Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5 squares,
like so: (crappy ASCII graphics)

______________________
|_A_|___|_A_|___|___|___|
|___|_A_|___|___|___|___|
|_A_|___|_A_|___|___|___|
|___|___|____|___|___|___|
|___|___|____|___|___|___|
|___|___|____|___|___|___|

Each block has one face painted with A, as shown.
A block moves as follows: it moves into an empty
adjacent cell. It rolls over an edge, accomplishing a 1/4 turn.

Your mission, Hunter, should you decide to accept,
is to align all blocks in a single row or column, with no
intermittent spaces, such that each ends up in the same
original orientation. Should you fail, your father the Supreme
Warlord will, as usual, disavow any knowledge of your activities.
Good luck, Hunter -

--
Rich

On Wednesday, July 12, 2023 at 4:34:05 PM UTC-4, RichD wrote:
> Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5 squares,
> like so: (crappy ASCII graphics)
>
> ______________________
> |_A_|___|_A_|___|___|___|
> |___|_A_|___|___|___|___|
> |_A_|___|_A_|___|___|___|
> |___|___|____|___|___|___|
> |___|___|____|___|___|___|
> |___|___|____|___|___|___|
>
> Each block has one face painted with A, as shown.
> A block moves as follows: it moves into an empty
> adjacent cell. It rolls over an edge, accomplishing a 1/4 turn.
>
> Your mission, Hunter, should you decide to accept,
> is to align all blocks in a single row or column, with no
> intermittent spaces, such that each ends up in the same
> original orientation. Should you fail, your father the Supreme
> Warlord will, as usual, disavow any knowledge of your activities.
> Good luck, Hunter -
>
>
> --
> Rich
With the least moves possible?

On July 12, RichD wrote:
> Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5 squares,
> like so: (crappy ASCII graphics)

of course I botched the diagram.
Re-try:
______________________
|___|_A_|____|___|___|___|
|_A_|_A_|_A_|___|___|___|
|___|_A_|____|___|___|___|
|___|___|____|___|___|___|
|___|___|____|___|___|___|
|___|___|____|___|___|___|

--
Rich

On Wednesday, July 12, 2023 at 5:23:35 PM UTC-4, RichD wrote:
> On July 12, RichD wrote:
> > Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5 squares,
> > like so: (crappy ASCII graphics)
> of course I botched the diagram.
> Re-try:
>
> ______________________
> |___|_A_|____|___|___|___|
> |_A_|_A_|_A_|___|___|___|
> |___|_A_|____|___|___|___|
> |___|___|____|___|___|___|
> |___|___|____|___|___|___|
> |___|___|____|___|___|___|
> --
> Rich
Did you also mean, with the least amount of moves?

On July 12, Dan joyce wrote:
>> > Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5 squares,
>> > like so: (crappy ASCII graphics)
>> of course I botched the diagram.
>> Re-try:
>
>> ______________________
>> |___|_A_|____|___|___|___|
>> |_A_|_A_|_A_|___|___|___|
>> |___|_A_|____|___|___|___|
>> |___|___|____|___|___|___|
>> |___|___|____|___|___|___|
>> |___|___|____|___|___|___|
>
> Did you also mean, with the least amount of moves?

For bonus credit, sure -

--
Rich

On 7/12/2023 1:34 PM, RichD wrote:
> Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5 squares,
> like so: (crappy ASCII graphics)
>
> ______________________
> |_A_|___|_A_|___|___|___|
> |___|_A_|___|___|___|___|
> |_A_|___|_A_|___|___|___|
> |___|___|____|___|___|___|
> |___|___|____|___|___|___|
> |___|___|____|___|___|___|

>
> Each block has one face painted with A, as shown.
> A block moves as follows: it moves into an empty
> adjacent cell. It rolls over an edge, accomplishing a 1/4 turn.
>
> Your mission, Hunter, should you decide to accept,
> is to align all blocks in a single row or column, with no
> intermittent spaces, such that each ends up in the same
> original orientation.

You mean all cells with A on them should be in a single row?

A are panted blocks and E are empty blocks?

AAAAAE
EEEEEE
EEEEEE
EEEEEE
EEEEEE
EEEEEE

? Or

AEEEEE
AEEEEE
AEEEEE
AEEEEE
AEEEEE
EEEEEE

?

Not exactly sure what you mean.

> Should you fail, your father the Supreme
> Warlord will, as usual, disavow any knowledge of your activities.
> Good luck, Hunter -

I must be totally misunderstanding your rules.

On 7/12/2023 3:07 PM, Chris M. Thomasson wrote:
> On 7/12/2023 1:34 PM, RichD wrote:
>> Given a 6 x 6 checkerboard.  At the top left, 5 blocks fill 5 squares,
>> like so:  (crappy ASCII graphics)
>>
>>   ______________________
>> |_A_|___|_A_|___|___|___|
>> |___|_A_|___|___|___|___|
>> |_A_|___|_A_|___|___|___|
>> |___|___|____|___|___|___|
>> |___|___|____|___|___|___|
>> |___|___|____|___|___|___|
>
>>
>> Each block has one face painted with A, as shown.
>> A block moves as follows: it moves into an empty
>> adjacent cell.  It rolls over an edge, accomplishing a 1/4 turn.
>>
>> Your mission, Hunter, should you decide to accept,
>> is to align all blocks in a single row or column, with no
>> intermittent spaces, such that each ends up in the same
>> original orientation.
>
> You mean all cells with A on them should be in a single row?
>
[...]
>
> Not exactly sure what you mean.
>
>
>> Should you fail, your father the Supreme
>> Warlord will, as usual, disavow any knowledge of your activities.
>> Good luck, Hunter -
>
> I must be totally misunderstanding your rules.
>

Sorry for the stupid question, buts its 3d where if I move a block with
an A on it to the right the the A face will be rotated 1/4 turn so that
the A is on the right hand side of the block, no longer facing me?

On July 12, Chris M. Thomasson wrote:
>>> Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5 squares,
>>> like so: (crappy ASCII graphics)
>>
> >> ______________________
> >> |_A_|___|_A_|___|___|___|
> >> |___|_A_|___|___|___|___|
> >> |_A_|___|_A_|___|___|___|
> >> |___|___|____|___|___|___|
> >> |___|___|____|___|___|___|
> >> |___|___|____|___|___|___|

Wrong diagram.
>>> Each block has one face painted with A, as shown.
>>> A block moves as follows: it moves into an empty
>>> adjacent cell. It rolls over an edge, accomplishing a 1/4 turn.
>
> Sorry for the stupid question, buts its 3d where if I move a block with
> an A on it to the right the the A face will be rotated 1/4 turn so that
> the A is on the right hand side of the block, no longer facing me?

Correct.
It's a 3-D context. End with 5 in a row. No tricks.

--
Rich

On 7/12/2023 2:23 PM, RichD wrote:
> On July 12, RichD wrote:
>> Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5 squares,
>> like so: (crappy ASCII graphics)
>
> of course I botched the diagram.
> Re-try:
>
> ______________________
> |___|_A_|____|___|___|___|
> |_A_|_A_|_A_|___|___|___|
> |___|_A_|____|___|___|___|
> |___|___|____|___|___|___|
> |___|___|____|___|___|___|
> |___|___|____|___|___|___|

Just to clear things up for me, lets say I moved the first A in the
first column second row, down. Then the A on it would be facing the
front of the block not the top, right? If I moved it down again then the
A would be on the bottom, another move the A would be on the back. One
more move down the A would be on the top again. So to move it down four
times it would be an A on the top in the lower left like:

_________________________
|___|_A_|____|___|___|___|
|__ |_A_|_A__|___|___|___|
|___|_A_|____|___|___|___|
|___|___|____|___|___|___|
|___|___|____|___|___|___|
|_A_|___|____|___|___|___|

Its this right?

On 7/12/2023 3:43 PM, RichD wrote:
> On July 12, Chris M. Thomasson wrote:
>>>> Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5 squares,
>>>> like so: (crappy ASCII graphics)
>>>
>>>> ______________________
>>>> |_A_|___|_A_|___|___|___|
>>>> |___|_A_|___|___|___|___|
>>>> |_A_|___|_A_|___|___|___|
>>>> |___|___|____|___|___|___|
>>>> |___|___|____|___|___|___|
>>>> |___|___|____|___|___|___|
>
> Wrong diagram.

Oh yeah. Shit. Sorry.

>>>> Each block has one face painted with A, as shown.
>>>> A block moves as follows: it moves into an empty
>>>> adjacent cell. It rolls over an edge, accomplishing a 1/4 turn.
>>
>> Sorry for the stupid question, buts its 3d where if I move a block with
>> an A on it to the right the the A face will be rotated 1/4 turn so that
>> the A is on the right hand side of the block, no longer facing me?
>
> Correct.
> It's a 3-D context. End with 5 in a row. No tricks.

Thanks. I wish I had more time, but I can get it like this. No Cigar at
all! ;^)

From here:
________________________
|___|_A_|____|___|___|___|
|_A_|_A_|_A__|___|___|___|
|___|_A_|____|___|___|___|
|___|___|____|___|___|___|
|___|___|____|___|___|___|
|___|___|____|___|___|___|

Move the top A to the right 4 times. This makes the A facing the top again:

________________________
|___|___|____|___|___|_A_|
|_A_|_A_|_A__|___|___|___|
|___|_A_|____|___|___|___|
|___|___|____|___|___|___|
|___|___|____|___|___|___|
|___|___|____|___|___|___|

Move the bottom A to the right 4 times:

________________________
|___|___|____|___|___|_A_|
|_A_|_A_|_A__|___|___|___|
|___|___|____|___|___|_A_|
|___|___|____|___|___|___|
|___|___|____|___|___|___|
|___|___|____|___|___|___|

Mode the right A down four times:

________________________
|___|___|____|___|___|_A_|
|_A_|_A_|____|___|___|___|
|___|___|____|___|___|_A_|
|___|___|____|___|___|___|
|___|___|____|___|___|___|
|___|___|__A_|___|___|___|

Move the middle A right 4 times:

________________________
|___|___|____|___|___|_A_|
|_A_|___|____|___|___|_A_|
|___|___|____|___|___|_A_|
|___|___|____|___|___|___|
|___|___|____|___|___|___|
|___|___|__A_|___|___|___|

Are these legitimate moves? Anyway, I have to start some work on other
things pretty soon. Shit happens!

;^)

On Wednesday, July 12, 2023 at 7:02:52 PM UTC-4, Chris M. Thomasson wrote:
> On 7/12/2023 3:43 PM, RichD wrote:
> > On July 12, Chris M. Thomasson wrote:
> >>>> Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5 squares,
> >>>> like so: (crappy ASCII graphics)
> >>>
> >>>> ______________________
> >>>> |_A_|___|_A_|___|___|___|
> >>>> |___|_A_|___|___|___|___|
> >>>> |_A_|___|_A_|___|___|___|
> >>>> |___|___|____|___|___|___|
> >>>> |___|___|____|___|___|___|
> >>>> |___|___|____|___|___|___|
> >
> > Wrong diagram.
> Oh yeah. Shit. Sorry.
> >>>> Each block has one face painted with A, as shown.
> >>>> A block moves as follows: it moves into an empty
> >>>> adjacent cell. It rolls over an edge, accomplishing a 1/4 turn.
> >>
> >> Sorry for the stupid question, buts its 3d where if I move a block with
> >> an A on it to the right the the A face will be rotated 1/4 turn so that
> >> the A is on the right hand side of the block, no longer facing me?
> >
> > Correct.
> > It's a 3-D context. End with 5 in a row. No tricks.
> Thanks. I wish I had more time, but I can get it like this. No Cigar at
> all! ;^)
>
> From here:
> ________________________
> |___|_A_|____|___|___|___|
> |_A_|_A_|_A__|___|___|___|
> |___|_A_|____|___|___|___|
> |___|___|____|___|___|___|
> |___|___|____|___|___|___|
> |___|___|____|___|___|___|
> Move the top A to the right 4 times. This makes the A facing the top again:
>
> ________________________
> |___|___|____|___|___|_A_|
> |_A_|_A_|_A__|___|___|___|
> |___|_A_|____|___|___|___|
> |___|___|____|___|___|___|
> |___|___|____|___|___|___|
> |___|___|____|___|___|___|
> Move the bottom A to the right 4 times:
>
> ________________________
> |___|___|____|___|___|_A_|
> |_A_|_A_|_A__|___|___|___|
> |___|___|____|___|___|_A_|
> |___|___|____|___|___|___|
> |___|___|____|___|___|___|
> |___|___|____|___|___|___|
> Mode the right A down four times:
>
> ________________________
> |___|___|____|___|___|_A_|
> |_A_|_A_|____|___|___|___|
> |___|___|____|___|___|_A_|
> |___|___|____|___|___|___|
> |___|___|____|___|___|___|
> |___|___|__A_|___|___|___|
>
> Move the middle A right 4 times:
>
> ________________________
> |___|___|____|___|___|_A_|
> |_A_|___|____|___|___|_A_|
> |___|___|____|___|___|_A_|
> |___|___|____|___|___|___|
> |___|___|____|___|___|___|
> |___|___|__A_|___|___|___|
>
>
> Are these legitimate moves? Anyway, I have to start some work on other
> things pretty soon. Shit happens!
>
> ;^)
Chris, we could probably figure an algorithm if the rules were a little more
straight forward. Also it should be the least amount of moves because
there is probably many different ways to accomplish the main objective
of the puzzle which is first to put them into a column or a row and then
place them back in their original positions.

On 7/12/2023 5:31 PM, Dan joyce wrote:
> On Wednesday, July 12, 2023 at 7:02:52 PM UTC-4, Chris M. Thomasson wrote:
>> On 7/12/2023 3:43 PM, RichD wrote:
>>> On July 12, Chris M. Thomasson wrote:
>>>>>> Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5 squares,
>>>>>> like so: (crappy ASCII graphics)
>>>>>
>>>>>> ______________________
>>>>>> |_A_|___|_A_|___|___|___|
>>>>>> |___|_A_|___|___|___|___|
>>>>>> |_A_|___|_A_|___|___|___|
>>>>>> |___|___|____|___|___|___|
>>>>>> |___|___|____|___|___|___|
>>>>>> |___|___|____|___|___|___|
>>>
>>> Wrong diagram.
>> Oh yeah. Shit. Sorry.
>>>>>> Each block has one face painted with A, as shown.
>>>>>> A block moves as follows: it moves into an empty
>>>>>> adjacent cell. It rolls over an edge, accomplishing a 1/4 turn.
>>>>
>>>> Sorry for the stupid question, buts its 3d where if I move a block with
>>>> an A on it to the right the the A face will be rotated 1/4 turn so that
>>>> the A is on the right hand side of the block, no longer facing me?
>>>
>>> Correct.
>>> It's a 3-D context. End with 5 in a row. No tricks.
>> Thanks. I wish I had more time, but I can get it like this. No Cigar at
>> all! ;^)
>>[...]
>> ;^)
> Chris, we could probably figure an algorithm if the rules were a little more
> straight forward. Also it should be the least amount of moves because
> there is probably many different ways to accomplish the main objective
> of the puzzle which is first to put them into a column or a row and then
> place them back in their original positions.

I think we have to get them into a row or column without any gaps and
all A's on top. A blocks original "orientation" is A on top right, or am
I missing something again? Humm...

One way to do this is to move a block four times say:
_________________________
|___|___|___|___|___|___|
|_A_|_1_|_2_|_3_|_4_|___|
|___|___|___|___|___|___|
|___|___|___|___|___|___|
|___|___|___|___|___|___|
|___|___|___|___|___|___|

A would go to the right, bottom, left and up faces during the 4 moves.
It would end up face up with an A looking at us.

This holds for the inverse of going left:
_________________________
|___|___|___|___|___|___|
|_4_|_3_|_2_|_1_|_4_|___|
|___|___|___|___|___|___|
|___|___|___|___|___|___|
|___|___|___|___|___|___|
|___|___|___|___|___|___|

It works going down and up:
_________________________
|___|___|___|___|___|___|
|_A_|___|___|___|_4_|___|
|_1_|___|___|___|_3_|___|
|_2_|___|___|___|_2_|___|
|_3_|___|___|___|_1_|___|
|_4_|___|___|___|_A_|___|

We can go say, down, down, right right and the A would be facing up again:

_________________________
|___|___|___|___|___|___|
|_A_|___|___|___|___|___|
|_1_|___|___|___|___|___|
|_2_|_3_|_4_|___|___|___|
|___|___|___|___|___|___|
|___|___|___|___|___|___|

On step 2 A would be at the bottom of the block. step 3 would mean the A
is on the left side, and step 4 means its back on top again. I think
that's how it works?

On Wednesday, July 12, 2023 at 5:23:35 PM UTC-4, RichD wrote:
> On July 12, RichD wrote:
> > Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5 squares,
> > like so: (crappy ASCII graphics)
> of course I botched the diagram.
> Re-try:
>
> ______________________
> |___|_A_|____|___|___|___|
> |_A_|_A_|_A_|___|___|___|
> |___|_A_|____|___|___|___|
> |___|___|____|___|___|___|
> |___|___|____|___|___|___|
> |___|___|____|___|___|___|
> --
> Rich
Okay, lets identify each A as the ones going across from left to right as--1,2,3
The top A as 4 and the bottom A as 5.
First we place them in a column across the bottom.
Flip 5 four 1/4 turns to the right.
Flip 123 four 1/4 turns down. Putting them with the A's all facing up on the
bottom.
Now rotate 5 two more 1/4 turns up.
Then rotate 5 two 1/4 turns left.
Then rotate 5 four 1/4 turns down.
This places 1235 all in a row.
Then place 4 one 1/4 turns down.
Then place 4 one 1/4 turn right.
Then place 4 one 1/4 turn down.
Then place 4 one 1/4 turn up.
Then place 4 one 1/4 turn right.
Then place 4 four 1/4 turns down.
All in a row.
Just reverse moves to put them all back to original positions.

;

On Wednesday, July 12, 2023 at 10:17:01 PM UTC-4, Dan joyce wrote:
> On Wednesday, July 12, 2023 at 5:23:35 PM UTC-4, RichD wrote:
> > On July 12, RichD wrote:
> > > Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5 squares,
> > > like so: (crappy ASCII graphics)
> > of course I botched the diagram.
> > Re-try:
> >
> > ______________________
> > |___|_A_|____|___|___|___|
> > |_A_|_A_|_A_|___|___|___|
> > |___|_A_|____|___|___|___|
> > |___|___|____|___|___|___|
> > |___|___|____|___|___|___|
> > |___|___|____|___|___|___|
> > --
> > Rich
> Okay, lets identify each A as the ones going across from left to right as--1,2,3
> The top A as 4 and the bottom A as 5.
> First we place them in a column across the bottom.
> Flip 5 four 1/4 turns to the right.
> Flip 123 four 1/4 turns down. Putting them with the A's all facing up on the
> bottom.
> Now rotate 5 two more 1/4 turns up.
> Then rotate 5 two 1/4 turns left.
> Then rotate 5 four 1/4 turns down.
> This places 1235 all in a row.
> Then place 4 one 1/4 turns down.
> Then place 4 one 1/4 turn right.
> Then place 4 one 1/4 turn down.
> Then place 4 one 1/4 turn up.
> Then place 4 one 1/4 turn right.
> Then place 4 four 1/4 turns down.
> All in a row.
> Just reverse moves to put them all back to original positions.
>
>
> ;
Got a problem with 5 but I will figure it out.

On 7/12/2023 7:16 PM, Dan joyce wrote:
> On Wednesday, July 12, 2023 at 5:23:35 PM UTC-4, RichD wrote:
>> On July 12, RichD wrote:
>>> Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5 squares,
>>> like so: (crappy ASCII graphics)
>> of course I botched the diagram.
>> Re-try:
>>
>> ______________________
>> |___|_A_|____|___|___|___|
>> |_A_|_A_|_A_|___|___|___|
>> |___|_A_|____|___|___|___|
>> |___|___|____|___|___|___|
>> |___|___|____|___|___|___|
>> |___|___|____|___|___|___|
>> --
>> Rich
> Okay, lets identify each A as the ones going across from left to right as--1,2,3
> The top A as 4 and the bottom A as 5.

> First we place them in a column across the bottom.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

How does a column go across the bottom? You mean the bottom row? Think
of row as x and a column as y. x goes left and right, and y goes up and
down. You have to start with the original positions at, x, y positions
right?

(1, 0), (0, 1), (1, 1), (1, 2), (1, 2)

This is:

__0___1___2____3___4___5__ x
0 |___|_A_|____|___|___|___|
1 |_A_|_A_|_A__|___|___|___|
2 |___|_A_|____|___|___|___|
3 |___|___|____|___|___|___|
4 |___|___|____|___|___|___|
5 |___|___|____|___|___|___|

y

So, your top A at point (1, 0) you are labeling 4, and the bottom A at
(1, 2) as 5, the others (0, 1), (1, 1), (1, 2) are 1, 2 and 3 in your
labeling?

So:

__0___1___2____3___4___5__ x
0 |___|_4_|____|___|___|___|
1 |_1_|_2_|_3__|___|___|___|
2 |___|_5_|____|___|___|___|
3 |___|___|____|___|___|___|
4 |___|___|____|___|___|___|
5 |___|___|____|___|___|___|

Right?

> Flip 5 four 1/4 turns to the right.
> Flip 123 four 1/4 turns down. Putting them with the A's all facing up on the
> bottom.
> Now rotate 5 two more 1/4 turns up.
> Then rotate 5 two 1/4 turns left.
> Then rotate 5 four 1/4 turns down.
> This places 1235 all in a row.
> Then place 4 one 1/4 turns down.
> Then place 4 one 1/4 turn right.
> Then place 4 one 1/4 turn down.
> Then place 4 one 1/4 turn up.
> Then place 4 one 1/4 turn right.
> Then place 4 four 1/4 turns down.
> All in a row.
> Just reverse moves to put them all back to original positions.
>
>
> ;
>
>
>
>
>
>
>

On Wednesday, July 12, 2023 at 10:29:18 PM UTC-4, Dan joyce wrote:
> On Wednesday, July 12, 2023 at 10:17:01 PM UTC-4, Dan joyce wrote:
> > On Wednesday, July 12, 2023 at 5:23:35 PM UTC-4, RichD wrote:
> > > On July 12, RichD wrote:
> > > > Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5 squares,
> > > > like so: (crappy ASCII graphics)
> > > of course I botched the diagram.
> > > Re-try:
> > >
> > > ______________________
> > > |___|_A_|____|___|___|___|
> > > |_A_|_A_|_A_|___|___|___|
> > > |___|_A_|____|___|___|___|
> > > |___|___|____|___|___|___|
> > > |___|___|____|___|___|___|
> > > |___|___|____|___|___|___|
> > > --
> > > Rich
> > Okay, lets identify each A as the ones going across from left to right as--1,2,3
> > The top A as 4 and the bottom A as 5.
> > First we place them in a column across the bottom.
> > Flip 5 four 1/4 turns to the right.
> > Flip 123 four 1/4 turns down. Putting them with the A's all facing up on the
> > bottom.
> > Now rotate 5 two more 1/4 turns up.
> > Then rotate 5 two 1/4 turns left.
> > Then rotate 5 four 1/4 turns down.
> > This places 1235 all in a row.
> > Then place 4 one 1/4 turns down.
> > Then place 4 one 1/4 turn right.
> > Then place 4 one 1/4 turn down.
> > Then place 4 one 1/4 turn up.
> > Then place 4 one 1/4 turn right.
> > Then place 4 four 1/4 turns down.
> > All in a row.
> > Just reverse moves to put them all back to original positions.
> >
> >
> > ;
> Got a problem with 5 but I will figure it out.
Okay.
After 5 is moved to the right two 1/4 turns, move 5 left 1/4 turn and
then down 1/4 turn then up two 1/4 turns the down four 1/4 turns.
Now placing 4 in a row.
Go on and now place number 1

.

On 13/07/2023 03:03, Chris M. Thomasson wrote:
> On 7/12/2023 5:31 PM, Dan joyce wrote:
>> On Wednesday, July 12, 2023 at 7:02:52 PM UTC-4, Chris M. Thomasson wrote:
>>> On 7/12/2023 3:43 PM, RichD wrote:
>>>> On July 12, Chris M. Thomasson wrote:
>>>>>>> Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5 squares,
>>>>>>> like so: (crappy ASCII graphics)
>>>>>>
>>>>>>> ______________________
>>>>>>> |_A_|___|_A_|___|___|___|
>>>>>>> |___|_A_|___|___|___|___|
>>>>>>> |_A_|___|_A_|___|___|___|
>>>>>>> |___|___|____|___|___|___|
>>>>>>> |___|___|____|___|___|___|
>>>>>>> |___|___|____|___|___|___|
>>>>
>>>> Wrong diagram.
>>> Oh yeah. Shit. Sorry.
>>>>>>> Each block has one face painted with A, as shown.
>>>>>>> A block moves as follows: it moves into an empty
>>>>>>> adjacent cell. It rolls over an edge, accomplishing a 1/4 turn.
>>>>>
>>>>> Sorry for the stupid question, buts its 3d where if I move a block with
>>>>> an A on it to the right the the A face will be rotated 1/4 turn so that
>>>>> the A is on the right hand side of the block, no longer facing me?
>>>>
>>>> Correct.
>>>> It's a 3-D context. End with 5 in a row. No tricks.
>>> Thanks. I wish I had more time, but I can get it like this. No Cigar at
>>> all! ;^)
>>> [...]
>>> ;^)
>> Chris, we could probably figure an algorithm if the rules were a little more
>> straight forward. Also it should be the least amount of moves because
>> there is probably many different ways to accomplish the main objective
>> of the puzzle which is first to put them into a column or a row and then
>> place them back in their original positions.
>
> I think we have to get them into a row or column without any gaps and all A's on top. A blocks
> original "orientation" is A on top right, or am I missing something again? Humm...
>
> One way to do this is to move a block four times say:
> _________________________
> |___|___|___|___|___|___|
> |_A_|_1_|_2_|_3_|_4_|___|
> |___|___|___|___|___|___|
> |___|___|___|___|___|___|
> |___|___|___|___|___|___|
> |___|___|___|___|___|___|
>
> A would go to the right, bottom, left and up faces during the 4 moves. It would end up face up with
> an A looking at us.
>
> This holds for the inverse of going left:
> _________________________
> |___|___|___|___|___|___|
> |_4_|_3_|_2_|_1_|_4_|___|
> |___|___|___|___|___|___|
> |___|___|___|___|___|___|
> |___|___|___|___|___|___|
> |___|___|___|___|___|___|
>
>
> It works going down and up:
> _________________________
> |___|___|___|___|___|___|
> |_A_|___|___|___|_4_|___|
> |_1_|___|___|___|_3_|___|
> |_2_|___|___|___|_2_|___|
> |_3_|___|___|___|_1_|___|
> |_4_|___|___|___|_A_|___|
>
>
> We can go say, down, down, right right and the A would be facing up again:
>
> _________________________
> |___|___|___|___|___|___|
> |_A_|___|___|___|___|___|
> |_1_|___|___|___|___|___|
> |_2_|_3_|_4_|___|___|___|
> |___|___|___|___|___|___|
> |___|___|___|___|___|___|
>
> On step 2 A would be at the bottom of the block. step 3 would mean the A is on the left side, and
> step 4 means its back on top again. I think that's how it works?

The A would be on top again, but upside down, i.e. not "in the same original orientation". I.e. it
would end up:

_________________________
|___|___|___|___|___|___|
|___|___|___|___|___|___|
|___|___|___|___|___|___|
|___|___|_∀_|___|___|___|
|___|___|___|___|___|___|
|___|___|___|___|___|___|

Also, a block could move like this:

_________________________
|___|___|___|___|___|___|
|___|___|___|___|___|___|
|___|___|_A_|_3_|___|___|
|___|___|_1_|_2_|___|___|
|___|___|___|___|___|___|
|___|___|___|___|___|___|

and the A is back on top, but rotated a quarter turn clockwise (from our viewpoint). Perhaps there
is a parity restriction which restricts how an A can end up given its start and end locations? E.g.
if the A starts and ends in the same location, how can it end up? Obviously it could be unrotated,
and using the above trick twice it could end up upside down, but are those the only possibilities?

If so, that would severely limit possible solutions, as a given starting block would be forced to
move an even number of cells (horizontal+vertical moves) for the final solution - meaning it could
only end up on squares of its own colour if we were using a chess board.

In fact, if that's how things are there would be no solution, as the starting position has 4 black
and 1 white square, but we are asked to end up with a 3-2 split of black/white, which would be
impossible.

But... I'm not sure about the whole parity question without some physical experimentation and it's
time for bed now...! :)

Mike.

On Wednesday, July 12, 2023 at 10:38:17 PM UTC-4, Dan joyce wrote:
> On Wednesday, July 12, 2023 at 10:29:18 PM UTC-4, Dan joyce wrote:
> > On Wednesday, July 12, 2023 at 10:17:01 PM UTC-4, Dan joyce wrote:
> > > On Wednesday, July 12, 2023 at 5:23:35 PM UTC-4, RichD wrote:
> > > > On July 12, RichD wrote:
> > > > > Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5 squares,
> > > > > like so: (crappy ASCII graphics)
> > > > of course I botched the diagram.
> > > > Re-try:
> > > >
> > > > ______________________
> > > > |___|_A_|____|___|___|___|
> > > > |_A_|_A_|_A_|___|___|___|
> > > > |___|_A_|____|___|___|___|
> > > > |___|___|____|___|___|___|
> > > > |___|___|____|___|___|___|
> > > > |___|___|____|___|___|___|
> > > > --
> > > > Rich
> > > Okay, lets identify each A as the ones going across from left to right as--1,2,3
> > > The top A as 4 and the bottom A as 5.
> > > First we place them in a column across the bottom.
> > > Flip 5 four 1/4 turns to the right.
> > > Flip 123 four 1/4 turns down. Putting them with the A's all facing up on the
> > > bottom.
> > > Now rotate 5 two more 1/4 turns up.
> > > Then rotate 5 two 1/4 turns left.
> > > Then rotate 5 four 1/4 turns down.
> > > This places 1235 all in a row.
> > > Then place 4 one 1/4 turns down.
> > > Then place 4 one 1/4 turn right.
> > > Then place 4 one 1/4 turn down.
> > > Then place 4 one 1/4 turn up.
> > > Then place 4 one 1/4 turn right.
> > > Then place 4 four 1/4 turns down.
> > > All in a row.
> > > Just reverse moves to put them all back to original positions.
> > >
> > >
> > > ;
> > Got a problem with 5 but I will figure it out.
> Okay.
> After 5 is moved to the right two 1/4 turns, move 5 left 1/4 turn and
> then down 1/4 turn then up two 1/4 turns the down four 1/4 turns.
> Now placing 4 in a row.
> Go on and now place number 1
I will get it right yet!
After 5 is moved to the right two 1/4 turns, move 5 left 1/4 turn and
then down 1/4 turn then right 1/4 turn, then down1/4 turns.
Now placing 4 in row. Now complete 1's path for 5 in a row and then
just reverse all moves paths starting with 1

On 7/12/2023 7:48 PM, Mike Terry wrote:
> On 13/07/2023 03:03, Chris M. Thomasson wrote:
>> On 7/12/2023 5:31 PM, Dan joyce wrote:
>>> On Wednesday, July 12, 2023 at 7:02:52 PM UTC-4, Chris M. Thomasson
>>> wrote:
>>>> On 7/12/2023 3:43 PM, RichD wrote:
>>>>> On July 12, Chris M. Thomasson wrote:
>>>>>>>> Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5
>>>>>>>> squares,
>>>>>>>> like so: (crappy ASCII graphics)
>>>>>>>
>>>>>>>> ______________________
>>>>>>>> |_A_|___|_A_|___|___|___|
>>>>>>>> |___|_A_|___|___|___|___|
>>>>>>>> |_A_|___|_A_|___|___|___|
>>>>>>>> |___|___|____|___|___|___|
>>>>>>>> |___|___|____|___|___|___|
>>>>>>>> |___|___|____|___|___|___|
>>>>>
>>>>> Wrong diagram.
>>>> Oh yeah. Shit. Sorry.
>>>>>>>> Each block has one face painted with A, as shown.
>>>>>>>> A block moves as follows: it moves into an empty
>>>>>>>> adjacent cell. It rolls over an edge, accomplishing a 1/4 turn.
>>>>>>
>>>>>> Sorry for the stupid question, buts its 3d where if I move a block
>>>>>> with
>>>>>> an A on it to the right the the A face will be rotated 1/4 turn so
>>>>>> that
>>>>>> the A is on the right hand side of the block, no longer facing me?
>>>>>
>>>>> Correct.
>>>>> It's a 3-D context. End with 5 in a row. No tricks.
>>>> Thanks. I wish I had more time, but I can get it like this. No Cigar at
>>>> all! ;^)
>>>> [...]
>>>> ;^)
>>> Chris, we could probably figure an algorithm if the rules were a
>>> little more
>>> straight forward. Also it should be the least amount of moves because
>>> there is probably many different ways to accomplish the main objective
>>> of the puzzle which is first to put them into a column or a row and then
>>> place them back in their original positions.
>>
>> I think we have to get them into a row or column without any gaps and
>> all A's on top. A blocks original "orientation" is A on top right, or
>> am I missing something again? Humm...
>>
>> One way to do this is to move a block four times say:
>> _________________________
>> |___|___|___|___|___|___|
>> |_A_|_1_|_2_|_3_|_4_|___|
>> |___|___|___|___|___|___|
>> |___|___|___|___|___|___|
>> |___|___|___|___|___|___|
>> |___|___|___|___|___|___|
>>
>> A would go to the right, bottom, left and up faces during the 4 moves.
>> It would end up face up with an A looking at us.
>>
>> This holds for the inverse of going left:
>> _________________________
>> |___|___|___|___|___|___|
>> |_4_|_3_|_2_|_1_|_4_|___|
>> |___|___|___|___|___|___|
>> |___|___|___|___|___|___|
>> |___|___|___|___|___|___|
>> |___|___|___|___|___|___|
>>
>>
>> It works going down and up:
>> _________________________
>> |___|___|___|___|___|___|
>> |_A_|___|___|___|_4_|___|
>> |_1_|___|___|___|_3_|___|
>> |_2_|___|___|___|_2_|___|
>> |_3_|___|___|___|_1_|___|
>> |_4_|___|___|___|_A_|___|
>>
>>
>> We can go say, down, down, right right and the A would be facing up
>> again:
>>
>> _________________________
>> |___|___|___|___|___|___|
>> |_A_|___|___|___|___|___|
>> |_1_|___|___|___|___|___|
>> |_2_|_3_|_4_|___|___|___|
>> |___|___|___|___|___|___|
>> |___|___|___|___|___|___|
>>
>> On step 2 A would be at the bottom of the block. step 3 would mean the
>> A is on the left side, and step 4 means its back on top again. I think
>> that's how it works?
>
> The A would be on top again, but upside down, i.e. not "in the same
> original orientation".  I.e. it would end up:

Yup. Well, shit.

>  _________________________
>  |___|___|___|___|___|___|
>  |___|___|___|___|___|___|
>  |___|___|___|___|___|___|
>  |___|___|_∀_|___|___|___|
>  |___|___|___|___|___|___|
>  |___|___|___|___|___|___|
>
> Also, a block could move like this:
>
>  _________________________
>  |___|___|___|___|___|___|
>  |___|___|___|___|___|___|
>  |___|___|_A_|_3_|___|___|
>  |___|___|_1_|_2_|___|___|
>  |___|___|___|___|___|___|
>  |___|___|___|___|___|___|
>
> and the A is back on top, but rotated a quarter turn clockwise (from our
> viewpoint).  Perhaps there is a parity restriction which restricts how
> an A can end up given its start and end locations?  E.g. if the A starts
> and ends in the same location, how can it end up?  Obviously it could be
> unrotated, and using the above trick twice it could end up upside down,
> but are those the only possibilities?

We need to get clarification if a solution just needs all A's up, their
rotations be dammed... Or if they all must be A, not some upside down A,
or a right or left A. Humm...

> If so, that would severely limit possible solutions, as a given starting
> block would be forced to move an even number of cells
> (horizontal+vertical moves) for the final solution - meaning it could
> only end up on squares of its own colour if we were using a chess board.
>
> In fact, if that's how things are there would be no solution, as the
> starting position has 4 black and 1 white square, but we are asked to
> end up with a 3-2 split of black/white, which would be impossible.

I need to ponder on that.

> But... I'm not sure about the whole parity question without some
> physical experimentation and it's time for bed now...!  :)

Yeah. I have already used up too much time on this puzzle. I need to go
to work on a GLSL shader of mine.

On Wednesday, July 12, 2023 at 10:50:07 PM UTC-4, Dan joyce wrote:
> On Wednesday, July 12, 2023 at 10:38:17 PM UTC-4, Dan joyce wrote:
> > On Wednesday, July 12, 2023 at 10:29:18 PM UTC-4, Dan joyce wrote:
> > > On Wednesday, July 12, 2023 at 10:17:01 PM UTC-4, Dan joyce wrote:
> > > > On Wednesday, July 12, 2023 at 5:23:35 PM UTC-4, RichD wrote:
> > > > > On July 12, RichD wrote:
> > > > > > Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5 squares,
> > > > > > like so: (crappy ASCII graphics)
> > > > > of course I botched the diagram.
> > > > > Re-try:
> > > > >
> > > > > ______________________
> > > > > |___|_A_|____|___|___|___|
> > > > > |_A_|_A_|_A_|___|___|___|
> > > > > |___|_A_|____|___|___|___|
> > > > > |___|___|____|___|___|___|
> > > > > |___|___|____|___|___|___|
> > > > > |___|___|____|___|___|___|
> > > > > --
> > > > > Rich
> > > > Okay, lets identify each A as the ones going across from left to right as--1,2,3
> > > > The top A as 4 and the bottom A as 5.
> > > > First we place them in a column across the bottom.
> > > > Flip 5 four 1/4 turns to the right.
> > > > Flip 123 four 1/4 turns down. Putting them with the A's all facing up on the
> > > > bottom.
> > > > Now rotate 5 two more 1/4 turns up.
> > > > Then rotate 5 two 1/4 turns left.
> > > > Then rotate 5 four 1/4 turns down.
> > > > This places 1235 all in a row.
> > > > Then place 4 one 1/4 turns down.
> > > > Then place 4 one 1/4 turn right.
> > > > Then place 4 one 1/4 turn down.
> > > > Then place 4 one 1/4 turn up.
> > > > Then place 4 one 1/4 turn right.
> > > > Then place 4 four 1/4 turns down.
> > > > All in a row.
> > > > Just reverse moves to put them all back to original positions.
> > > >
> > > >
> > > > ;
> > > Got a problem with 5 but I will figure it out.
> > Okay.
> > After 5 is moved to the right two 1/4 turns, move 5 left 1/4 turn and
> > then down 1/4 turn then up two 1/4 turns the down four 1/4 turns.
> > Now placing 4 in a row.
> > Go on and now place number 1
> I will get it right yet!
> After 5 is moved to the right two 1/4 turns, move 5 left 1/4 turn and
> then down 1/4 turn then right 1/4 turn, then down1/4 turns.
> Now placing 4 in row. Now complete 1's path for 5 in a row and then
> just reverse all moves paths starting with 1
The last move for 5 should be two 1/4 turns down, that puts 4 in a row.
Seeing I screwed up so many times moving #5, here it is again in order,
The top A as 4 and the bottom A as 5.
First we place them in a row across the bottom.
Flip 5 two 1/4 turns to the right.
Flip 123 four 1/4 turns down. Putting them with the A's all facing up on the
bottom in a row.
After 5 is moved to the right two 1/4 turns, move 5 left 1/4 turn and
then down 1/4 turn then right 1/4 turn, then down two1/4 turns.
Placing 1235 or 4 A's in a row on the bottom left.
Now move number 4, very top one, down 1/4 turn then right 1/4 turn then
down 1/4 turn then right 1/4 turn then up 1/4 turn then right 1/4 turn then
down four 1/4 turns putting 12354 or all A's in a row.
Just reverse moves starting with #4.
.

On Wednesday, July 12, 2023 at 11:13:08 PM UTC-4, Dan joyce wrote:
> On Wednesday, July 12, 2023 at 10:50:07 PM UTC-4, Dan joyce wrote:
> > On Wednesday, July 12, 2023 at 10:38:17 PM UTC-4, Dan joyce wrote:
> > > On Wednesday, July 12, 2023 at 10:29:18 PM UTC-4, Dan joyce wrote:
> > > > On Wednesday, July 12, 2023 at 10:17:01 PM UTC-4, Dan joyce wrote:
> > > > > On Wednesday, July 12, 2023 at 5:23:35 PM UTC-4, RichD wrote:
> > > > > > On July 12, RichD wrote:
> > > > > > > Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5 squares,
> > > > > > > like so: (crappy ASCII graphics)
> > > > > > of course I botched the diagram.
> > > > > > Re-try:
> > > > > >
> > > > > > ______________________
> > > > > > |___|_A_|____|___|___|___|
> > > > > > |_A_|_A_|_A_|___|___|___|
> > > > > > |___|_A_|____|___|___|___|
> > > > > > |___|___|____|___|___|___|
> > > > > > |___|___|____|___|___|___|
> > > > > > |___|___|____|___|___|___|
> > > > > > --
> > > > > > Rich
> > > > > Okay, lets identify each A as the ones going across from left to right as--1,2,3
> > > > > The top A as 4 and the bottom A as 5.
> > > > > First we place them in a column across the bottom.
> > > > > Flip 5 four 1/4 turns to the right.
> > > > > Flip 123 four 1/4 turns down. Putting them with the A's all facing up on the
> > > > > bottom.
> > > > > Now rotate 5 two more 1/4 turns up.
> > > > > Then rotate 5 two 1/4 turns left.
> > > > > Then rotate 5 four 1/4 turns down.
> > > > > This places 1235 all in a row.
> > > > > Then place 4 one 1/4 turns down.
> > > > > Then place 4 one 1/4 turn right.
> > > > > Then place 4 one 1/4 turn down.
> > > > > Then place 4 one 1/4 turn up.
> > > > > Then place 4 one 1/4 turn right.
> > > > > Then place 4 four 1/4 turns down.
> > > > > All in a row.
> > > > > Just reverse moves to put them all back to original positions.
> > > > >
> > > > >
> > > > > ;
> > > > Got a problem with 5 but I will figure it out.
> > > Okay.
> > > After 5 is moved to the right two 1/4 turns, move 5 left 1/4 turn and
> > > then down 1/4 turn then up two 1/4 turns the down four 1/4 turns.
> > > Now placing 4 in a row.
> > > Go on and now place number 1
> > I will get it right yet!
> > After 5 is moved to the right two 1/4 turns, move 5 left 1/4 turn and
> > then down 1/4 turn then right 1/4 turn, then down1/4 turns.
> > Now placing 4 in row. Now complete 1's path for 5 in a row and then
> > just reverse all moves paths starting with 1
> The last move for 5 should be two 1/4 turns down, that puts 4 in a row.
> Seeing I screwed up so many times moving #5, here it is again in order,
> The top A as 4 and the bottom A as 5.
> First we place them in a row across the bottom.
> Flip 5 two 1/4 turns to the right.
> Flip 123 four 1/4 turns down. Putting them with the A's all facing up on the
> bottom in a row.
> After 5 is moved to the right two 1/4 turns, move 5 left 1/4 turn and
> then down 1/4 turn then right 1/4 turn, then down two1/4 turns.
> Placing 1235 or 4 A's in a row on the bottom left.
> Now move number 4, very top one, down 1/4 turn then right 1/4 turn then
> down 1/4 turn then right 1/4 turn then up 1/4 turn then right 1/4 turn then
> down four 1/4 turns putting 12354 or all A's in a row.
> Just reverse moves starting with #4.
> .
You are right Chris, the first 3 of my A's in a row on the bottom are good but
the 4th and and 5th are facing ok but not in the right directions (point at the top).
When you return them to the original positions all A's are right.

On 7/12/2023 8:34 PM, Dan joyce wrote:
> On Wednesday, July 12, 2023 at 11:13:08 PM UTC-4, Dan joyce wrote:
>> On Wednesday, July 12, 2023 at 10:50:07 PM UTC-4, Dan joyce wrote:
>>> On Wednesday, July 12, 2023 at 10:38:17 PM UTC-4, Dan joyce wrote:
>>>> On Wednesday, July 12, 2023 at 10:29:18 PM UTC-4, Dan joyce wrote:
>>>>> On Wednesday, July 12, 2023 at 10:17:01 PM UTC-4, Dan joyce wrote:
>>>>>> On Wednesday, July 12, 2023 at 5:23:35 PM UTC-4, RichD wrote:
>>>>>>> On July 12, RichD wrote:
>>>>>>>> Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5 squares,
>>>>>>>> like so: (crappy ASCII graphics)
>>>>>>> of course I botched the diagram.
>>>>>>> Re-try:
>>>>>>>
>>>>>>> ______________________
>>>>>>> |___|_A_|____|___|___|___|
>>>>>>> |_A_|_A_|_A_|___|___|___|
>>>>>>> |___|_A_|____|___|___|___|
>>>>>>> |___|___|____|___|___|___|
>>>>>>> |___|___|____|___|___|___|
>>>>>>> |___|___|____|___|___|___|
[...]
> You are right Chris, the first 3 of my A's in a row on the bottom are good but
> the 4th and and 5th are facing ok but not in the right directions (point at the top).
> When you return them to the original positions all A's are right.

Labeled as:

________________________
|___|_0_|____|___|___|___|
|_1_|_2_|_3__|___|___|___|
|___|_4_|____|___|___|___|
|___|___|____|___|___|___|
|___|___|____|___|___|___|
|___|___|____|___|___|___|

(1) down 4
(3) down 4
(4) right 4
(2) down 4

gets me to:

________________________
|___|_0_|____|___|___|___|
|___|___|____|___|___|___|
|___|___|____|___|___|_4_|
|___|___|____|___|___|___|
|___|___|____|___|___|___|
|_1_|_2_|_3__|___|___|___|

All A up in the right rotation.

On Wednesday, July 12, 2023 at 11:48:42 PM UTC-4, Chris M. Thomasson wrote:
> On 7/12/2023 8:34 PM, Dan joyce wrote:
> > On Wednesday, July 12, 2023 at 11:13:08 PM UTC-4, Dan joyce wrote:
> >> On Wednesday, July 12, 2023 at 10:50:07 PM UTC-4, Dan joyce wrote:
> >>> On Wednesday, July 12, 2023 at 10:38:17 PM UTC-4, Dan joyce wrote:
> >>>> On Wednesday, July 12, 2023 at 10:29:18 PM UTC-4, Dan joyce wrote:
> >>>>> On Wednesday, July 12, 2023 at 10:17:01 PM UTC-4, Dan joyce wrote:
> >>>>>> On Wednesday, July 12, 2023 at 5:23:35 PM UTC-4, RichD wrote:
> >>>>>>> On July 12, RichD wrote:
> >>>>>>>> Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5 squares,
> >>>>>>>> like so: (crappy ASCII graphics)
> >>>>>>> of course I botched the diagram.
> >>>>>>> Re-try:
> >>>>>>>
> >>>>>>> ______________________
> >>>>>>> |___|_A_|____|___|___|___|
> >>>>>>> |_A_|_A_|_A_|___|___|___|
> >>>>>>> |___|_A_|____|___|___|___|
> >>>>>>> |___|___|____|___|___|___|
> >>>>>>> |___|___|____|___|___|___|
> >>>>>>> |___|___|____|___|___|___|
> [...]
> > You are right Chris, the first 3 of my A's in a row on the bottom are good but
> > the 4th and and 5th are facing ok but not in the right directions (point at the top).
> > When you return them to the original positions all A's are right.
> Labeled as:
>
> ________________________
> |___|_0_|____|___|___|___|
> |_1_|_2_|_3__|___|___|___|
> |___|_4_|____|___|___|___|
> |___|___|____|___|___|___|
> |___|___|____|___|___|___|
> |___|___|____|___|___|___|
> (1) down 4
> (3) down 4
> (4) right 4
> (2) down 4
>
> gets me to:
>
> ________________________
> |___|_0_|____|___|___|___|
> |___|___|____|___|___|___|
> |___|___|____|___|___|_4_|
> |___|___|____|___|___|___|
> |___|___|____|___|___|___|
> |_1_|_2_|_3__|___|___|___|
> All A up in the right rotation.
Your 0 is my 4 and your 4 is my 5.
My bottom row =12354 but 4 and 5 the A's are not pointed in an upward position.
I am not sure if when placed in a row on the bottom the last two, 5 and 4 can
ever be where the A's are orientated in the right position. 123 are fine.

On Thursday, July 13, 2023 at 12:14:23 AM UTC-4, Dan joyce wrote:
> On Wednesday, July 12, 2023 at 11:48:42 PM UTC-4, Chris M. Thomasson wrote:
> > On 7/12/2023 8:34 PM, Dan joyce wrote:
> > > On Wednesday, July 12, 2023 at 11:13:08 PM UTC-4, Dan joyce wrote:
> > >> On Wednesday, July 12, 2023 at 10:50:07 PM UTC-4, Dan joyce wrote:
> > >>> On Wednesday, July 12, 2023 at 10:38:17 PM UTC-4, Dan joyce wrote:
> > >>>> On Wednesday, July 12, 2023 at 10:29:18 PM UTC-4, Dan joyce wrote:
> > >>>>> On Wednesday, July 12, 2023 at 10:17:01 PM UTC-4, Dan joyce wrote:
> > >>>>>> On Wednesday, July 12, 2023 at 5:23:35 PM UTC-4, RichD wrote:
> > >>>>>>> On July 12, RichD wrote:
> > >>>>>>>> Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5 squares,
> > >>>>>>>> like so: (crappy ASCII graphics)
> > >>>>>>> of course I botched the diagram.
> > >>>>>>> Re-try:
> > >>>>>>>
> > >>>>>>> ______________________
> > >>>>>>> |___|_A_|____|___|___|___|
> > >>>>>>> |_A_|_A_|_A_|___|___|___|
> > >>>>>>> |___|_A_|____|___|___|___|
> > >>>>>>> |___|___|____|___|___|___|
> > >>>>>>> |___|___|____|___|___|___|
> > >>>>>>> |___|___|____|___|___|___|
> > [...]
> > > You are right Chris, the first 3 of my A's in a row on the bottom are good but
> > > the 4th and and 5th are facing ok but not in the right directions (point at the top).
> > > When you return them to the original positions all A's are right.
> > Labeled as:
> >
> > ________________________
> > |___|_0_|____|___|___|___|
> > |_1_|_2_|_3__|___|___|___|
> > |___|_4_|____|___|___|___|
> > |___|___|____|___|___|___|
> > |___|___|____|___|___|___|
> > |___|___|____|___|___|___|
> > (1) down 4
> > (3) down 4
> > (4) right 4
> > (2) down 4
> >
> > gets me to:
> >
> > ________________________
> > |___|_0_|____|___|___|___|
> > |___|___|____|___|___|___|
> > |___|___|____|___|___|_4_|
> > |___|___|____|___|___|___|
> > |___|___|____|___|___|___|
> > |_1_|_2_|_3__|___|___|___|
> > All A up in the right rotation.
> Your 0 is my 4 and your 4 is my 5.
> My bottom row =12354 but 4 and 5 the A's are not pointed in an upward position.
> I am not sure if when placed in a row on the bottom the last two, 5 and 4 can
> ever be where the A's are orientated in the right position. 123 are fine.
Looking at it further, with the proper moves it probably can be done.
I will attack it later.

On 13/07/2023 04:06, Chris M. Thomasson wrote:
> On 7/12/2023 7:48 PM, Mike Terry wrote:
>> On 13/07/2023 03:03, Chris M. Thomasson wrote:
>>> On 7/12/2023 5:31 PM, Dan joyce wrote:
>>>> On Wednesday, July 12, 2023 at 7:02:52 PM UTC-4, Chris M. Thomasson wrote:
>>>>> On 7/12/2023 3:43 PM, RichD wrote:
>>>>>> On July 12, Chris M. Thomasson wrote:
>>>>>>>>> Given a 6 x 6 checkerboard. At the top left, 5 blocks fill 5 squares,
>>>>>>>>> like so: (crappy ASCII graphics)
>>>>>>>>
>>>>>>>>> ______________________
>>>>>>>>> |_A_|___|_A_|___|___|___|
>>>>>>>>> |___|_A_|___|___|___|___|
>>>>>>>>> |_A_|___|_A_|___|___|___|
>>>>>>>>> |___|___|____|___|___|___|
>>>>>>>>> |___|___|____|___|___|___|
>>>>>>>>> |___|___|____|___|___|___|
>>>>>>
>>>>>> Wrong diagram.
>>>>> Oh yeah. Shit. Sorry.
>>>>>>>>> Each block has one face painted with A, as shown.
>>>>>>>>> A block moves as follows: it moves into an empty
>>>>>>>>> adjacent cell. It rolls over an edge, accomplishing a 1/4 turn.
>>>>>>>
>>>>>>> Sorry for the stupid question, buts its 3d where if I move a block with
>>>>>>> an A on it to the right the the A face will be rotated 1/4 turn so that
>>>>>>> the A is on the right hand side of the block, no longer facing me?
>>>>>>
>>>>>> Correct.
>>>>>> It's a 3-D context. End with 5 in a row. No tricks.
>>>>> Thanks. I wish I had more time, but I can get it like this. No Cigar at
>>>>> all! ;^)
>>>>> [...]
>>>>> ;^)
>>>> Chris, we could probably figure an algorithm if the rules were a little more
>>>> straight forward. Also it should be the least amount of moves because
>>>> there is probably many different ways to accomplish the main objective
>>>> of the puzzle which is first to put them into a column or a row and then
>>>> place them back in their original positions.
>>>
>>> I think we have to get them into a row or column without any gaps and all A's on top. A blocks
>>> original "orientation" is A on top right, or am I missing something again? Humm...
>>>
>>> One way to do this is to move a block four times say:
>>> _________________________
>>> |___|___|___|___|___|___|
>>> |_A_|_1_|_2_|_3_|_4_|___|
>>> |___|___|___|___|___|___|
>>> |___|___|___|___|___|___|
>>> |___|___|___|___|___|___|
>>> |___|___|___|___|___|___|
>>>
>>> A would go to the right, bottom, left and up faces during the 4 moves. It would end up face up
>>> with an A looking at us.
>>>
>>> This holds for the inverse of going left:
>>> _________________________
>>> |___|___|___|___|___|___|
>>> |_4_|_3_|_2_|_1_|_4_|___|
>>> |___|___|___|___|___|___|
>>> |___|___|___|___|___|___|
>>> |___|___|___|___|___|___|
>>> |___|___|___|___|___|___|
>>>
>>>
>>> It works going down and up:
>>> _________________________
>>> |___|___|___|___|___|___|
>>> |_A_|___|___|___|_4_|___|
>>> |_1_|___|___|___|_3_|___|
>>> |_2_|___|___|___|_2_|___|
>>> |_3_|___|___|___|_1_|___|
>>> |_4_|___|___|___|_A_|___|
>>>
>>>
>>> We can go say, down, down, right right and the A would be facing up again:
>>>
>>> _________________________
>>> |___|___|___|___|___|___|
>>> |_A_|___|___|___|___|___|
>>> |_1_|___|___|___|___|___|
>>> |_2_|_3_|_4_|___|___|___|
>>> |___|___|___|___|___|___|
>>> |___|___|___|___|___|___|
>>>
>>> On step 2 A would be at the bottom of the block. step 3 would mean the A is on the left side, and
>>> step 4 means its back on top again. I think that's how it works?
>>
>> The A would be on top again, but upside down, i.e. not "in the same original orientation".  I.e.
>> it would end up:
>
> Yup. Well, shit.
>
>
>>   _________________________
>>   |___|___|___|___|___|___|
>>   |___|___|___|___|___|___|
>>   |___|___|___|___|___|___|
>>   |___|___|_∀_|___|___|___|
>>   |___|___|___|___|___|___|
>>   |___|___|___|___|___|___|
>>
>> Also, a block could move like this:
>>
>>   _________________________
>>   |___|___|___|___|___|___|
>>   |___|___|___|___|___|___|
>>   |___|___|_A_|_3_|___|___|
>>   |___|___|_1_|_2_|___|___|
>>   |___|___|___|___|___|___|
>>   |___|___|___|___|___|___|
>>
>> and the A is back on top, but rotated a quarter turn clockwise (from our viewpoint).  Perhaps
>> there is a parity restriction which restricts how an A can end up given its start and end
>> locations?  E.g. if the A starts and ends in the same location, how can it end up?  Obviously it
>> could be unrotated, and using the above trick twice it could end up upside down, but are those the
>> only possibilities?
>
> We need to get clarification if a solution just needs all A's up, their rotations be dammed... Or if
> they all must be A, not some upside down A, or a right or left A. Humm...
>
>
>> If so, that would severely limit possible solutions, as a given starting block would be forced to
>> move an even number of cells (horizontal+vertical moves) for the final solution - meaning it could
>> only end up on squares of its own colour if we were using a chess board.
>>
>> In fact, if that's how things are there would be no solution, as the starting position has 4 black
>> and 1 white square, but we are asked to end up with a 3-2 split of black/white, which would be
>> impossible.
>
> I need to ponder on that.
>

I thought about it in bed before I finally dropped off, and decided there was indeed such a parity
constraint, assuming the As must all end up the right way up, not rotated or upside down. So the
problem has no solution.

Here's a quick outline of a proof:

1) first just think of a cube with no A, but marked with what I'll call "parity lines":
Labelling the faces T,B,N,E,S,W (for top, bottom, north, east, south and west).
Bisect the faces T,B with a north/south line.
Bisect the faces N,S with a horizontal (east/west) line.
Bisect the faces E,W with a vertical line.
2) So each face has a bisecting line, and if we roll the cube around on a grid, as in
the OP's problem, there are just two distinguished orientations for the cube:
- all the parity lines match up with the lines in the starting position.
Call this "even parity".
- all the parity lines are perpendicular to the lines in the starting position.
Call this "odd parity".
3) Key observation: every individual "roll" of the cube about one of its
edges rotates it by 90 degrees, AND FLIPS THE PARITY between even/odd parity.
4) If the cube is rolled along a path through the grid, the parity of the cube
is fully determined by whether an even/odd number of rolls has been performed.
So if the cube starts with even parity on a black chess-board square, it will
be even parity when it gets to any black square, and odd parity on any white
square.
5) Now marking the A on one face, as in Rich's problem... Obviously the A stays
alligned with its parity line as the cube rolls along a path.
6) So if we require the final A to be "the right way up" as we view it, the
cube must end up with the even parity orientation (which it started with)
otherwise if the final parity were odd the A would be showing
sideways (along with it's associated parity line).
7) As noted in (4) the even parity positions are the squares of the same colour
as the cube's starting square.
8) Er, that's it!

Mike.