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The Genius
Posted: Monday, January 14, 2013 6:03:31 AM
Rank: Newbie

Joined: 1/14/2013
Posts: 15
Neurons: 45
Location: Hong Kong
Ladies and gentlemen,welcome to 'Ahem!!'
In fact I am not quite sure if my topic goes in this category,but whatever.
Yesterday I suddenly thought of a truly vexing problem-here it goes!

There are six 2x4 lego blocks,and they all look exactly the same. How many combinations can we make with the blocks,without any repetitions?

How was that?I couldn't sleep well yesterday because of this.So,hope you,yea,YOU could offer me a helping hand.
(I want the sum,the calculation,not the number-that'll be enormous!)
Tovarish
Posted: Monday, January 14, 2013 6:23:19 AM
Rank: Advanced Member

Joined: 9/2/2009
Posts: 11,101
Neurons: 39,933
Location: Booligal, New South Wales, Australia
I dont know wether or not to mark you as spam.

Is this a genuine question?

Be careful, you are a Newbie.
IMcRout
Posted: Monday, January 14, 2013 7:10:49 AM
Rank: Advanced Member

Joined: 5/27/2011
Posts: 35,380
Neurons: 563,379
Location: Lübeck, Schleswig-Holstein, Germany


[image not available]
thar
Posted: Monday, January 14, 2013 7:27:47 AM

Rank: Advanced Member

Joined: 7/8/2010
Posts: 22,793
Neurons: 92,591
love the hobbit with the smirk (can't remember names)

genie - ah, the manifold pleasures of permutations and combinations! I am glad I was into all that before I had my brain surgery, (so I must have been about six or seven) because I can't remember any of what I was working on then and probably wouldn't understand it now post-surgery! Combinatorics is evil. Rocks are good. Much more fun to mark lego blocks and play until you can produce no more unique combinations... then realise it would have been easier using maths....

But you have to do at least some of the work yourself, or it is no fun....
The Genius
Posted: Monday, January 14, 2013 7:58:48 AM
Rank: Newbie

Joined: 1/14/2013
Posts: 15
Neurons: 45
Location: Hong Kong
To Tovarish:Well,why?If you don't know the solution,just don't post your 'answer'.And,why should I be careful because of being a 'Newbie'?
Thar: Ha,you're there again.Okay,let me investigate,and of course I would,but if I could learn to CALCULATE it,it would be much faster,and of course I'd love to listen to a solution...I mean,has anyone thought of the same question?No?Thar,do you really know the solution,or are you simply pretending?You SEEM to be very smart,but,well,do you really know the answer?
For IMcRout:??????????????????

ThX
Ravindra
Posted: Monday, January 14, 2013 8:12:24 AM
Rank: Advanced Member

Joined: 3/23/2009
Posts: 733
Neurons: 63,152
Location: Bangalore, Karnataka, India
This forum, I am sure, has no room for the rude.

Courtesy begets courtesy.
thar
Posted: Monday, January 14, 2013 9:40:57 AM

Rank: Advanced Member

Joined: 7/8/2010
Posts: 22,793
Neurons: 92,591
1- no, I don't know the anwer
2- what is the point of just giving an answer? THe fun in maths is trying to work it out!
Tovarish
Posted: Monday, January 14, 2013 8:06:32 PM
Rank: Advanced Member

Joined: 9/2/2009
Posts: 11,101
Neurons: 39,933
Location: Booligal, New South Wales, Australia
Oh, another one.

The Genuis, if you notice on the top right hand corner of your post there is a Spam Button, look again and you will see, only on yours.

You will also notice all the other members who answered your post are regarded as Advance Members.

You as a Newbie are on probation.

Do not be rude to other members, or we will push your button!

Ask sensible questions and you will get sensible answers.
Epiphileon
Posted: Monday, January 14, 2013 8:13:25 PM

Rank: Advanced Member

Joined: 3/22/2009
Posts: 4,287
Neurons: 166,581
Tov I don't think new members see the spam button.

The Genius, welcome to the forum, there are a great bunch of people here, I would advise that you not take offense but listen to the advice you've been offered.
Jyrkkä Jätkä
Posted: Monday, January 14, 2013 8:17:45 PM

Rank: Advanced Member

Joined: 9/21/2009
Posts: 43,131
Neurons: 595,331
Location: Helsinki, Southern Finland Province, Finland
I'm not at the Lego-age any more, nor are my children.
My granddaughter will be at that age soon, but we'll let her figure out the combinations with her own hands.

Genius,
you could try pressing the spacebar in your keyboard after every comma and period. That way your writing would be more readable.

Ray41
Posted: Monday, January 14, 2013 9:49:08 PM

Rank: Advanced Member

Joined: 9/9/2010
Posts: 1,937
Neurons: 45,980
Location: Orange, New South Wales, Australia
JJ wrote;
Genius,
you could try pressing the spacebar in your keyboard after every comma and period. That way your writing would be more readable.


I would have thought that even a genius with only a small 'g' would have worked that out.
Maybe he should be demoted for awhile, like to 'budding genius' or 'wannabe genius'.Whistle

Mmmm! 12 years old?Think
Tovarish
Posted: Monday, January 14, 2013 10:07:00 PM
Rank: Advanced Member

Joined: 9/2/2009
Posts: 11,101
Neurons: 39,933
Location: Booligal, New South Wales, Australia
I cant remember being 12 years old, I think I skipped right over than one.


"Well, why?"
The Genius
Posted: Tuesday, January 15, 2013 5:44:37 AM
Rank: Newbie

Joined: 1/14/2013
Posts: 15
Neurons: 45
Location: Hong Kong
Okay,okay.I really don't get it-this is not a 'sensible' question? And,I'm sorry,if I was really rude(In fact I don't really recognize it),I do apologize-only if I REALLY was rude.PLZ tell me why.
If this isn't a 'sensible' question,well then,tell me what a 'sensible' one is like.It doesn't mean it's nonsense just because you don't understand,guys(Is this forum very formal actually?Then I shall erase the 'guys'...). Simply ignore if you are uncertain. Who said answering me is compulsory?
If I am sounding rude again,please forgive me for my courtesy.You know,Hong Kongers(Duh!) are not very nice-I wish I was born in another country despite the good education system in HK. Well,I really hate people with bad manners and etiquette. Thanks for reminding me then,and it is really great to find out this forum.THX.
Back to topic-I know this is a tricky one. Yea. My foolish Math teacher didn't know the answer(She just yakked out something totally different,in which I could see that she knows nothing.),and though that was just what I expected. I think I should investigate by myself and maybe... find something NEW?
-So now you'd believe that I'm 12. The childishness-Finding something new!Ha! But I can't stop thinking about it.So... yea.
THX again.
Drag0nspeaker
Posted: Tuesday, January 15, 2013 7:51:24 AM

Rank: Advanced Member

Joined: 9/12/2011
Posts: 34,427
Neurons: 228,163
Location: Livingston, Scotland, United Kingdom

Well - I think it's a good puzzle.

Of course, you have to really listen to the question, and ignore the other 'rambling words' that are given around it.

The question in full is: "There are six 2x4 lego blocks,and they all look exactly the same. How many combinations can we make with the blocks,without any repetitions?"

Ignoring all the redundant words, the question is: "There are six blocks. How many combinations can we make with the blocks?"

Combination
In mathematics a combination is a way of selecting several things out of a larger group, where (unlike permutations) order does not matter.


The answer must be the sum of the number of combinations of:
just one block - that is = 6
two blocks (= 6!/2!(6-2)!) = 15
three blocks (= 6!/3!(6-3)!) = 20
four blocks (= 6!/4!(6-4)!) = 15
five blocks (= 6!/5!(6-5)!) = 6
six blocks (= 6!/6!(6-6)!) = 1
TOTAL.........................= 63

If you need the calculations showing you how the formula [n!/k!(n-k)!] comes from, it would do you good to work it out yourself using lego blocks.

Or you could try to read this.
The Genius
Posted: Wednesday, January 16, 2013 3:37:51 AM
Rank: Newbie

Joined: 1/14/2013
Posts: 15
Neurons: 45
Location: Hong Kong
Hi Drag0nspeaker,
THX for reply,but I 'm sorry to tell you that your answer is not the one I want. Maybe you misunderstood my question.
First,there are no redundant words written in the question,Drag0nspeaker.'2x4 lego block' is an important information because the number of combinations varies with the block.The number of combinations of different blocks (Say,2x2 and 2x4)are TOTALLY different.This one is not unneeded.
Also,'without repetition' is a required information. I do not know how to explain you this cuz I can't show you a lego block.But if you(PLZ don't think I'm looking down on you)try to work this out with lego blocks,after rotating the blocks,you'll find many repeated patterns.
And,Drag0nspeaker,sorry to be rude,but I really don't get why one block has 6 combinations. That's impossible(Unless you 'chop' the block-but,still it 'doesn't make sense' because the number of combinations would be INFINITE if so(You can chop it as small as you can if you have a really sharp equipment to help you!))1 block(Whatever the block is,this is always right)has 1 combination.
However 6 blocks only have 1 combination.(?)That's again quite impossible(please excuse me for my courtesy).
I'm not quite sure if you got me. How did you make the calculations actually?
Thank you but I'm sorry,I don't think you are right.
THX anyway.
Epiphileon
Posted: Wednesday, January 16, 2013 4:39:59 AM

Rank: Advanced Member

Joined: 3/22/2009
Posts: 4,287
Neurons: 166,581
Hi Genius, well I knew that number would be huge considering the number of ways just two of these blocks could be put together, so while having no intent on trying to figure out exactly how you would compute the final answer, I went looking for an example of how just two would go together. Guess what. I found the answer to your original question. Here's a quote from the top of the page.

Quote:
It is commonly believed that the number of ways to combine six two-by-four studded lego bricks of the same color is

102,981,500

This number was computed at LEGO in 1974 and has been systematically repeated by LEGO Company since, for instance in LEGO Company Profile 2004. It also appears in books like The ultimate LEGO book (Dorling Kindersley, 1999). Consequently, the number can be found in several "fun fact" books and on more than 250 home pages in a multitude of languages. However, the number is wrong - very wrong. By an extensive, but rather straightforward computer calculation we have found that the correct number is

915,103,765

Questions and answers

Why is 102,981,500 wrong?
How does one arrive at the number 102,981,500?
How does one arrive at the number 915,103,765?
Why didn't LEGO compute the correct number?
Why is it so hard?
Why is it interesting?
What are our results?
Who are we?
How to contact us?



All of the above questions are linked to answers, and the page is, A LEGO Counting problem
The Genius
Posted: Wednesday, January 16, 2013 5:02:18 AM
Rank: Newbie

Joined: 1/14/2013
Posts: 15
Neurons: 45
Location: Hong Kong
THX Epiphileon.In fact just now I found that article too. That's coincidence. But I 'm sorry,what I want,is the calculation,not the answer. Knowing the answer means nothing to me. What I want to know is actually the way how people calculated it.By a computer program,your answer would probably be,but...Is it that this cannot be calculated by a human brain?I mean,Math?Einstein... what would you say?
So there's really NO solution at all if we should use our brains?

Now I am even more confused than ever.No one tend to calculate it-by Math?

But I thank you. Thanks.

Hey guys please don't misunderstand me.I'm not blaming anyone here. All sorts of technologies are improving in today's world...but are men's brains evolving... or degenerating?

Haha. Please don't laugh.12-year-old children are indeed naive.
Ray41
Posted: Wednesday, January 16, 2013 6:35:41 AM

Rank: Advanced Member

Joined: 9/9/2010
Posts: 1,937
Neurons: 45,980
Location: Orange, New South Wales, Australia
Genius wrote;
And,Drag0nspeaker,sorry to be rude,but I really don't get why one block has 6 combinations. That's impossible(Unless you 'chop' the block-but,still it 'doesn't make sense' because the number of combinations would be INFINITE if so(You can chop it as small as you can if you have a really sharp equipment to help you!))1 block(Whatever the block is,this is always right)has 1 combination.



Have a close look at the Lego blocks. Yes, I do know that they are 8 X 2, but are they flat?
In order to find out the number of combinations possible in your question then we must first establish how many surfaces there are to continue working out the possible number of combinations, correct?

The Lego blocks are three dimensional. That gives us six surfaces. Does that clarify what DragOn has written?



[image not available]


Mathematics has advanced at a such rate the brain does not have the time to calculate every single individual calculation and record it on paper(a formulae, which is what you are asking for here).
For example, when I went to school we used a book of logarithm tables otherwise we would spent so much time working on logs that we would never have time to solve the problem.
Students today have scientific calculators that have logs, plus sines, cosines, tangents, etc. all at the press of the buttons (in the right order of course). Who designed the calculators? and also programmed them so that they give the correct answer?
This basically answers your question here;
So there's really NO solution at all if we should use our brains?
Hey guys please don't misunderstand me.I'm not blaming anyone here. All sorts of technologies are improving in today's world...but are men's brains evolving... or degenerating?


What do you think? Where does the improving technology come from if not from the human brain?


Please, I know that the younger generation use text speak a lot, but, this is an English forum so the use of THX, etc. is best avoided. Thank you.
early_apex
Posted: Wednesday, January 16, 2013 9:38:53 AM
Rank: Advanced Member

Joined: 4/20/2009
Posts: 2,281
Neurons: 12,855
Location: Spindletop, Texas, United States
The Genius wrote:
THX Epiphileon.In fact just now I found that article too. That's coincidence. But I 'm sorry,what I want,is the calculation,not the answer. Knowing the answer means nothing to me. What I want to know is actually the way how people calculated it.By a computer program,your answer would probably be,but...Is it that this cannot be calculated by a human brain?I mean,Math?Einstein... what would you say?
So there's really NO solution at all if we should use our brains?

Now I am even more confused than ever.No one tend to calculate it-by Math?

But I thank you. Thanks.

Hey guys please don't misunderstand me.I'm not blaming anyone here. All sorts of technologies are improving in today's world...but are men's brains evolving... or degenerating?


Haha. Please don't laugh.12-year-old children are indeed naive.


TG, I looked at the web site Epi posted, and their approach to the problem seems rigorous. It appears that this is a counting problem, and no one has yet discovered how to reduce it to a mathematical formula.

Years ago, I read an article about the so-called traveling salesman problem. The salesman has to travel to a certain number of cities, and wants to know the shortest path to visit them all. As far as I know, this is still an unsolved problem, and the answer can only be found by exhaustive brute force search.

My point is that people have thought of problems that existing mathematical techniques cannot solve. Seven Bridges of Königsberg is another example.

Most likely, our brains are not evolving at this point. There seems to be only a weak connection between intelligence and survival after a certain level is reached. If we were evolving, it would not be noticeable within a few generations. Our brains may be degenerating, but this is more on an individual basis, and alcohol could be involved.
The Genius
Posted: Thursday, January 17, 2013 12:07:39 AM
Rank: Newbie

Joined: 1/14/2013
Posts: 15
Neurons: 45
Location: Hong Kong
Thank you everyone.This is really a good forum which is quite formal and your answers are so straight-forward.I like this,and I think I shall thank you again. After all my knowledge is very insufficient for this hard problem. However my adoration for math (And all sorts of that things)has again increased. I think I shall read more if I really want to solve this,or determine that this is an unsolvable problem.

Lastly, I want to ask if my English was poor. People in Hong Kong always compliment me for my English, but I don't agree.
Were there any strange mistakes written that made you think I 'm not native?

Thank you and don't reply if you think this question is frivolous.
Ray41
Posted: Thursday, January 17, 2013 1:21:10 AM

Rank: Advanced Member

Joined: 9/9/2010
Posts: 1,937
Neurons: 45,980
Location: Orange, New South Wales, Australia
Hi Genius.
There is no such thing as a frivolous question, or a question which may answer to any other description.
If we do not ask, then we will never learn.Think

You have learnt quite a lot in a short period, no text speak, word spacing has improved and your last post is very polite. Also, you are talking to us, not at us.Applause

Your written English is good but, it is easy to pick you as a non-native by the way you construct a sentence.
Do not let that deter you in any way as you are easily understood and, the more you post, the better you will become.

I cannot comment re: "People in Hong Kong always compliment me for my English,but I don't agree", as I assume that you are referring to spoken English.Anxious

For a twelve year old, your writing to me, is reasonably mature.

Keep posting.

Drag0nspeaker
Posted: Thursday, January 17, 2013 7:22:29 AM

Rank: Advanced Member

Joined: 9/12/2011
Posts: 34,427
Neurons: 228,163
Location: Livingston, Scotland, United Kingdom

Hello - not so much a genius!

The word you used in your question has a very specific meaning.

Quote:
Combination:
5. Mathematics One or more elements selected from a set without regard to the order of selection.
6. a group formed in this way. The number of combinations of n objects taken r at a time is n!/[(n - r)!r!]. Symbol: nCr Compare permutation


Since you posted this in the Science and Technology forum, I assumed you knew the real definition of "combinations".

What you are talking about is how many different shapes can be produced from the different combinations of blocks from a set of six identical 2x4 lego blocks - not just how many combinations there are. A very different question.

For example "How many combinations of ONE block can be found in a set of SIX blocks?"


r =1
n = 6
n!/[(n - r)!r!] = 6 combinations
The Genius
Posted: Thursday, January 17, 2013 11:05:57 PM
Rank: Newbie

Joined: 1/14/2013
Posts: 15
Neurons: 45
Location: Hong Kong
Drag0nspeaker:
First,'genius' is my username,I'm not really one(Or...not yet recognized?)You can't and shouldn't talk that way because if I were really stupid,I'll feel hurt.I hope you get that.

Once again,I'd say you misunderstood my problem.I'm not here to argue with you about this little thing.I'm just hoping that you will read more carefully before you give an answer.

You yourself said,"How many combinations of ONE block can be found in a set of SIX blocks?". That's totally a different thing,and I didn't ask so.That definitely was not my question,friend.

If I'm sounding rude (Okay,in fact I feel it myself.I know it now) again,I'd apologize again,but I really had to say that since you replied me that way-I'm very sorry.

Anyway,thanks for letting me know what a combination is,Drag0nspeaker.

I don't want anymore arguments. Thank you.

********************************************************************************************************************************
Ray,

Thanks for your kind reply.

That's all I can say and umm,I agree with you.I think I did improve because of all the people here.(In fact I don't use and honestly,I 've NEVER used a phone to text.My mobile phone is just a phone,not a 'smart' one.I don't use them at all.I think they spoil teenagers.A simple one is reasonably sufficient.)
Thank you and I thank you again(I don't have anything else to say).

from 'The Genius'
pedro
Posted: Friday, January 18, 2013 5:54:19 AM
Rank: Advanced Member

Joined: 5/21/2009
Posts: 13,057
Neurons: 63,022
I wonder if Lego workers are ever given compassionate leave, or at least counselling.
Drag0nspeaker
Posted: Friday, January 18, 2013 9:32:08 AM

Rank: Advanced Member

Joined: 9/12/2011
Posts: 34,427
Neurons: 228,163
Location: Livingston, Scotland, United Kingdom
The Genius wrote:
Quote:
First,'genius' is my username,I'm not really one

Oh, I don't know.
You are obviously brighter than most 12-year-olds I've met.

To answer your original question in the way you meant it, my answer is:
"Errr... Duh! d'oh! my brain hurts. Too many for me to count. I only have ten fingers and ten toes."
"Give us an easier one."


[image not available]

leonAzul
Posted: Saturday, January 19, 2013 1:49:55 PM

Rank: Advanced Member

Joined: 8/11/2011
Posts: 8,589
Neurons: 31,086
Location: Miami, Florida, United States
The Genius wrote:
Ladies and gentlemen,welcome to 'Ahem!!'
In fact I am not quite sure if my topic goes in this category,but whatever.
Yesterday I suddenly thought of a truly vexing problem-here it goes!

There are six 2x4 lego blocks,and they all look exactly the same. How many combinations can we make with the blocks,without any repetitions?

How was that?I couldn't sleep well yesterday because of this.So,hope you,yea,YOU could offer me a helping hand.
(I want the sum,the calculation,not the number-that'll be enormous!)


You will need to provide a better definition of the problem in order to arrive at the desired solution.

The word "combination" has a very specific meaning in mathematics, and the evidence of the replies so far demonstrates that the number of unique combinations is not what you are looking for.

Instead, let me invite you to discuss further what it is you mean to solve.

I suspect that the problem will involve identifying combinatorial modes, including a certain isotropic sense to each lego oblong, and the number of points of attachment. Additionally, we will need to define what sort of symmetries to consider when judging a particular structural arrangement as "unique." The topological notion of "tessellation" will probably not be directly helpful in the development of a solution, but some of the mathematics might serve as a point of departure.
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