Some Challenging logical questions

[Hm2]

Edit:
Problems have been edited to be a bit more generic. The questions remains the same in spirit.

1. You have been given an infinite number of ropes, each of which burns completely in exactly one hour. However, the rates at which different parts of any given rope varies. For example, if you cut a rope in half and burn one of those halves you may find it takes one minute. Or it might take 59 minutes. You can't predict how long it will take for a partial segment of a rope to burn to completion. How can you burn ropes, some perhaps simultaneously, so as to measure exactly one hour and 15 minutes?


2. Devise a set of four test masses with integer masses that total to 40 kg such that the test masses can be used on a balance scale to measure the mass of any object with integral mass between 1 and 40 kg, inclusive.


3. Using this image:

Design a path through the lands so that you cross each of the seven bridges once and only once. The path must be uni-directional, i.e. you cannot double-back and move the way you came.

 

[Hm2]

Can any1 be kind enough to help?

 

[gneill]

Can any1 be kind enough to help?

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.

 

[SammyS]

Can any1 be kind enough to help?

You have waited all of 5 minutes.

Wait 24 hours before "bumping" your thread. That's one of the rules for these Forums.

As to your 1st solution, you are not taking this statement into account, "Some parts burn faster than the others. "

 

[D H]

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.

 

[dacruick]

2. You are given a stone, weighing 40 kg. Your job is to break it in four different parts such that they can then be used in a balance to accurately weigh all objects in the range from 1-40kg. What would be the individual weight of the four parts? (Assume that all objects to be measured have integer weights, i.e. no decimal numbers)

Answer: I have no idea as to how to approach this question. It seems so vague

You will need to be able to place weights on either side of the balance to get all numbers.

EDIT: But im trying to do this right now and I can't...haha

EDIT2: Wait! If I break it in 4 different parts does that mean I break it 4 times, into 5 pieces?

 

[kromkat]

Do not post answers to these questions. They are the questions to an entry test.
The guy is plagiarizing.

 

[D H]

You will need to be able to place weights on either side of the balance to get all numbers.

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.

 

[D H]

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.

 

[D H]

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.

 

[PeterO]

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.

In that case..
light both ends of a string - it will take 1/2 hour to burn
At that time light both ends of a second string, plus one end of a third.
When the second string has burned, light the other end of what is left of the 3rd sting, with "1/2 hour" of the string left, it will only take a further 15 mins to burn from each end.

 

[D H]

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.

 

[D H]

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.

 

[Hamzah28]

The first question is quite easy, it can be done with 2 ropes as well as 3.
The second question definitely a trick question, as I myself have not found the four individual weight values.
Same goes for the third. Can't figure it out, even with the above given diagram. :P

 

[D H]

The second question definitely a trick question, as I myself have not found the four individual weight values.

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?

Same goes for the third. Can't figure it out, even with the above given diagram. :P

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)?

 

[Hamzah28]

I fully agree with your method of shifting weights to the left and right side of the pan. But there must be a formula, through which we can determine the 4 integral weight values...or do we have to try out random weight value combinations again and again until we hit the right one?

 

[D H]

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.)

 

[Hamzah28]

This is the problem which im facing. Have spent 4 hours on this question trying out random weight combinations, but no luck. There are always 2 or 3 mass values which cannot be determined using the 4 weight combinations. This has led me to believe that this question cannot be solved using only 4 parts.

 

[Hamzah28]

-40kg? is that even possible?

 

[Hamzah28]

Explain the term 'base' to me please. :P

 

[Jimmy Snyder]

Spoiler

1, 3, 9, 27
1: 1
2: 3 - 1
3: 3
4: 3 + 1
5: 9 - 3 - 1
6: 9 - 3
7: 9 + 1 - 3
8: 9 - 1
9: 9
10: 9 + 1
11: 9 + 3 - 1
12: 9 + 3
13: 9 + 3 + 1
14: 27 - 9 - 3 -1
etc.

 

[D H]

We use base 10 because we have ten fingers, not because there is something magical about ten. There is nothing special about ten. For example, your computer uses base 2. Here is how to count from 1 to what we call 15 in bases 10, 4, 3, and 2:

  Base 10  Base 4  Base 3   Base 2
      0        0       0        0   
      1        1       1        1   
      2        2       2       10  
      3        3      10       11  
      4       10      11      100 
      5       11      12      101 
      6       12      20      110 
      7       13      21      111 
      8       20      22     1000
      9       21     100     1001
     10       22     101     1010
     11       23     102     1011
     12       30     110     1100
     13       31     111     1101
     14       32     112     1110
     15       33     120     1111

 

[D H]

Jimmy Snyder nailed it. This is anything but a trick question.

 

[Hamzah28]

Indeed. Snyder nailed it! :P

 

[Jimmy Snyder]

Spoiler

This is a famous puzzle, I didn't figure it out. There are several regions separated by rivers. If there are an even number of bridges that reach a region, then you can cross one bridge into the region and one bridge out of the region in pairs. However, if there is an odd number of bridges that reach a region, then that region must be either a starting point or an ending point in your journey since you must cross one bridge and have no offsetting bridge to 'undo' the crossing. Therefore, if there are zero, one, or two regions with an odd number of bridges, it may be possible to start at one region and end at another. It still may be impossible to solve the problem, but at least a start and end point can be established. However, if there are 3 or more regions with an odd number of bridges, then the problem cannot be solved. In this case there are 4 regions with an odd number of bridges and so the problem cannot be solved.

 

[D H]

To see how this relates to base 3, suppose we have a balance scale and weights of 1, 3, 9, and 27 kilograms. Normally when representing numbers in base 3 one uses the digits 0, 1, and 2. There's nothing wrong with using -1, 0, and 1. This will lead to a slightly different counting scheme, but it is still essentially base 3. To make this alternative scheme a bit more compact, I'll use R, 0, and L in lieu of -1, 0, and 1. For short, let's call this alternative base 3 system the scale system. Here is how to count from 1 to 40 in base 10, base 3, and the scale system:

Base 10  Base 3  Scale
    1     0001    000L
    2     0002    00LR
    3     0010    00L0
    4     0011    00LL
    5     0012    0LRR
    6     0020    0LR0
    7     0021    0LRL
    8     0022    0L0R
    9     0100    0L00
   10     0101    0L0L
   11     0102    0LLR
   12     0110    0LL0
   13     0111    0LLL
   14     0112    LRRR
   15     0120    LRR0
   16     0121    LRRL
   17     0122    LR0R
   18     0200    LR00
   19     0201    LR0L
   20     0202    LRLR
   21     0210    LRL0
   22     0211    LRLL
   23     0212    L0RR
   24     0220    L0R0
   25     0221    L0RL
   26     0222    L00R
   27     1000    L000
   28     1001    L00L
   29     1002    L0LR
   30     1010    L0L0
   31     1011    L0LL
   32     1012    LLRR
   33     1020    LLR0
   34     1021    LLRL
   35     1022    LL0R
   36     1100    LL00
   37     1101    LL0L
   38     1102    LLLR
   39     1110    LLL0
   40     1111    LLLL

Note that each value from 1 to 40 has a unique 'scale' representation. Place the object to be weighed on the right pan. Find the unique combination of 27, 9, 3, and 1 kg weights that such that a weight is on the left pan (L), omitted (0), or right pan (R) than makes the balance balance. Encode the combination that yields a balance per the position of the 27 kg weight, then the 9 kg weight, then the 3 kg weight, and finally the 1 kg weight. This is the 'scale' representation of the subject object's weight; you will find the corresponding base 10 representation in the above table.

 

[gneill]

Note that the set of weights is not unique! There is at least one other split of the 40lb rock that will do the job. It takes advantage of the fact that the balance is a "comparison engine' that yields three possible results when comparing weights A and B, namely: A < B, A = B, or A > B. The original problem did not place any restrictions on the number of weighings allowed.

If you allow more than one weighing for determining the weight of the test object of integral weight, then it suffices, for example, to determine that if the test weight is less than 12 and greater than 10, then it must be 11. So you don't need a weight combination that produces 11 explicitly!

So the challenge now is to find another split of the 40lb rock that will allow you to determine the weight of a test object (of integral weight). More than one weighing operation is allowed per object.

 

[D H]

Note that the set of weights is not unique! There is at least one other split of the 40lb rock that will do the job. It takes advantage of the fact that the balance is a "comparison engine' that yields three possible results when comparing weights A and B, namely: A < B, A = B, or A > B. The original problem did not place any restrictions on the number of weighings allowed.

If you allow more than one weighing for determining the weight of the test object of integral weight, then it suffices, for example, to determine that if the test weight is less than 12 and greater than 10, then it must be 11. So you don't need a weight combination that produces 11 explicitly!

So the challenge now is to find another split of the 40lb rock that will allow you to determine the weight of a test object (of integral weight). More than one weighing operation is allowed per object.

Be that way then! (Grumble, grumble, makes me rewrite the problem statement again.)

Devise a set of four test masses with integer masses that total to 40 kg such that for every integral mass w between 1 and 40 kg (inclusive) there exists a way to achieve an exact balance on a balance scale with the mass w and some or all of the test weights placed on the pans of the balance scale.​

Sans such a rewrite there are 43 additional solutions.