From Newsgroup: rec.puzzles

On Fri, 4 Jul 2025 15:47:04 +0000, Carl G. wrote:

On 7/1/2025 7:36 PM, IlanMayer wrote:

On Tue, 1 Jul 2025 19:01:29 +0000, Richard Tobin wrote:

A well-known puzzle is to divide an L-shape - a square with one square
quarter removed - into four identical pieces.

But what about a square where the quarter removed is an isosceles
right-angled triangle with one of the sides as its hypotenuse?

This problem was set in Peter Parley's Annual, 1877, but I fear that
they are no longer available to provide the answer:

https://i.ebayimg.com/images/g/BZUAAOSwPqVlS8OY/s-l1600.jpg
or
https://web.archive.org/web/20250701185837/https://i.ebayimg.com/images/g/BZUAAOSwPqVlS8OY/s-l1600.jpg

-- Richard

SPOILER <snipped>

Ilan,

I had considered writing a computer program to solve this puzzle and
similar puzzles. Did you write a program to solve this puzzle? If so,
what algorithm did it use?

No program; I solved this manually by dividing the shape into right
triangles and grouping them together.

--
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From Newsgroup: rec.puzzles

On Tue, 1 Jul 2025 19:01:29 -0000 (UTC), Richard Tobin wrote:

A well-known puzzle is to divide an L-shape - a square with one square quarter removed - into four identical pieces.

I haven't attempted your original question yet, but for anyone who has.
Henry Ernest Dudeney had a variation of that first puzzle in "Amusements
in Mathematics". He also has a variation on the puzzle with the square,
with a triangle removed. More on that later. Here's the first variation...

180. THE FOUR SONS

Readers will recognize the diagram as a familiar friend of their youth. A
man possessed a square shaped-estate. He bequeathed to his widow the
quarter of it that is shaded off. The remainder was to be divided
equitably amongst his four sons, so that each should receive land of
exactly the same area and exactly similar in shape. We are shown how this
was done. But the remainder of the story is not so generally known. In the centre of the estate was a well, indicated by the dark spot, and benjamin, Charles and David complained that the division was not "equitable," since Alfred had access to this well, while they could not reach it without trespassing on somebody else's land.The puzzle is to show how the estate
is to be apportioned so that each son shall have land of the same shape
and area, and each have access to the well without going off his own land.

The accompanying image is available here:

http://www.puzzles.50webs.org/pics/q180.png

Most of the puzzles from Dudeney's book are available here:

http://www.puzzles.50webs.org/index.html
--
David Entwistle
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From Newsgroup: rec.puzzles

On Wed, 2 Jul 2025 02:36:29 +0000, IlanMayer wrote:

A well-known puzzle is to divide an L-shape - a square with one square
quarter removed - into four identical pieces.

But what about a square where the quarter removed is an isosceles
right-angled triangle with one of the sides as its hypotenuse?

This problem was set in Peter Parley's Annual, 1877, but I fear that
they are no longer available to provide the answer:

https://i.ebayimg.com/images/g/BZUAAOSwPqVlS8OY/s-l1600.jpg or
https://web.archive.org/web/20250701185837/https://i.ebayimg.com/ images/g/BZUAAOSwPqVlS8OY/s-l1600.jpg

-- Richard

SPOILER

Hi Ilyan,

I failed to find a solution after an hour or so of looking (shuffling bits
of paper), so was fascinated to see a solution.

Are we sure those shapes are identical? Two look the same, one looks to be
a reflection of the first two, and the third looks to be a different
shape. They all have the same area, but I don't think you would describe
them as the same shape.

Apologies if I have this wrong, which I may well do.

I think the problem is very much related to the second of Dudeney's
problems, which I'll post shortly.
--
David Entwistle
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

In article <104d918$21g4p$1@dont-email.me>,
David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> wrote:

I failed to find a solution after an hour or so of looking (shuffling bits >of paper), so was fascinated to see a solution.

Are we sure those shapes are identical? Two look the same, one looks to be
a reflection of the first two, and the third looks to be a different
shape. They all have the same area, but I don't think you would describe >them as the same shape.

You are correct. One of the parts is a different shape, so it is not
a solution.

If the puzzle is worded in terms of cutting a sheet of paper into four
pieces, all the same size and shape, then I think we can accept
reflections.

-- Richard
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

On 06/07/2025 08:36, David Entwistle wrote:

On Wed, 2 Jul 2025 02:36:29 +0000, IlanMayer wrote:

A well-known puzzle is to divide an L-shape - a square with one square
quarter removed - into four identical pieces.

But what about a square where the quarter removed is an isosceles
right-angled triangle with one of the sides as its hypotenuse?

This problem was set in Peter Parley's Annual, 1877, but I fear that
they are no longer available to provide the answer:

https://i.ebayimg.com/images/g/BZUAAOSwPqVlS8OY/s-l1600.jpg or
https://web.archive.org/web/20250701185837/https://i.ebayimg.com/

images/g/BZUAAOSwPqVlS8OY/s-l1600.jpg

-- Richard

SPOILER

Hi Ilyan,

I failed to find a solution after an hour or so of looking (shuffling bits
of paper), so was fascinated to see a solution.

Are we sure those shapes are identical? Two look the same, one looks to be
a reflection of the first two, and the third looks to be a different
shape. They all have the same area, but I don't think you would describe
them as the same shape.

When I first saw Ilyan's "solution" I thought "ah, right, that's clever" and never spotted that the
shapes didn't match. Now I see that the top right shape doesn't even have the right number of
edges!! (It has 6 instead of 5. Doh!) That makes it quite amusing that it took so long for
someone to notice - a subtle yet not so subtle difference! :)

Mike.

Apologies if I have this wrong, which I may well do.

I think the problem is very much related to the second of Dudeney's
problems, which I'll post shortly.

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

On Tue, 1 Jul 2025 19:01:29 -0000 (UTC), Richard Tobin wrote:

This problem was set in Peter Parley's Annual, 1877, but I fear that
they are no longer available to provide the answer:

I'm not sure but, in connection with the Children's annuals, it seems
Peter Parley was a pseudonym of George Mogridge; rather than the American author Samuel Griswold Goodrich who established that pseudonym. However,
both would have been dead by 1877.

https://en.wikipedia.org/wiki/George_Mogridge_(writer) https://en.wikipedia.org/wiki/Samuel_Griswold_Goodrich

It seems a bit of a coincidence that publisher Faber and Faber are now
based at 51 Hatton Garden, London, but they haven't been around long
enough to be involved in this particular publication.

The Hatton Garden safe deposit burglary took place a few doors away in
April 2015. At the time the value of the goods taken was -u14 million.

I'm suspecting there isn't a solution to this puzzle, but hope to be
proved wrong. I'm continuing to work on it.
--
David Entwistle
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

On Mon, 7 Jul 2025 12:39:22 +0000, David Entwistle wrote:

On Tue, 1 Jul 2025 19:01:29 -0000 (UTC), Richard Tobin wrote:

This problem was set in Peter Parley's Annual, 1877, but I fear that
they are no longer available to provide the answer:

I'm not sure but, in connection with the Children's annuals, it seems
Peter Parley was a pseudonym of George Mogridge; rather than the
American
author Samuel Griswold Goodrich who established that pseudonym. However,
both would have been dead by 1877.

https://en.wikipedia.org/wiki/George_Mogridge_(writer) https://en.wikipedia.org/wiki/Samuel_Griswold_Goodrich

It seems a bit of a coincidence that publisher Faber and Faber are now
based at 51 Hatton Garden, London, but they haven't been around long
enough to be involved in this particular publication.

The Hatton Garden safe deposit burglary took place a few doors away in
April 2015. At the time the value of the goods taken was -u14 million.

I'm suspecting there isn't a solution to this puzzle, but hope to be
proved wrong. I'm continuing to work on it.

Probably not the intended solution as each piece is two triangles joined
by a single point, and there is flipping involved.

-------------+-------------.
|\ /| /|
| \ D / | / |
| \ / | D / |
| A . | . B |
| / C | / \ |
| / | / C \ |
|/____________./___________\|
| / \ |
| / \ |
| A / \ B |
| / \ |
| / \ |
| / \ |
|/ \|

--
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

On 7/7/2025 6:45 PM, IlanMayer wrote:

On Mon, 7 Jul 2025 12:39:22 +0000, David Entwistle wrote:

On Tue, 1 Jul 2025 19:01:29 -0000 (UTC), Richard Tobin wrote:

This problem was set in Peter Parley's Annual, 1877, but I fear that
they are no longer available to provide the answer:

I'm not sure but, in connection with the Children's annuals, it seems
Peter Parley was a pseudonym of George Mogridge; rather than the
American
author Samuel Griswold Goodrich who established that pseudonym. However,
both would have been dead by 1877.

https://en.wikipedia.org/wiki/George_Mogridge_(writer)
https://en.wikipedia.org/wiki/Samuel_Griswold_Goodrich

It seems a bit of a coincidence that publisher Faber and Faber are now
based at 51 Hatton Garden, London, but they haven't been around long
enough to be involved in this particular publication.

The Hatton Garden safe deposit burglary took place a few doors away in
April 2015. At the time the value of the goods taken was -u14 million.

I'm suspecting there isn't a solution to this puzzle, but hope to be
proved wrong. I'm continuing to work on it.

Probably not the intended solution as each piece is two triangles joined
by a single point, and there is flipping involved.

-------------+-------------.
|\-a-a-a-a-a-a-a-a-a-a /|-a-a-a-a-a-a-a-a-a-a-a /|
|-a \-a-a D-a-a /-a |-a-a-a-a-a-a-a-a-a /-a |
|-a-a-a \-a-a /-a-a-a |-a-a D-a-a-a /-a-a-a |
|-a A-a-a .-a-a-a-a-a |-a-a-a-a-a .-a-a B-a |
|-a-a-a /-a-a-a C-a-a |-a-a-a /-a-a \-a-a-a |
|-a /-a-a-a-a-a-a-a-a-a |-a /-a-a C-a-a \-a |
|/____________./___________\|
|-a-a-a-a-a-a-a-a-a-a-a / \-a-a-a-a-a-a-a-a-a-a-a |
|-a-a-a-a-a-a-a-a-a /-a-a-a-a \-a-a-a-a-a-a-a-a-a |
|-a-a A-a-a-a /-a-a-a-a-a-a-a-a \-a-a-a B-a-a |
|-a-a-a-a-a /-a-a-a-a-a-a-a-a-a-a-a-a \-a-a-a-a-a |
|-a-a-a /-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a \-a-a-a |
|-a /-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a \-a | |/-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a-a \|

--

Ilan: I think that you may have found the intended solution. When I
saw this puzzle, I had a faint recollection that the solution needed a
trick just like the one you found ("shapes connected by one point"). I
may have seen the solution in some puzzle book. It's the kind of puzzle Martin Gardner would have in his Mathematical Games column in Scientific American.
--
Carl G.


--
This email has been checked for viruses by AVG antivirus software.
www.avg.com
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From Newsgroup: rec.puzzles

On Tue, 8 Jul 2025 01:45:23 +0000, IlanMayer wrote:

Probably not the intended solution as each piece is two triangles joined
by a single point, and there is flipping involved.

I agree with Carl, I think it probably is the intended solution. Well
done.

I thought there should be a non-contiguous solution, but couldn't find it.
And there it is, as plain as day. Nice.
--
David Entwistle
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

In article <5dc8321866bd27ea9328f1af0cd5bf96@www.novabbs.com>,
IlanMayer <ilan_no_spew@hotmail.com> wrote:

Probably not the intended solution as each piece is two triangles joined
by a single point, and there is flipping involved.

That's the only solution I know of.

-- Richard




--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> writes:

A man possessed a square shaped-estate. He bequeathed to his widow the quarter of it that is shaded off. The remainder was to be divided
equitably amongst his four sons, so that each should receive land of
exactly the same area and exactly similar in shape.

The accompanying image is available here: http://www.puzzles.50webs.org/pics/q180.png

I love how those fit together. It's not new, but I still find it
elegant. I think I shall try to create a sudoku variant puzzle based
somehow on that geometry.

Is there a name for a shape that tesselates in such a way that it
can create a scaled-up version of itself?

Phil
--
We are no longer hunters and nomads. No longer awed and frightened, as we have gained some understanding of the world in which we live. As such, we can cast aside childish remnants from the dawn of our civilization.
-- NotSanguine on SoylentNews, after Eugen Weber in /The Western Tradition/
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

Phil Carmody <pc+usenet@asdf.org> writes:

David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> writes:

A man possessed a square shaped-estate. He bequeathed to his widow the
quarter of it that is shaded off. The remainder was to be divided
equitably amongst his four sons, so that each should receive land of
exactly the same area and exactly similar in shape.

The accompanying image is available here:
http://www.puzzles.50webs.org/pics/q180.png

I love how those fit together. It's not new, but I still find it
elegant. I think I shall try to create a sudoku variant puzzle based
somehow on that geometry.

Is there a name for a shape that tesselates in such a way that it
can create a scaled-up version of itself?

Rep-tile, apparently.

Phil
--
We are no longer hunters and nomads. No longer awed and frightened, as we have gained some understanding of the world in which we live. As such, we can cast aside childish remnants from the dawn of our civilization.
-- NotSanguine on SoylentNews, after Eugen Weber in /The Western Tradition/
--- Synchronet 3.21a-Linux NewsLink 1.2