Five balls in a scale Suppose that I have 5 balls with different weights. A scale tells me which one ball is heavier than another. I have to write down the the pairs of balls I use before I use the scale. Is it possible to write down at most nine pair of balls which tells me the order of balls weights?
I guess the answer is no as 4+3+2+1=10>9 but I have no proof.

As you point out, there are 10 pairs of balls.  If we renumber the balls, any weighing of 9 pairs will omit one pair, so let us say we write down all the pairs except 4 vs 5.  Any ordering of the weights that has 4 and 5 neighbors will be indistinguishable from the same ordering with 4 and 5 reversed.  So you are right that if you have to specify the weighings in advance there are cases you cannot resolve with 9 weighings.  You could tell 14253 from any other order, but could not tell 14523 from 15423.
