Maria has a balance scale that can indicate which of its pans is heavier or whether they have equal weight. She also has 4 weights that look the same but have masses of 1001, 1002, 1004 and 1005g. Can Maria determine the mass of each weight in 4 weightings? The weights for a new weighing may be picked when the result of the previous ones is known. The Jury (For the senior paper) The same question when the left pan of the scale is lighter by 1g than the right one, so the scale indicates equality when the mass on the left pan is heavier by 1g than the mass on the right pan. Alexey Tolpygo
Problem
Source: 42nd International Tournament of Towns, Junior O-Level P3 & Senior O-Level P2, Spring 2021
Tags: combinatorics, Tournament of Towns
QueenArwen
05.10.2024 11:11
Yes. For her first weighing, Maria puts two weights on each pan. The possible combinations are $1001, 1002$ on one pan and $1004, 1005$ on the other; $1001, 1004$ on one pan and $1002, 1005$ on the other; and $1001, 1005$ on one pan and $1002, 1004$ on the other. In the first two cases, one pan will be lighter than the other and in the last one both will be equal.
If the pans are equal, Maria knows that one of them has $1001, 1005$ and the other has $1002, 1004$. For her second weighing she weighs the weights from one pan against each other and does the same for the third weighing. She now knows that the lighter ones from each weighing are $1001$ and $1002$ and the heavier ones are $1004$ and $1005$. For her last weighing, she weighs the two lighter ones against each other. The lighter of these is $1001$ and the other is $1002$. She also knows that the weight that was with $1001$ in the first weighing is $1005$ and the weight that was with $1002$ in the first weighing is $1004$.
If the pans are unequal, Maria knows that the lighter one contains $1001$ and the heavier one contains $1005$. For her second weighing, she weighs the weights of the lighter pan against each other. The lighter weight will be $1001$. For her third weighing, she weighs the weights of the heavier pan against each other. The heavier weight will be $1005$. For her fourth weighing, she weighs the two unknown weights against each other. The lighter one will be $1002$ and the heavier one will be $1004$. Hence Maria can determine the mass of each weight in four weightings.