Carpet cutting
Jan Kristian Haugland
Is this carpet cutting puzzle well known?
http://www.neutreeko.net/carpet.htm
Thanks!
---
J K Haugland
http://www.neutreeko.net
Nick Wedd
<6e8eaa33-4e2c-4d25...@g23g2000vbr.googlegroups.com>, Jan
Kristian Haugland <jank...@hotmail.com> writes
Here's another answer, it's not at all interesting. Take the first
solution in the bottom row, and transfer the first square in its sixth
row from the 2x1 rectangle to the big piece above it.
Nick
--
Nick Wedd ni...@maproom.co.uk
Jan Kristian Haugland
> Take the first
> solution in the bottom row, and transfer the first square in its sixth
> row from the 2x1 rectangle to the big piece above it.
Thanks, it's been added.
I've found a relevant web page:
http://www.ics.uci.edu/~eppstein/junkyard/bao-dissection-challenges.html
Mensanator
wrote:
> On 24 Sep, 12:28, Nick Wedd <n...@maproom.co.uk> wrote:
>
> > Take the first
> > solution in the bottom row, and transfer the first square in its sixth
> > row from the 2x1 rectangle to the big piece above it.
>
> Thanks, it's been added.
How come you don't have the solution that starts by cutting
the carpet into 49 1x1 squares?
Jan Kristian Haugland
> On Sep 24, 7:04 am, Jan Kristian Haugland <jankri...@hotmail.com>
> wrote:
>
> > On 24 Sep, 12:28, Nick Wedd <n...@maproom.co.uk> wrote:
>
> > > Take the first
> > > solution in the bottom row, and transfer the first square in its sixth
> > > row from the 2x1 rectangle to the big piece above it.
>
> > Thanks, it's been added.
>
> How come you don't have the solution that starts by cutting
> the carpet into 49 1x1 squares?
Because the carpet is to be cut in only five pieces.
Mensanator
wrote:
> On 24 Sep, 18:43, Mensanator <mensana...@aol.com> wrote:
>
> > On Sep 24, 7:04 am, Jan Kristian Haugland <jankri...@hotmail.com>
> > wrote:
>
> > > On 24 Sep, 12:28, Nick Wedd <n...@maproom.co.uk> wrote:
>
> > > > Take the first
> > > > solution in the bottom row, and transfer the first square in its sixth
> > > > row from the 2x1 rectangle to the big piece above it.
>
> > > Thanks, it's been added.
>
> > How come you don't have the solution that starts by cutting
> > the carpet into 49 1x1 squares?
>
> Because the carpet is to be cut in only five pieces.
Duh, didn't see that. I thought I was missing something.
Philippe 92
>> solution in the bottom row, and transfer the first square in its sixth
>> row from the 2x1 rectangle to the big piece above it.
>
> Thanks, it's been added.
>
> I've found a relevant web page:
>
> http://www.ics.uci.edu/~eppstein/junkyard/bao-dissection-challenges.html
>
relevant, euh... same subject "dissections" but other puzzles.
The problem of the carpet is from Sam Loyd (1908)
Greg Frederickson's "Dissections: Plane and Fancy" chapters 7 and 8
are fully devoted to this kind of dissections, especially chapter 8.
In which we find your carpet puzzle and many variants.
For a similar problem of dissecting a 5x5 square into 3x3 and 4x4
squares, there was recently a discussion in de.rec.denksport which
resulted into 72 different dissections (24 requiring pieces flip)
and even infinitely many if we allow slanted cuts.
I summarized these on a web page :
<http://mathafou.free.fr/pbg_en/sol110d.html>
The 'parent' button in the menu gives access to many other dissection
puzzles.
Regards.
--
Philippe Ch., mail : chephi...@free.fr
site : http://mathafou.free.fr/ (recreational mathematics)
Jan Kristian Haugland
> The problem of the carpet is from Sam Loyd (1908)
> Greg Frederickson's "Dissections: Plane and Fancy" chapters 7 and 8
> are fully devoted to this kind of dissections, especially chapter 8.
> In which we find your carpet puzzle and many variants.
Thanks, I have updated the web page.
---
J K Haugland
http://ww.neutreeko.net
Jan Kristian Haugland
> Greg Frederickson's "Dissections: Plane and Fancy" chapters 7 and 8
> are fully devoted to this kind of dissections, especially chapter 8.
> In which we find your carpet puzzle and many variants.
Btw, perhaps you could tell me how many solutions are given there? :-)
Philippe 92
> On 25 Sep, 11:14, "Philippe 92" <nos...@free.invalid> wrote:
>
>> The problem of the carpet is from Sam Loyd (1908)
>
> Thanks, I have updated the web page.
>
You wrote there : "I do not know how many solutions they give"
Just one (which is your 20th solution, rotated). The one originally
given by Sam Loyd.
Greg Frederickson just gives more of the x^2 + y^2 + z^2 = w^2
dissections of a square into 3 squares with just 5 pieces.
Like the 1^2 + 4^2 + 8^2 = 9^2, 2^2 + 5^2 + 14^2 = 15^2,
2^2 + 12^2 + 36^2 = 38^2 etc.
It doesn't seem he was interrested in finding several and how many
solutions for a given puzzle, just at least one solution.
Sam Loyd's puzzle was on a checkerboard, and his solution preserves
the checkerboard property.
Many of yours also do. But not all of them. For instance your 2nd
solution violates the checkerboard on the 6x6 square.
Another class of solutions is those which are "translational"
That is the pieces are only translated without rotations.
Your 5th is the first example of such a solution in your list.
The 9th is both translational and preserving the parity (checkerboard)
For solutions with cuts not on the grid lines, I suspect there are
none if fliping pieces is not allowed.
By fliping pieces, a "non grid line" solution may be derived from
your 18th solution as <http://cjoint.com/?jzseQgEs3X>
The 1x2 rectangular piece is morphed into a piece which must now be
flipped to fit in the 6x6.
The cutting line is any curve symmetric through the 45ￜ diagonal.
Hence a infinite number of variants of this solution.