- Hello, I was wondering if the next quote from Carroll's Phantasmagoria

is a mathematical puzzle. If so, then I'd have to come up with a

similiar one to translate it into my language, as I am in the midst of

translating Phantasmagoria:

And I remember nothing more

That I can clearly fix,

Till I was sitting on the floor,

Repeating "Two and five are four,

But five and two are six."

Any help, telling me whether it's a puzzle or not, and if so, helping

me solve it, will be much appreciated.

Thanks a lot,

Hagay - --- In lewiscarroll@yahoogroups.com, "hagaysc" <hschurr@g...> wrote:
>

Just make it into a pair of equations.....

> Hello, I was wondering if the next quote from Carroll's Phantasmagoria

> is a mathematical puzzle. If so, then I'd have to come up with a

> similiar one to translate it into my language, as I am in the midst of

> translating Phantasmagoria:

>

> And I remember nothing more

> That I can clearly fix,

> Till I was sitting on the floor,

> Repeating "Two and five are four,

> But five and two are six."

>

> Any help, telling me whether it's a puzzle or not, and if so, helping

> me solve it, will be much appreciated.

>

> Thanks a lot,

>

2X + 5Y = 4

5X + 2Y = 6

You can solve these by the use of determinates, and CLD wrote a book

on the theory of determinates as well as a paper of his was read

before the Royal Society of London.

You can pretend that you are CLD and "play" with the equations.

Add them...... and you get

7X + 7Y = 10 ....... which is kind of cute or you can get

X + Y = (10/7)

You can subtract the equations and you can then get

3X - 3Y = 2 ....... which is kind of cute also or you can get

X - Y = (2/3)

or

X + Y = A .......[A = 10/7]

X - Y = B .......[B = 2/3]

And by adding the equations you solve for X and by subtracting the

equations you solve for Y.

Adding gives:

2X = (A + B)

X = (A + B)/2

Subtracting gives:

2Y = (A - B)

and finally

Y = (A - B)/2

X = (A + B)/2

This is even more cute, or pretty or might have a little algebraic

beauty.....

I am arithmetically challenged, and usually mess up the fractional

reduction of sticking in the known values of A and B... so I leave it

to the reader......

Leaving it to the reader is a common ploy used in mathematical

writings to avoid doing what the author find tedious, and shifts the

burden to the reader......

When the fractions are done, the actual fractional numbers may suggest

physical things in Dodgson's life, or may not do so.

He could have just been fascinated that the simple rhyme led to such

an interesting form of mathematical solution as....

Y = (A - B)/2

X = (A + B)/2

Now you can work backwards, maybe, and find out how to invent more

rhymes that lead to pretty equations, or maybe not (which is what I

think is the result).

Play like Dodgson, and you may think like Dodgson. Or maybe not.

Jim

> Hagay

Try solving the simultaneous algebra equations....

>

2X + 5Y = 4

5X + 2Y = 6

Then maybe given X and Y you could puzzle out something physical or

logical that corresponds to those two numbers.

Or just leave them alone as an intrepretation of the mathematical

phrasing of simultaneous linear algebraic equations.

You can formally solve these equations with the application of

determinates, and the mathematician Dodgson wrote a book on

determinates and a mathematical paper which was read (by his member

correspondent) before the Royal Society of London --- usually a mark

of significant technical merit.

So, we made this into a mathematical thing.... and then what?

We can play further.

By adding the two equations we obtain a third equation...

7X + 7Y = = 10

or X + Y = 10/7

and by subtracting the first equation from the second we obtain

3X - 3Y = 2

or X - Y = 2/3

Or.. X + Y = A [A = 10/7]

And. X - Y = B [B = 2/3]

Adding the equations gives...

2X = (A + B)

or

X = (A + B)/2

Subtracting the equations gives...

2Y = (A - B)

or

Y = (A - B)/2

X = (A + B)/2

Which is very pretty, isn't it. Some would call this a kind of

mathematical beauty, of the algebraic sort.

I am arithmetically challanged, so I won't put in the fractions for A

and B.

Use of a calculator is cheating.

Jim - --- In lewiscarroll@yahoogroups.com, "Jim Buch" <jbuch@r...> wrote:
>

Thank you so very much Jim. That's very helpful. I should have figured

> --- In lewiscarroll@yahoogroups.com, "hagaysc" <hschurr@g...> wrote:

> >

> > Hello, I was wondering if the next quote from Carroll's Phantasmagoria

> > is a mathematical puzzle. If so, then I'd have to come up with a

> > similiar one to translate it into my language, as I am in the midst of

> > translating Phantasmagoria:

> >

> > And I remember nothing more

> > That I can clearly fix,

> > Till I was sitting on the floor,

> > Repeating "Two and five are four,

> > But five and two are six."

> >

> > Any help, telling me whether it's a puzzle or not, and if so, helping

> > me solve it, will be much appreciated.

> >

> > Thanks a lot,

> >

>

> Just make it into a pair of equations.....

>

> 2X + 5Y = 4

> 5X + 2Y = 6

>

>

> You can solve these by the use of determinates, and CLD wrote a book

> on the theory of determinates as well as a paper of his was read

> before the Royal Society of London.

>

>

> You can pretend that you are CLD and "play" with the equations.

>

> Add them...... and you get

>

> 7X + 7Y = 10 ....... which is kind of cute or you can get

>

> X + Y = (10/7)

>

> You can subtract the equations and you can then get

>

> 3X - 3Y = 2 ....... which is kind of cute also or you can get

>

> X - Y = (2/3)

>

> or

>

> X + Y = A .......[A = 10/7]

> X - Y = B .......[B = 2/3]

>

> And by adding the equations you solve for X and by subtracting the

> equations you solve for Y.

>

> Adding gives:

>

> 2X = (A + B)

> X = (A + B)/2

>

> Subtracting gives:

>

> 2Y = (A - B)

>

> and finally

>

> Y = (A - B)/2

> X = (A + B)/2

>

>

> This is even more cute, or pretty or might have a little algebraic

> beauty.....

>

> I am arithmetically challenged, and usually mess up the fractional

> reduction of sticking in the known values of A and B... so I leave it

> to the reader......

>

> Leaving it to the reader is a common ploy used in mathematical

> writings to avoid doing what the author find tedious, and shifts the

> burden to the reader......

>

> When the fractions are done, the actual fractional numbers may suggest

> physical things in Dodgson's life, or may not do so.

>

> He could have just been fascinated that the simple rhyme led to such

> an interesting form of mathematical solution as....

>

> Y = (A - B)/2

> X = (A + B)/2

>

> Now you can work backwards, maybe, and find out how to invent more

> rhymes that lead to pretty equations, or maybe not (which is what I

> think is the result).

>

> Play like Dodgson, and you may think like Dodgson. Or maybe not.

>

> Jim

>

>

>

>

>

> > Hagay

> >

>

> Try solving the simultaneous algebra equations....

>

> 2X + 5Y = 4

> 5X + 2Y = 6

>

> Then maybe given X and Y you could puzzle out something physical or

> logical that corresponds to those two numbers.

>

> Or just leave them alone as an intrepretation of the mathematical

> phrasing of simultaneous linear algebraic equations.

>

> You can formally solve these equations with the application of

> determinates, and the mathematician Dodgson wrote a book on

> determinates and a mathematical paper which was read (by his member

> correspondent) before the Royal Society of London --- usually a mark

> of significant technical merit.

>

> So, we made this into a mathematical thing.... and then what?

>

> We can play further.

>

> By adding the two equations we obtain a third equation...

>

> 7X + 7Y = = 10

>

> or X + Y = 10/7

>

> and by subtracting the first equation from the second we obtain

>

> 3X - 3Y = 2

>

> or X - Y = 2/3

>

> Or.. X + Y = A [A = 10/7]

> And. X - Y = B [B = 2/3]

>

>

> Adding the equations gives...

> 2X = (A + B)

> or

> X = (A + B)/2

>

> Subtracting the equations gives...

> 2Y = (A - B)

> or

> Y = (A - B)/2

> X = (A + B)/2

>

> Which is very pretty, isn't it. Some would call this a kind of

> mathematical beauty, of the algebraic sort.

>

> I am arithmetically challanged, so I won't put in the fractions for A

> and B.

>

> Use of a calculator is cheating.

>

> Jim

out myself I have to put them into a pair of equations.>

> --- In lewiscarroll@yahoogroups.com, "hagaysc" <hschurr@g...> wrote:

of> translating Phantasmagoria:

> Hello, I was wondering if the next quote from Carroll's Phantasmagoria

> is a mathematical puzzle. If so, then I'd have to come up with a

> similiar one to translate it into my language, as I am in the midst

And I remember nothing more

That I can clearly fix,

Till I was sitting on the floor,

Repeating "Two and five are four,

But five and two are six."> > >

helping

> > > Any help, telling me whether it's a puzzle or not, and if so,

> > > me solve it, will be much appreciated.

And I remember from two men

some clever arithmetic tricks.

And wrote them down with my pen.

"How can five and one be ten,

but one and five are twenty six."

If I were American, what would I be talking about, using common

everyday items?

This has a solution, but I don't think it is "mathematically pretty"

as was the one by CLD.

If you can play like Dodgson, it might help you think like Dodgson, or

maybe not.

Jim- Hi, I'm on the scrounge again. If any member could

email me a jpeg of the cover to "Rhyme?And Reason?"

and one of a sample interior illustration to my

personal address, I would be very grateful (and I'd

give you an acknowledgement in the book).

Bryan