Some Challenging logical questions

Your puzzles might fair better in the General Math Forum, as they don't appear to be specifically physics-oriented.

Also, keep in mind that forum rules strongly discourage "bumping" a thread before 24 hours has elapsed.

 

To Hm2: you posted this in physics homework; why?

Some questions:
1. Is this homework?
2a. If the answer to question 1 is yes, why didn't you follow the template, and what made you think this is physics homework?
2b. If the answer to question 1 is no, what made you post this here rather than in Brain Teasers?

Eventually this thread needs to be moved to the Precalculus Mathematics homework section or to Brain Teasers.

 

Or not use some of the weights at all. For example, if one of the lopped-off chunks of stone has a mass of 1 kg, the remaining three chunks are not needed to weigh a 1 kg mass.

 

Mentor's note
The OP was indeed cheating and was plagiarizing as charged.

These questions however are old and do not belong to the group that levied the charges. That group's specific wording and specific imagery does belong to them. I have rephrased the original post to get around the plagiarism problem. The OP has already received full answers to each of these questions elsewhere on the internet, so there is no need to worry that in answering the questions you would be aiding and abetting the OP's cheating.


Feel free to have fun answering the questions as modified.

 

Hints:

1. This is relatively easy with three ropes. It can also be done with only two ropes. That solution isn't so easy.

2. The goal is to come up with a set of four integer masses that total to 40 kg (example: 1 kg, 1 kg, 1 kg, and 37 kg, but that doesn't work) that will let you determine the mass of a given object whose mass is anywhere from 1 to 40 kg, in steps of 1 kg. Assume the object to be weighed is placed on the right hand pan of the balance scale. Each one of the test masses can be placed on the left hand pan, the right hand pan, or not used.

3. The image that I supplied should be a dead giveaway.

 

Now find a solution using only two ropes.

Hint: Start by lighting one rope at one end. Wait until that rope burns to completion. Now do something with the other rope to mark the remaining time.

 

The two rope solution:

Spoiler

Light rope #1 at one end and let it burn to completion. This will happen at t=60 minutes. At this time, light rope #2 at both ends and somewhere in the middle. That internal burn will cut the rope into strands, each of which is burning at both ends. One strand will take 15-x1 minutes to completion, the other in 15+x1 minutes, where 0≤x1<15. That first strand will burn to completion at time t=75-x1 minutes. At this time, the other strand has 2*x1 minutes remaining before it burns to completion; light that other strand somewhere in the middle at this time. Now we once again have two strands each of which is burning at both ends. One strand will take x1-x2 minutes to burn to completion, the other x1+x2 minutes, where 0≤x2<x1. That first strand will burn to completion at time t=75-x2 minutes. At this time, the other strand has 2*x1 minutes remaining before it burns to completion; light that other strand somewhere in the middle at this time. Continue this process until both strands burn to completion at the same time (to within your ability to see any difference).

Because 15>x1>x2>x3>...≥0, the sequence {75-x1, 75-x2, 75-x3, ...} will converge monotonically to 75.

 

Not at all a trick question!

Each one of the individual weights can be on the left pan of the scale, the right pan, or not used. Three choices. What base does that suggest you should be working in?

This one is a trick question. Click on the image. What is the name of the image file (the part after the last slash and before the .png suffix)?

 

The selection of weights is anything but random.

Another hint: If the object's mass could be negative, the correct selection of (positive) test weights would be able to handle everything from -40 kg to +40 kg, a total of 81 values.

What is the prime factorization of 81? What base should you be working in? (Hint: It's not base 10.)