Well let's try to look at the problem in a logical way.
I am assuming that the minister has a standard set of weights to keep on one side of the balance.
What would the PM do if there was no constraint of weighing once ?
Of course he/she would weigh each landlord's lumps and then would identify who's the culprit. But what is the implication of this constraint ?
This constraint implies that
1.) Whatever be the technique we cannot leave any two landlord's gift lump out of the one measurement that we are allowed to make because that would lead to ambiguity.
2.) We have to devise a technique to index the measurement that we get to the culprit.
Naturally we can use different amount of lumps from each of the landlord's gifts for our measurement. This would satisfy our need of not leaving any two from the measurement and then having an index of everyone's identity in the measurement. Let me explain how we would index the measurement with the
identity of landlord.
We keep 1 lump from 1st landlord's gift, 2 lumps from 2nd landlord's gift, 3 lumps from 3rd landlord's gift … 12 lumps from 12th landlord's gift. A
So if measurement weighs x*100 grams less from the expected weight of 144 lumps then xth landlord is the culprit.
Hope my answer helps !