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**bill_ghatis****Guest**

http://nrich.maths.org/786

I can get up to 42. But what is the maximum number and how can I prove it? :s so confused help please

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,436

Hi Bill;

42 seems pretty good. I believe the upper bound is 63, but am not sure. Anyway, with the geometric stipulation of only using neighbors I couldn't do better than 37.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

You can improve the upper bound by considering how many pieces can be made.

There are 32 that include the centre. (No restrictions on which of the 5 petals can be included, so the amount is 2^5)

For arrangements excluding the centre, there are also 5 ways of taking one petal, 5 of taking 2, 5 of 3, 5 of 4 and 1 of 5. This is another 21.

Therefore, there are only 53 different combinations available and so that is an upper bound.

My first interpretation of the puzzle was that any two petals had to be connected via the centre. If we use that interpretation, then we have an upper bound of 37.

Why did the vector cross the road?

It wanted to be normal.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,436

Hi mathsyperson;

I meant 63 as the upper bound for 1,2,4,8,16,32 as the sum for 6 numbers. He can represent 1 to 42 by his six mystery numbers he has found using the rules.

Happy holiday!

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,847

I tried it with 1,2,4,8,16,32 but couldn't do it. The best I got was 44, using these numbers:

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,436

Hi phrontister;

What positions did you use those numbers in?

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,847

Hi Bobby,

This is it:

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,436

Hi phrontister;

That's close but you can't make a 41.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,847

Hi Bobby,

Yes, 41 is possible: 8 + 17 + 12 + 4 = 41

Here's how I got them all:

I used T&E to find the six numbers.

I think 1, 2, 4 & 8 are essential for the first four numbers, and a central 1, surrounded by 8 > 2 > 4, gives the highest score: 11.

So that gives 12 (or something lower) for the fifth number.

12 succeeds right up to 19, and I then tested for the sixth number, starting with 28 (one greater than the sum of the other numbers) and working down. 17 is the first one that works up to the sum of all six numbers.

I doubt that number 1 would succeed anywhere but in the centre, as probably all the other numbers need access to it at some stage or other, which would not be possible if it were placed on the outer ring.

I wonder what the max is.

*Last edited by phrontister (2009-12-24 19:25:41)*

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,436

Hi phrontister;

Yes, I just got that now. You did go up to 44 a new record!

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,436

This yields 45! What is unique is that the 1 is not in the center.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,847

Xlnt, Bobby!

I thought of trying the 2 in the centre but didn't give it much thought, and gave up at the first hurdle.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,436

Hi;

46 !

Nope! Mathsyperson found an error. Upon checking the program I had a logic error. Corrected that. So the 45 is good, the 46 is not!

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

The 45 flower is very impressive!

Unfortunately, I don't think the next one can make 21.

Why did the vector cross the road?

It wanted to be normal.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,436

True! Have corrected the program and added to the the incorrect post.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,847

I think I found a 46!

*Last edited by phrontister (2009-12-26 01:29:32)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,436

You sure did! I think that is maximum.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**nombredaisy****Guest**

i have this problem too,but how do you prove this? i cant get over 46...................

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,436

Hi nombredaisy;

No one could beat the 46 and we think that is maximum. Welcome to the forum!

If you have a different 46, then please post it.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**drdangerlove****Member**- Registered: 2014-06-03
- Posts: 2

phrontister wrote:

I think I found a 46!

How did you go about finding this solution? Did you work in some systematic way?

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,847

Hi drdangerlove, and welcome to the forum!

I used T&E, sprinkled with a smattering of logic and system...as I described in post #9.

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**drdangerlove****Member**- Registered: 2014-06-03
- Posts: 2

phrontister wrote:

Hi drdangerlove, and welcome to the forum!

I used T&E, sprinkled with a smattering of logic and system...as I described in post #9.

Thanks. I'm trying to explain different ways of working mathematically to some maths students so the ways in which these types of problems are tackled is of great interest to me.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

Hi guys, wouldn't it be better if there was a 20 instead of a 19 there? You can then get up to 47.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**zetafunc****Member**- Registered: 2014-05-21
- Posts: 87

anonimnystefy wrote:

Hi guys, wouldn't it be better if there was a 20 instead of a 19 there? You can then get up to 47.

How would you make 22 and 24?

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,522

Hm, I don't know, but how do you make a 21 with that one?

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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