IA (HL) Portfolio Type I -- How Many Pieces

[Ben92]

Hi

Does anyone have any idea how to cut a cube into 15 pieces using only 4 cuts?

Thanks

[pieee]

Hi

Does anyone have any idea how to cut a cube into 15 pieces using only 4 cuts?

Thanks

The idea is that when you're cutting the nth cut, you should try to cross n-1 cuts that are already on the cube.

I'd recommend using a software cuz then you could tweak the points that make the plane (I used Cabri 3D which works pretty well)

by the 5th cut, you'll have pieces that are so small they might seem like a dot, but it's there.

[Guest JimmyK]

Hi

Does anyone have any idea how to cut a cube into 15 pieces using only 4 cuts?

Thanks

The idea is that when you're cutting the nth cut, you should try to cross n-1 cuts that are already on the cube.

I'd recommend using a software cuz then you could tweak the points that make the plane (I used Cabri 3D which works pretty well)

by the 5th cut, you'll have pieces that are so small they might seem like a dot, but it's there.

Google sketch up 8 works perfect too. The only problem is intersecting the planes though you can do that by first creating a square in 2D, intersecting the square as if you were doing for 2D and then there is a function in the Sketch Up which makes it to 3D by pushing it upwards

[Marvin]

I'm having difficulty for the investigation on 2D, 3D, and 4D objects.

Recursive rule, cuts made, everything!!!! Can anyone help?

Thanks!

[Ben92]

I'm having difficulty for the investigation on 2D, 3D, and 4D objects.

Recursive rule, cuts made, everything!!!! Can anyone help?

Thanks!

Hi,

can you be more specific?

I can give you a hint to help you along with finding the maximum number of cuts in 2D. when you make a cut, to maximize the number of regions, you have to cut all the previous lines you've used to cut the circle in new distinct points of intersection.

Hope this helps. If you ask a more specific question I'll be happy to help

[therationalist]

Hey guys,

Here are the things I'm having trouble with. Can anyone help me out (Ben 92?)

1). I got the conjecture for the one-dimensional one but it also says comment on your results? What do you comment about? I mean what else do you say beside the conjecture?

2). For the 2-dimensional one I got the conjecture, the diagrams, and the technology part. But I'm having trouble with the recursive formula because doesn't the recursive formula for the 2-dimensional one have to use the the results from the one-dimensional one. In fact, my doubts here continue on because the 3-dimensional one uses both 2 and 1 dimensions and the 4th uses the 3,2 and 1 dimensions?

3). Moreover for the 2-dimensional one, I know you prove it by induction. But how? Do you use n+1 or n-1? Please help here.

4). For the 3rd dimension one, I know the conjecture will be cubic but I'm having trouble finding it. Do you use regression? And moreover, how do you prove it?

5). Which software do you use to draw the 3D cube? I tried using Capri but I don't know how to add cuts on it. Can anyone please help?

6). I aim on getting a 20/20 and so to get this I understand that one needs to do extra. Can anyone tell me what additional things or extra features I should add. I know one could be coming up a conjecture for the n-dimensions ? Anything else?

Anyways, thanks to anyone that reads this or helps.

Edited February 20, 2011 by therationalist

[Marvin]

I'm having difficulty for the investigation on 2D, 3D, and 4D objects.

Recursive rule, cuts made, everything!!!! Can anyone help?

Thanks!

Hi,

can you be more specific?

I can give you a hint to help you along with finding the maximum number of cuts in 2D. when you make a cut, to maximize the number of regions, you have to cut all the previous lines you've used to cut the circle in new distinct points of intersection.

Hope this helps. If you ask a more specific question I'll be happy to help

Actually, I can't get the recursive rule for the 3D object:

A cuboid is a finite three-dimensional object. Investigate the maximum number of parts (P) that are obtained with n cuts.

oUse geometry software to investigate the maximum number of parts for n =1, 2, 3, 4 and 5 (free

three-dimensional geometry software is readily available online).

o Tabulate your results.

o Find a recursive rule to generate the maximum number of parts.

o Use technology to make a conjecture for the relationship between the maximum number of

parts (P) and the number of cuts (N ).

o Prove your conjecture.

o Rewrite your formula in the form P = Y + X + S , where Y is an algebraic expression in n.

The one in bold is my question.

[Guest JimmyK]

I'm having difficulty for the investigation on 2D, 3D, and 4D objects.

Recursive rule, cuts made, everything!!!! Can anyone help?

Thanks!

Hi,

can you be more specific?

I can give you a hint to help you along with finding the maximum number of cuts in 2D. when you make a cut, to maximize the number of regions, you have to cut all the previous lines you've used to cut the circle in new distinct points of intersection.

Hope this helps. If you ask a more specific question I'll be happy to help

Actually, I can't get the recursive rule for the 3D object:

A cuboid is a finite three-dimensional object. Investigate the maximum number of parts (P) that are obtained with n cuts.

oUse geometry software to investigate the maximum number of parts for n =1, 2, 3, 4 and 5 (free

three-dimensional geometry software is readily available online).

o Tabulate your results.

o Find a recursive rule to generate the maximum number of parts.

o Use technology to make a conjecture for the relationship between the maximum number of

parts (P) and the number of cuts (N ).

o Prove your conjecture.

o Rewrite your formula in the form P = Y + X + S , where Y is an algebraic expression in n.

The one in bold is my question.

If you do the sequence of your results, you'll be able to reach to the conclusion that if follows a recursive rule that you have to find out. The recursive rule can be found out through the method of finite differentiation... That's basically what you have to do. I can tell you the recursive but it's of no use if you can't prove it

[Guest JimmyK]

Hey guys,

Here are the things I'm having trouble with. Can anyone help me out (Ben 92?)

1). I got the conjecture for the one-dimensional one but it also says comment on your results? What do you comment about? I mean what else do you say beside the conjecture?

2). For the 2-dimensional one I got the conjecture, the diagrams, and the technology part. But I'm having trouble with the recursive formula because doesn't the recursive formula for the 2-dimensional one have to use the the results from the one-dimensional one. In fact, my doubts here continue on because the 3-dimensional one uses both 2 and 1 dimensions and the 4th uses the 3,2 and 1 dimensions?

3). Moreover for the 2-dimensional one, I know you prove it by induction. But how? Do you use n+1 or n-1? Please help here.

4). For the 3rd dimension one, I know the conjecture will be cubic but I'm having trouble finding it. Do you use regression? And moreover, how do you prove it?

5). Which software do you use to draw the 3D cube? I tried using Capri but I don't know how to add cuts on it. Can anyone please help?

6). I aim on getting a 20/20 and so to get this I understand that one needs to do extra. Can anyone tell me what additional things or extra features I should add. I know one could be coming up a conjecture for the n-dimensions ? Anything else?

Anyways, thanks to anyone that reads this or helps.

1. You comment it in terms of the results that you will further obtain from it by saying if it follows your logic (which you need to explain)

2. Yes, all dimensions use the results obtained in the previous dimensions because if you actually see, the dimension you'll be studying is an expansion of the previous ones. Now, what happens is that the recursive is found through the use of the sequence of parts that you obtained solely from the results of that particular dimension. You do not use the recursive rules of the other dimensions.

3. Induction is always proved through n+1

4. Don't know that one to be honest...

5. I used Google Sketch Up and I recommend it

6. Commentaries and try to broaden to other areas and try to investigate more about dimensions and geometry. That might help

[therationalist]

Hey Jimmy,

Thanks for your answer. It helped a lot. For the Google Sketch Pad thing, can you please guide me for the basic steps like how to draw a cuboid, and how to draw lines to cut it in a way that you can see the individual steps. Perhaps you can inbox me the basic steps. Thanks again.

[Ben92]

Hey guys,

Here are the things I'm having trouble with. Can anyone help me out (Ben 92?)

1). I got the conjecture for the one-dimensional one but it also says comment on your results? What do you comment about? I mean what else do you say beside the conjecture?

2). For the 2-dimensional one I got the conjecture, the diagrams, and the technology part. But I'm having trouble with the recursive formula because doesn't the recursive formula for the 2-dimensional one have to use the the results from the one-dimensional one. In fact, my doubts here continue on because the 3-dimensional one uses both 2 and 1 dimensions and the 4th uses the 3,2 and 1 dimensions?

3). Moreover for the 2-dimensional one, I know you prove it by induction. But how? Do you use n+1 or n-1? Please help here.

4). For the 3rd dimension one, I know the conjecture will be cubic but I'm having trouble finding it. Do you use regression? And moreover, how do you prove it?

5). Which software do you use to draw the 3D cube? I tried using Capri but I don't know how to add cuts on it. Can anyone please help?

6). I aim on getting a 20/20 and so to get this I understand that one needs to do extra. Can anyone tell me what additional things or extra features I should add. I know one could be coming up a conjecture for the n-dimensions ? Anything else?

Anyways, thanks to anyone that reads this or helps.

1. You comment it in terms of the results that you will further obtain from it by saying if it follows your logic (which you need to explain)

2. Yes, all dimensions use the results obtained in the previous dimensions because if you actually see, the dimension you'll be studying is an expansion of the previous ones. Now, what happens is that the recursive is found through the use of the sequence of parts that you obtained solely from the results of that particular dimension. You do not use the recursive rules of the other dimensions.

3. Induction is always proved through n+1

4. Don't know that one to be honest...

5. I used Google Sketch Up and I recommend it

6. Commentaries and try to broaden to other areas and try to investigate more about dimensions and geometry. That might help

I agree with everything, plus the 3-D you have to try for n=1,2,3 on your own and you'll see a pattern emerging.... Based on that conjecture a recursive.. based on the recursive generate more maximum number of parts for n=4,5,6,,,100 say. Then, use a cubic regression to find the formula... Also to find out why cubic look up the method of finite differences.

good luck!

[therationalist]

Thanks for your help once again guys. Anyways, for the 4th dimension one it says that you have to use a spreadsheet to find a pattern. How? I mean I find a pattern and everything but how will a spreadsheet help. Also for the 1-dimensional one and the 4th dimensional one it says "comment." Comment about what? Is it the same thing as elaborate. I mean for the 1-dimensional one, after you've found the conjecture, what else do you say. Also, I shall ask this question once again. Getting a 19-20/20 is very rare. What additional or extra features can I bring in to IA to make it reach that 20/20 level. Do I talk about what the 4th dimension is (perhaps time) or do I bring in the 10-dimensional string theory, etc? Thanks once again.

[Ben92]

Thanks for your help once again guys. Anyways, for the 4th dimension one it says that you have to use a spreadsheet to find a pattern. How? I mean I find a pattern and everything but how will a spreadsheet help. Also for the 1-dimensional one and the 4th dimensional one it says "comment." Comment about what? Is it the same thing as elaborate. I mean for the 1-dimensional one, after you've found the conjecture, what else do you say. Also, I shall ask this question once again. Getting a 19-20/20 is very rare. What additional or extra features can I bring in to IA to make it reach that 20/20 level. Do I talk about what the 4th dimension is (perhaps time) or do I bring in the 10-dimensional string theory, etc? Thanks once again.

The 4th dimension in this portfolio is not time, it is spatial. It's hard to visualize, but mathematically it exists. For a 20, I would recommend you find a general formula for any dimension, say the k'th dimension with n cuts. Also,when it says comment you are to summarize your findings perhaps say what you find interesting. But you don't need to both about the instructions, they are there to help you, you should not think in terms of questions and answers. You should aim to produce a standalone report.

[Guest JimmyK]

Thanks for your help once again guys. Anyways, for the 4th dimension one it says that you have to use a spreadsheet to find a pattern. How? I mean I find a pattern and everything but how will a spreadsheet help. Also for the 1-dimensional one and the 4th dimensional one it says "comment." Comment about what? Is it the same thing as elaborate. I mean for the 1-dimensional one, after you've found the conjecture, what else do you say. Also, I shall ask this question once again. Getting a 19-20/20 is very rare. What additional or extra features can I bring in to IA to make it reach that 20/20 level. Do I talk about what the 4th dimension is (perhaps time) or do I bring in the 10-dimensional string theory, etc? Thanks once again.

The 4th dimension in this portfolio is not time, it is spatial. It's hard to visualize, but mathematically it exists. For a 20, I would recommend you find a general formula for any dimension, say the k'th dimension with n cuts. Also,when it says comment you are to summarize your findings perhaps say what you find interesting. But you don't need to both about the instructions, they are there to help you, you should not think in terms of questions and answers. You should aim to produce a standalone report.

I agree with Ben though let me mention the method of finite differences again. Through this method, you're able to see the changes in each dimension. However, if you put together every data that you came up with for the first 3 dimensions, you will clearly notice a trend and if not, you did something wrong lol After this, you can comment whether it is noticeable in reality (as of "transportation" of the n-1 dimension to the nth one) or if it makes sense or not... that's how you comment it. As for what Ben said about the way you write, it is true. It's not an exam paper, it's an investigation... Oh! and make sure you compare with the evaluation criteria your working and see if everything is at the top scores because what really matters is if you are able to follow the criteria properly and not how far you reach this investigation or come up with something that no one has ever came up with.

[ishaan1993]

Does anyone knows 3D geometry software for drawing 3D cube in IB HL IA how many pieces ?

you can use "CABRI" or "GEOGEBRA"

hey guys i need a little help with the general form for an 'n' dimensional object.. If someone could help it would be greatly appreciated..

Moreover, i am also having difficulty on using the spreadsheet part..can we use a GDC? if no, how do we use a GDC?? please guys help...

Thank You...

Edited March 11, 2011 by Desy ♫
do not type in CAPS

[Guest Pingg]

So I am working on this portfolio too and I am stuck on the technology part. What are we supposed to use for technology? a software or something?? Some help will be really appreciated!!

I strongly suggest using Cabri. You can download a 30 day free trial directly from their website.

I know there are other ways of getting your hands on the software... such as school computers.

But if your school doesn't offer it, i would suggest it to them and wait to see if they get it, or just try the trial.

Also, if anyone wants to know how to use Cabri just let me know. Although, it is pretty simple.

[RAL]

Hi,

I am having trouble solving doing proving my conjectures through induction for the 3D and 4D parts (mainly the 4D part). I have my recursive formulas and my conjectures but when I get to the n=k+1 case I do not know what to do. Some help would be greatly appreciated.

Edited March 13, 2011 by RAL

[Guest Pingg]

Hi,

I am having trouble solving doing proving my conjectures through induction for the 3D and 4D parts (mainly the 4D part). I have my recursive formulas and my conjectures but when I get to the n=k+1 case I do not know what to do. Some help would be greatly appreciated.

Well for 3D on the Left hand side (LHS), you have P(k+1) but you have to rewrite that as P(k) + the rule.

And the rule is your equation for 2D minus the rule for 1D.

Then for 4D, the LHS, is P(k+1) which gets rewritten as p(k) + 3D equation minus the rule for the 2D equation...

Obviously, for the RHS of each proof it's just your equations which you can easily get by plugging all your points in excel and finding a trendline.

Just remember to have the order as 3 for 3D and 4 for 4D in excel.

Sorry, I will make my last post more clear.

Just remember your LHS of your proof is always p(k) + the rule.

So for 3D it's p(k) + .5k^2 + .5k + 1.

And then you rewrite p(k) as your inductive hypothesis. That's a pretty big hint.

[Ptolomy7]

Alright, so I am also stuck on 3d, with the the n=4. I got 8 pieces with 3 planar cuts for the cuboid.

Also, i dont understand how to create my recursive pattern into a conjecture. I have a conjecture for 1d (n+1 pieces) 2 d(sum of n +1) but 3d?

Would graphing the values and finding the curve of a quadratic function work as a general rule?

Any help would be extremely appreciated.

Thanks for your help once again guys. Anyways, for the 4th dimension one it says that you have to use a spreadsheet to find a pattern. How? I mean I find a pattern and everything but how will a spreadsheet help. Also for the 1-dimensional one and the 4th dimensional one it says "comment." Comment about what? Is it the same thing as elaborate. I mean for the 1-dimensional one, after you've found the conjecture, what else do you say. Also, I shall ask this question once again. Getting a 19-20/20 is very rare. What additional or extra features can I bring in to IA to make it reach that 20/20 level. Do I talk about what the 4th dimension is (perhaps time) or do I bring in the 10-dimensional string theory, etc? Thanks once again.

The 4th dimension in this portfolio is not time, it is spatial. It's hard to visualize, but mathematically it exists. For a 20, I would recommend you find a general formula for any dimension, say the k'th dimension with n cuts. Also,when it says comment you are to summarize your findings perhaps say what you find interesting. But you don't need to both about the instructions, they are there to help you, you should not think in terms of questions and answers. You should aim to produce a standalone report.

I agree with Ben though let me mention the method of finite differences again. Through this method, you're able to see the changes in each dimension. However, if you put together every data that you came up with for the first 3 dimensions, you will clearly notice a trend and if not, you did something wrong lol After this, you can comment whether it is noticeable in reality (as of "transportation" of the n-1 dimension to the nth one) or if it makes sense or not... that's how you comment it. As for what Ben said about the way you write, it is true. It's not an exam paper, it's an investigation... Oh! and make sure you compare with the evaluation criteria your working and see if everything is at the top scores because what really matters is if you are able to follow the criteria properly and not how far you reach this investigation or come up with something that no one has ever came up with.

Hey JimmyK, thanks a ton for the explanation, saved me a few hours of hair pulling. But what exactly is the method of finite differences, could you give me a small summary or something? The internet isnt being helpful, and my report is due in a few hours = i have a very closed mind.

All help is appreciated, Thanks...

[Guest JimmyK]

Alright, so I am also stuck on 3d, with the the n=4. I got 8 pieces with 3 planar cuts for the cuboid.

Also, i dont understand how to create my recursive pattern into a conjecture. I have a conjecture for 1d (n+1 pieces) 2 d(sum of n +1) but 3d?

Would graphing the values and finding the curve of a quadratic function work as a general rule?

Any help would be extremely appreciated.

Hey JimmyK, thanks a ton for the explanation, saved me a few hours of hair pulling. But what exactly is the method of finite differences, could you give me a small summary or something? The internet isnt being helpful, and my report is due in a few hours = i have a very closed mind.

All help is appreciated, Thanks...

Hey mate, no worries. The method of finite differentiation is just making the differences between each value and these differences (if it is a sequence or a series) will end up in a constant value which will not change no matter how the actual term of the sequence is varying. If you don't understand just take this example:

For n=1, Term=9

n=2, term=27

n=3, term=81

Doing the differentiation,

1st.27-9=18, 81-27=54

2nd 54-18=36

and this would go on until you'd reach a constant number. The number of differentiations that you needed is the degree of your polynomial.

However, you can make a graph out of the sequence and determine the trendline with the r^2 closest to 1 which gives the accuracy of the trendline and then, you'd get a polynomial of that line which would fit for your sequence.

Hope this was useful...