Help needed for doubt in diagram on equivalent grid rotations.

Question

In the book titled: "Combinatorics - A guided tour', by David Mazur; there is in chapter 5, titled: 'Counting Under Equivalence', a diagram showing up $4$ equivalent grids (under rotation) of a $3*3$ square grid, colored using two colors - black, white.

According to me, the diagram's last two grids are flawed, if take grid$1$ or grid $2$ as reference.

To easily explain my doubt, if assume that the grid has following structure:

$A1 \ \ A2 \ \ A3$
$B1 \ \ B2 \ \ B3$
$C1 \ \ C2 \ \ C3$

Then the $4$ grids are having colors as:
$\ Grid 1 \ \ \ \ \ \ \ \ \ \ Grid 2 \ \ \ \ \ \ \ Grid 3 \ \ \ \ \ \ \ \ \ Grid4$
$W \ B \ \ W \ \ \ \ W \ W \ W \ \ \ \ B \ W \ W\ \ \ \ W \ W \ B$
$W \ W \ W\ \ \ \ W \ W \ B \ \ \ \ W \ W \ W\ \ \ \ B \ W \ W$
$W \ W \ B \ \ \ \ \ B \ W \ W \ \ \ \ W \ B \ W\ \ \ \ W \ W \ W$

As per me, the $4$ grids that are equivalent under rotation,
should have different colors in the last two grids,
as shown below:
$\ Grid 1 \ \ \ \ \ \ \ \ \ \ Grid 2 \ \ \ \ \ \ \ Grid 3 \ \ \ \ \ \ \ \ \ Grid4$
$W \ B \ \ W \ \ \ \ W \ W \ W \ \ \ \ B \ W \ W\ \ \ \ W \ W \ B$
$W \ W \ W\ \ \ \ W \ W \ B \ \ \ \ W \ W \ B\ \ \ \ W \ W \ W$
$W \ W \ B \ \ \ \ \ B \ W \ W \ \ \ \ W \ W \ W\ \ \ \ W \ B \ W$

score 1 · Accepted Answer · answered Feb 29 '20 at 15:22

1

The third picture can be obtained by rotating the second picture clockwise $90^\circ$.
The fourth picture can be obtained by rotating the third picture clockwise $90^\circ$.

answered Feb 29 '20 at 15:22

Siong Thye Goh

153,832

Please help with my today's post at: https://math.stackexchange.com/q/3594018/424260 – jiten Mar 25 '20 at 05:00
Thanks for providing answer, although deleted now. But request to state the error in your answer. I mean is it for any value of $i|p$, for non-prime $p$. – jiten Mar 25 '20 at 06:38
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I did not consider division by $0$. The counterexample has shown that the result need not hold for non-prime. – Siong Thye Goh Mar 25 '20 at 06:39
I don't have a concrete attack plan for now. Let's hope the rest of them figure out :) – Siong Thye Goh Mar 25 '20 at 06:43
If you don't mind, it is the first problem at https://www.worldscientific.com/doi/pdf/10.1142/9789813238312_0001; but am unable to understand the approach given there completely and hence asked the question. – jiten Mar 25 '20 at 06:45
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will read at home later. – Siong Thye Goh Mar 25 '20 at 06:49
seems to be beyond me. – Siong Thye Goh Mar 25 '20 at 07:07
There is a new answer to my post by @Karl. Please see my last comment to that. It sums up my confusion to understand his answer. Also, your answer divided by $0!=1$ and not $0$, although the counterexample is fine. – jiten Mar 25 '20 at 07:44
Please respond to my last comment - in particular to how your answer was wrong. – jiten Mar 25 '20 at 08:42
Let us continue this discussion in chat. – Siong Thye Goh Mar 25 '20 at 08:43

Help needed for doubt in diagram on equivalent grid rotations.

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