# Minimize cuts to an object to obtain weights which can be used to measure 1-40kg

The problem is as follows:

Rick has a small store and a two pan scale which allows him to weight the coffee he sells. Certain day he forgotten the weights that he uses in his truck, however in his toolbox he had found the following:

An iron bar with the shape of a paralellepiped whose large is 40 centimeters and it is of 40 kilograms in weight.

A tape measure

A battery operated angle grinder.

Assuming that without joining, aligning or overlapping the iron pieces. How many straight cuts as a minimum will you make to the iron bar, to obtain a system of weights that allows you to weigh, in a single weighing trial, all weights from one to forty kilograms?

The choices given are:

1. 4 cuts
2. 3 cuts
3. 2 cuts
4. 1 cut

This puzzle seems to be a modification from the usual puzzle which involves finding all the objects which can be measured by using a set of weights of known amounts.

But in this case the set is not given, and instead must be found.

Thus I think it the intent is to split the iron bar into series of small weights so they can be used as an auxiliary weights. But I don't know which weights are needed to get 40 kilograms.

Upon reading this answer and also comparing with what is explained in this Wikipedia entry

I was able to spot that to get 40 kilograms you need at least $$3^n$$

To my mind it comes a balanced ternary, which in short means that one can measure from an interval from zero to that power of three.

To get to 40 kilograms would require at least four weights, $$1+3+9+27$$. But the thing here is that this question is not asking specifically this. But instead the number of cuts to the parallelepiped. How can I find that?

I'm assuming that if a congruent piece of 40 kilograms can be split in those required weights then the problem would be solved. But how to dissect that object to match those required weights?

Can someone help me? Please try to describe the steps as necessary, because I don't know how the dissection should be made to the object to obtain these weights.

Regarding the source, this seems to be an adaptation from an IQ APA exam of the mid 1980s which is based on a puzzle created by Konstantin Knop and mixed with similar puzzles developed in psychometry by Leon Thurstone in the 1960s.

By the way, I've also like to note that regarding the clue about not joining, aligning or overlapping it seems to that the cut might be straight. But since this might be a bit misleading, or leading to a different interpretation, please show what sort of interpretation would fit in the given choices. I assume that it is just doing straight cuts.

And if it that were the case you could use the measuring tape to say, each segment which will cut will be proportional to the weight thus 1 cm is 1 kilogram.

Therefore Rick would end up cutting the pieces by those lengths, and since four of these are required he would only make three cuts therefore the answer. I conclude is choice 2.

But is this the right interpretation? It would help me a lot if someone could explain this in an answer and provide the steps or details.

• Instead of "large is 40 centimeters", did you mean "length is 40 centimeters"? Also, please PLEASE stop using MathJax to make numbered lists. Markdown works perfectly fine, as you have been told many times now. Mar 11, 2021 at 22:43