B.

Weigh group A if the weight is 500 lbs Group B has the heavy box

If Group A weighs 510 then it is the group with the heave box.

Let's say Group B has the heavy box, Break group b into 2 groups, one group will have 3 boxes the other group 2 boxes. Weigh the group with 2 boxes, if it weighs 210 one of the 2 boxes is the heavy one. If the group with the 3 boxes weighs 310 then the heavy box is in that group.

If you have just an old weighing scale without graduations...(i.e) don't have graduated weighing machine or modern equipments the below method is the simplest and easiest way...if you have a modern weighing machine then go ahead with RJJ's answer

METHOD:

Split 10 boxes into 2 groups "X" and "y"....X should contain 5 boxes and Y should contain 5 boxes.

then one of them(example -X) is heavier...leave the lighter group...and take the heavier group of balls(5 boxes)

now take one box randomly out from the 5 boxes and make 2 groups from the remaining boxes as U and V..name the random box taken out as "L"...if U(2 boxes) and V(2 boxes) are equal then "L" is the heaviest box of all...thus u got the answer...if U and V are unequal...then take the heavier one(example-U) then separate the two boxes and measure them...thus now the heavier box is the one which is the heaviest of all boxes you will be able to find the correct box...

Actually you can do it with only a single weighing

1.First name the boxes as 1,2,3...,10.

2. Then you take 1 ball from box 1 , 2 balls from box 2 and so on.

3. Weigh them all at once.

4. If all boxes had balls of 10lbs then you should have got 10x10=100 lbs ,

But since a box is having balls of 11 lbs you get more weight.

5. This excess weight gives you your box number i.e , if you get 113 lbs then the box is 113-110 = box no. 3 .( As you have taken 3 balls from box 3) .

I know I am posting it late but hope this helps you.