What is the largest value of the integer $n$, if the mass of any object can be determined using four suitable weights and a two-pan balance, given that we know the mass of the object in kilograms is an integer between 1 and $n$? (With the two-pan balance, we can perform as many measurements as we like, but we can only place the weights and the object to be measured on the pans, and we cannot cut either the weights or the objects to be measured.)
Solution. The solution to the original version of the problem hinges on the observation that four weights can be arranged in $3^{4}=81$ different ways on a two-pan balance. Each weight can independently be placed in the left pan, the right pan, or left out of the measurement. Thus, the difference in the contents of the left and right pans can take on at most 81 different values, regardless of the weights' values. Among these, there is the 0 - if no weights are placed - and the other eighty can be paired into opposites by swapping the contents of the pans in each arrangement. Including 0, this results in a total of 41 non-negative values that can be measured, and if the weights are 1, 3, 9, and 27 kilograms, respectively, then these values are precisely the integers from 0 to $1+3+9+27=40$. Any arbitrary arrangement of these weights will clearly measure a non-negative value within these bounds, and if there were two arrangements that yielded the same value, then setting the resulting equality to zero would imply that a number has two different representations in the ternary (base-3) system.

The classic problem is now modified in that a body's mass is also "determinable" in cases where we cannot find weights among ours that can balance the body. For example, if after a series of measurements it turns out that the mass of the body is greater than 3 kilograms and less than 5 kilograms, then according to the condition, it can only be 4 kilograms.

It is still true that with four weights, at most 40 different positive masses can be balanced on a two-pan balance. If every integer in the interval $[1; n]$ is either measurable or can be split into such numbers - as the measurable 3 and 5 values separate the 4 from the other integers in the example above - then the mass of any body with an integer mass from 1 to $n$ can be determined.

This implies that there cannot be consecutive integers in the interval that are both unmeasurable. Thus, the largest measurable value cannot be greater than 80, since, as we have seen, there can be at most 40 different positive measurable numbers. This limit is achievable if we double the weights that are the solution to the classic problem. Following the original problem, with weights of 2, 6, 18, and 54 kilograms, every even integer between 0 and 80 can be measured. If the mass of a body is an odd integer less than 80, then a series of measurements can determine the two consecutive even numbers that bracket the body's mass; thus, the odd number is also determined.

Finally, if the maximum mass of the body to be measured is predetermined, then $n=81$ is also possible, for in this case it is justified to conclude that if the body is heavier than 80 kilograms, then its mass is 81 kilograms. If the problem statement is interpreted to mean that before the measurement begins, we only know that the mass of the body is an integer, then we cannot undertake to determine the mass of a body heavier than 80 kilograms.