In article
62941788-7532-4b60-8db6-c01335e51...@h1g2000prh.googlegroups.com>,



Albert <albert.xtheunkno...@gmail.com> wrote:
'We place some positive whole numbers around a circle. The sum of each
pair of neighbouring numbers is written in a box drawn in the space
between them. We call this arrangement a sum circle.
Like a bracelet, a sum circle is not changed by rotations and
reflections.
If all the boxes contain primes, the sum circle is called a prime sum
circle.

a Find all the different ways of arranging 1, 2, 3, 4, 5, 6, 7, 8, 9,
10 into a prime sum circle with seven different prime numbers in the
boxes. Explain why there are no more such prime sum circles.'

From '2008 MATHS CHALLENGE STAGE MATHEMATICS CHALLENGE FOR YOUNG
AUSTRALIANS MARCH-JUNE INTERMEDIATE STUDENT PROBLEMS AN ACTIVITY OF
THE AUSTRALIAN MATHEMATICAL OLYMPIAD COMMITTEE A SUBCOMMITTEE OF THE
AUSTRALIAN MATHEMATICS TRUST IN ASSOCIATION WITH THE AUSTRALIAN
ACADEMY OF SCIENCE AND THE UNIVERSITY OF CANBERRA'

Now you guys (for a fact which I know for sure) are really good. As
you guys found a way to reduce the guessing and checking in the I1.
Ice creams part c, I'm sure you guys will find a way of narrowing down
the number of possibilities to test out here. Now for I1. Ice creams
part c (http://groups.google.com/group/sci.math/browse_thread/thread/
ca5d8fcb28b870e7/4d1d71ae7d839146?lnk=gst&q=%22copycats
%22#4d1d71ae7d839146 thank you to quasi), I would consider the hint to
be: convert the percentages into fractions.

Could somebody give me a hint here?

Make a graph where the vertices are 1, 2, ..., 10
and where two vertices are joined by an edge
if and only if the two vertices add to a prime.
Then try to find a Hamiltonian cycle in this graph.

--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)

In article
330617d4-4ff7-4d69-8637-0e94d96f0...@j33g2000pri.googlegroups.com>,



Albert <albert.xtheunkno...@gmail.com> wrote:
On May 1, 10:55 am, Gerry Myerson <ge...@maths.mq.edi.ai.i2u4email
wrote:
In article
62941788-7532-4b60-8db6-c01335e51...@h1g2000prh.googlegroups.com>,

Albert <albert.xtheunkno...@gmail.com> wrote:
'We place some positive whole numbers around a circle. The sum of each
pair of neighbouring numbers is written in a box drawn in the space
between them. We call this arrangement a sum circle.
Like a bracelet, a sum circle is not changed by rotations and
reflections.
If all the boxes contain primes, the sum circle is called a prime sum
circle.

a Find all the different ways of arranging 1, 2, 3, 4, 5, 6, 7, 8, 9,
10 into a prime sum circle with seven different prime numbers in the
boxes. Explain why there are no more such prime sum circles.'

From '2008 MATHS CHALLENGE STAGE MATHEMATICS CHALLENGE FOR YOUNG
AUSTRALIANS MARCH-JUNE INTERMEDIATE STUDENT PROBLEMS AN ACTIVITY OF
THE AUSTRALIAN MATHEMATICAL OLYMPIAD COMMITTEE A SUBCOMMITTEE OF THE
AUSTRALIAN MATHEMATICS TRUST IN ASSOCIATION WITH THE AUSTRALIAN
ACADEMY OF SCIENCE AND THE UNIVERSITY OF CANBERRA'

Now you guys (for a fact which I know for sure) are really good. As
you guys found a way to reduce the guessing and checking in the I1.
Ice creams part c, I'm sure you guys will find a way of narrowing down
the number of possibilities to test out here. Now for I1. Ice creams
part c (http://groups.google.com/group/sci.math/browse_thread/thread/
ca5d8fcb28b870e7/4d1d71ae7d839146?lnk=gst&q=%22copycats
%22#4d1d71ae7d839146 thank you to quasi), I would consider the hint to
be: convert the percentages into fractions.

Could somebody give me a hint here?

Make a graph where the vertices are 1, 2, ..., 10
and where two vertices are joined by an edge
if and only if the two vertices add to a prime.
Then try to find a Hamiltonian cycle in this graph.

--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)

Is there a way of doing this with 'a Hamiltonian cycle' and 'graphs'?
I'm only 14.

You mean *without*?

Yes, undoubtedly, but look: "graph" is just a fancy word
for a bunch of dots together with a bunch of lines joining
some pairs of dots.

So you've got 10 dots, labeled 1, 2, ..., 10.

Now 1 + 2 is a prime, so you draw a line (or a curve, doesn't
matter) joining 1 and 2. 1 + 3 isn't prime, so you don't draw
a connection between 1 and 4. And so on.

Now a Hamiltonian cycle is just a fancy way of saying start at
some dot and follow lines in such a way that you visit every dot
exactly once and wind up at the dot where you started. For
big graphs (with hundreds of dots & thousands of lines) it can
be very difficult to decide whether there even is one of these
cycles, but for a little graph with 10 dots you should be able
to find all these cycles just by educated trial and error.

Go for it.

--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)

On May 1, 10:55 am, Gerry Myerson <ge...@maths.mq.edi.ai.i2u4email
wrote:
In article
62941788-7532-4b60-8db6-c01335e51...@h1g2000prh.googlegroups.com>,



Albert <albert.xtheunkno...@gmail.com> wrote:
'We place some positive whole numbers around a circle. The sum of each
pair of neighbouring numbers is written in a box drawn in the space
between them. We call this arrangement a sum circle.
Like a bracelet, a sum circle is not changed by rotations and
reflections.
If all the boxes contain primes, the sum circle is called a prime sum
circle.

a Find all the different ways of arranging 1, 2, 3, 4, 5, 6, 7, 8, 9,
10 into a prime sum circle with seven different prime numbers in the
boxes. Explain why there are no more such prime sum circles.'

From '2008 MATHS CHALLENGE STAGE MATHEMATICS CHALLENGE FOR YOUNG
AUSTRALIANS MARCH-JUNE INTERMEDIATE STUDENT PROBLEMS AN ACTIVITY OF
THE AUSTRALIAN MATHEMATICAL OLYMPIAD COMMITTEE A SUBCOMMITTEE OF THE
AUSTRALIAN MATHEMATICS TRUST IN ASSOCIATION WITH THE AUSTRALIAN
ACADEMY OF SCIENCE AND THE UNIVERSITY OF CANBERRA'

Now you guys (for a fact which I know for sure) are really good. As
you guys found a way to reduce the guessing and checking in the I1.
Ice creams part c, I'm sure you guys will find a way of narrowing down
the number of possibilities to test out here. Now for I1. Ice creams
part c (http://groups.google.com/group/sci.math/browse_thread/thread/
ca5d8fcb28b870e7/4d1d71ae7d839146?lnk=gst&q=%22copycats
%22#4d1d71ae7d839146 thank you to quasi), I would consider the hint to
be: convert the percentages into fractions.

Could somebody give me a hint here?

Make a graph where the vertices are 1, 2, ..., 10
and where two vertices are joined by an edge
if and only if the two vertices add to a prime.
Then try to find a Hamiltonian cycle in this graph.

--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)

Is there a way of doing this with 'a Hamiltonian cycle' and 'graphs'?
I'm only 14.

'We place some positive whole numbers around a circle. The
sum of each pair of neighbouring numbers is written in a box
drawn in the space between them. We call this arrangement a
sum circle. Like a bracelet, a sum circle is not changed by
rotations and reflections. If all the boxes contain primes,
the sum circle is called a prime sum circle.

a Find all the different ways of arranging 1, 2, 3, 4, 5, 6, 7, 8, 9,
10 into a prime sum circle with seven different prime numbers in the
boxes. Explain why there are no more such prime sum circles.'


Make a graph where the vertices are 1, 2, ..., 10
and where two vertices are joined by an edge
if and only if the two vertices add to a prime.
Then try to find a Hamiltonian cycle in this graph.

Is there a way of doing this with 'a Hamiltonian cycle' and 'graphs'?
I'm only 14.

You mean *without*?

Yes, undoubtedly, but look: "graph" is just a fancy word
for a bunch of dots together with a bunch of lines joining
some pairs of dots.

So you've got 10 dots, labeled 1, 2, ..., 10.

Now 1 + 2 is a prime, so you draw a line (or a curve, doesn't
matter) joining 1 and 2. 1 + 3 isn't prime, so you don't draw
a connection between 1 and 4. And so on.

Now a Hamiltonian cycle is just a fancy way of saying start at
some dot and follow lines in such a way that you visit every dot
exactly once and wind up at the dot where you started. For
big graphs (with hundreds of dots & thousands of lines) it can
be very difficult to decide whether there even is one of these
cycles, but for a little graph with 10 dots you should be able
to find all these cycles just by educated trial and error.

Surely there is some method that involves a bit more mathematics...

And I think that that is implied because

b Show that when 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 are arranged into a
prime sum circle, the boxes must contain at least three different
prime numbers.

Are you suggesting that I find every single prime sum circle and say
'all possible prime circles contain at least three different prime
numbers'?