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More Homework 6

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More Homework 6

Postby JPF » Wed Nov 23, 2022 11:34 pm

Here is a totally useless exercise but challenging!
consider the following pattern:
Code: Select all
+-------+-------+-------+
| . . . | . . . | . . . |
| . x . | . x . | . x . |
| x x . | x x . | x x . |
+-------+-------+-------+
| . . . | . . . | . . . |
| . x . | . x . | . x . |
| x x . | x x . | x x . |
+-------+-------+-------+
| . . . | . . . | . . . |
| . x . | . x . | . x . |
| x x . | x x . | x x . |
+-------+-------+-------+


The value of a box is the sum of its digits.
Find valid puzzles with the highest number of boxes having the same value.
Here is an example with 5 equal-value boxes:
Code: Select all
+-------+-------+-------+
| . . . | . . . | . . . |
| . 5 . | . 9 . | . 4 . |
| 9 4 . | 5 2 . | 8 6 . |
+-------+-------+-------+
| . . . | . . . | . . . |
| . 9 . | . 5 . | . 2 . |
| 5 2 . | 4 6 . | 9 7 . |
+-------+-------+-------+
| . . . | . . . | . . . |
| . 7 . | . 8 . | . 9 . |
| 3 8 . | 9 7 . | 4 5 . |
+-------+-------+-------+

5 boxes have the same value equal to 18 : B1,B3,B6,B7,B9

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Re: More Homework 6

Postby marek stefanik » Thu Nov 24, 2022 3:18 pm

14 all, non-minimal
Code: Select all
+---------+---------+---------+
| .  .  . | .  .  . | .  .  . |
| .  2  . | .  6  . | .  1  . |
| 4  8  . | 5  3  . | 6  7  . |
+--–------+---------+---------+
| .  .  . | .  .  . | .  .  . |
| .  6  . | .  1  . | .  2  . |
| 5  3  . | 6  7  . | 4  8  . |
+---–-----+---------+---------+
| .  .  . | .  .  . | .  .  . |
| .  4  . | .  2  . | .  6  . |
| 9  1  . | 4  8  . | 5  3  . |
+---–-----+---------+---------+


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Re: More Homework 6

Postby m_b_metcalf » Thu Nov 24, 2022 3:41 pm

marek stefanik wrote:14 all, non-minimal
Marek

... from which one can immediately obtain, BINGO!,
all 14, minimal:
Code: Select all
 . . . . . . . . .
 . 2 . . 6 . . 5 .
 4 8 . 5 3 . 2 7 .
 . . . . . . . . .
 . 6 . . 1 . . 2 .
 5 3 . 6 7 . 4 8 .
 . . . . . . . . .
 . 7 . . 2 . . 6 .
 6 1 . 4 8 . 5 3 .


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Re: More Homework 6

Postby JPF » Thu Nov 24, 2022 4:58 pm

Congratulations to both of you!

Let's see how it works for this one:
Code: Select all
+-------+-------+-------+
| . . . | x . x | . . . |
| x . x | . . . | x . x |
| . . . | x . x | . . . |
+-------+-------+-------+
| x . x | . . . | x . x |
| . . . | x . x | . . . |
| x . x | . . . | x . x |
+-------+-------+-------+
| . . . | x . x | . . . |
| x . x | . . . | x . x |
| . . . | x . x | . . . |
+-------+-------+-------+

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Re: More Homework 6

Postby m_b_metcalf » Thu Nov 24, 2022 5:32 pm

1st. try:
Code: Select all
 . . . 4 . 2 . . .
 2 . 8 . . . 9 . 1
 . . . 3 . 1 . . .
 6 . 2 . . . 4 . 3
 . . . 7 . 3 . . .
 7 . 5 . . . 8 . 6
 . . . 9 . 7 . . .
 9 . 1 . . . 2 . 8
 . . . 8 . 5 . . .   6 x 10, non-minimal
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Re: More Homework 6

Postby JPF » Thu Nov 24, 2022 6:07 pm

Code: Select all
+-------+-------+-------+
| . . . | 9 . 6 | . . . |
| 4 . 7 | . . . | 6 . 5 |
| . . . | 7 . 5 | . . . |
+-------+-------+-------+
| 1 . 3 | . . . | 8 . 2 |
| . . . | 3 . 8 | . . . |
| 2 . 5 | . . . | 7 . 6 |
+-------+-------+-------+
| . . . | 1 . 2 | . . . |
| 5 . 6 | . . . | 2 . 9 |
| . . . | 5 . 3 | . . . |
+-------+-------+-------+   7x11 non minimal

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Re: More Homework 6

Postby JPF » Thu Nov 24, 2022 8:49 pm

Code: Select all
+-------+-------+-------+
| . . . | 1 . 4 | . . . |
| 9 . 4 | . . . | 5 . 8 |
| . . . | 3 . 5 | . . . |
+-------+-------+-------+
| 1 . 7 | . . . | 6 . 4 |
| . . . | 5 . 8 | . . . |
| 5 . 9 | . . . | 2 . 1 |
+-------+-------+-------+
| . . . | 4 . 1 | . . . |
| 4 . 2 | . . . | 7 . 6 |
| . . . | 2 . 6 | . . . |
+-------+-------+-------+   7x13 minimal

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Re: More Homework 6

Postby jovi_al01 » Fri Nov 25, 2022 2:25 am

back to the original question, i was curious as to if the most "extreme" sums were possible.
they are!

all 11, minimal.

this is the lowest possible, and of course can be relabeled to obtain 19, which is the highest possible.


Code: Select all
.........
.8..1..2.
21.73.54.
.........
.7..4..1.
13.25.46.
.........
.2..6..3.
54.32.17.
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Re: More Homework 6

Postby shye » Fri Nov 25, 2022 12:43 pm

Code: Select all
..........7..9..2.24.13.56...........9..2..7.13.58.24...........2..7..9.56.24.13.

8 boxes with a sum of 13, non-minimal
not as impressive as the others but this has some interesting symmetry to it
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Re: More Homework 6

Postby coloin » Fri Nov 25, 2022 2:31 pm

jovi_al01 wrote:...this is the lowest possible, and of course can be relabeled to obtain 19, which is the highest possible.

Very good - i see why 11 is the lowest possible [ there must be an 8 [plus 2 plus 1]], and 19 the highest [ there must be a 2 [plus 8 plus 9]]

but I cant achieve all boxes to be 19 replacing the 1 with a 9 however
Code: Select all
+---+---+---+
|...|...|...|
|.8.|.9.|.2.|
|29.|73.|54.|
+---+---+---+
|...|...|...|
|.7.|.4.|.9.|
|93.|25.|46.|
+---+---+---+
|...|...|...|
|.2.|.6.|.3.|
|54.|32.|97.|
+---+---+---+   6 boxes @19

Code: Select all
+---+---+---+
|...|...|...|
|.8.|.9.|.5.|
|29.|73.|68.|
+---+---+---+
|...|...|...|
|.7.|.6.|.9.|
|93.|58.|46.|
+---+---+---+
|...|...|...|
|.6.|.2.|.3.|
|85.|64.|97.|
+---+---+---+     8 boxes @19

Code: Select all
+---+---+---+
|...|...|...|
|.8.|.9.|.5.|
|29.|73.|68.|
+---+---+---+
|...|...|...|
|.7.|.5.|.9.|
|93.|87.|46.|
+---+---+---+
|...|...|...|
|.6.|.8.|.3.|
|85.|46.|97.|
+---+---+---+     not an improvement 7 boxes @19  box 5 @20 box 8 @18 !!
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Re: More Homework 6

Postby blue » Fri Nov 25, 2022 5:11 pm

shye wrote:
Code: Select all
..........7..9..2.24.13.56...........9..2..7.13.58.24...........2..7..9.56.24.13.

8 boxes with a sum of 13, non-minimal
not as impressive as the others but this has some interesting symmetry to it

The solution grid has automorphisms, but not the givens :(

Here are two with symmetry in the givens as well:

Code: Select all
+-------+-------+-------+
| . . . | . . . | . . . |
| . . 1 | . . 3 | . . 4 |
| . 3 9 | . 4 6 | . 2 7 |
+-------+-------+-------+
| . . . | . . . | . . . |
| . . 7 | . . 4 | . . 1 |
| . 1 5 | . 7 2 | . 3 9 |
+-------+-------+-------+
| . . . | . . . | . . . |
| . . 2 | . . 1 | . . 6 |
| . 7 4 | . 3 9 | . 5 2 |   9 x 13, diagonal reflection symmetry, minimal
+-------+-------+-------+

+-------+-------+-------+
| . . . | 7 . 9 | . . . |
| 9 . 1 | . . . | 4 . 6 |
| . . . | 8 . 6 | . . . |
+-------+-------+-------+
| 2 . 4 | . . . | 7 . 9 |
| . . . | 2 . 8 | . . . |
| 1 . 3 | . . . | 6 . 8 |
+-------+-------+-------+
| . . . | 4 . 2 | . . . |
| 4 . 6 | . . . | 9 . 1 |
| . . . | 1 . 3 | . . . |   7 x 10, 180 degree rotatioal symmetry, non-minimal
+-------+-------+-------+

Added: one more, similar to yours ...

Code: Select all
+-------+-------+-------+
| . . . | . . . | . . . |
| . 1 . | . 6 . | . 8 . |
| 9 7 . | 2 5 . | 1 4 . |
+-------+-------+-------+
| . . . | . . . | . . . |
| . 8 . | . 1 . | . 6 . |
| 1 4 . | 3 9 . | 2 5 . |
+-------+-------+-------+
| . . . | . . . | . . . |
| . 6 . | . 8 . | . 1 . |
| 2 5 . | 1 4 . | 7 3 . |   7 x 13, Jumping diagonals (BS) symmetry, non-minimal
+-------+-------+-------+
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Re: More Homework 6

Postby jovi_al01 » Fri Nov 25, 2022 6:20 pm

coloin wrote:
jovi_al01 wrote:...this is the lowest possible, and of course can be relabeled to obtain 19, which is the highest possible.

but I cant achieve all boxes to be 19 replacing the 1 with a 9 however


1 --> 9
2 --> 8
3 --> 7
4 --> 6

should do the job-- we're reflecting over the average sum of three cells (15) so our relabeling must reflect over the average cell value (5)
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Re: More Homework 6

Postby coloin » Fri Nov 25, 2022 6:28 pm

jovi_al01 wrote:1 --> 9
2 --> 8
3 --> 7
4 --> 6
should do the job-- we're reflecting over the average sum of three cells (15) so our relabeling must reflect over the average cell value (5)

I see now ... very good !

Code: Select all
+---+---+---+
|...|...|...|
|.2.|.9.|.8.|
|89.|37.|56.|
+---+---+---+
|...|...|...|
|.3.|.6.|.9.|
|97.|85.|64.|
+---+---+---+
|...|...|...|
|.8.|.4.|.7.|
|56.|78.|93.|
+---+---+---+      9 x 19  - isomorph of the 9 x 11


Good job it wasnt a pattern in the patterns game, despite many many hardish puzzles
only one proper diamond !!
Code: Select all
..........8..9..2.29.67.54...........7..4..8.43.95.71...........2..6..5.64.38.19. ED=7.1/7.1/7.1
..........1..9..3.69.73.54...........7..4..6.23.67.49...........2..6..1.54.92.67. ED=9.2/9.2/3.8
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Re: More Homework 6

Postby JPF » Sat Nov 26, 2022 1:06 pm

Back to pattern 2.

It is easy to see that the common value (p) of the boxes is between 10 to 15.
Here are examples of 7 x (p) for p=10,..,15
Code: Select all
+---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
|...|1.2|...|   |...|6.1|...|   |...|1.6|...|   |...|6.8|...|   |...|1.2|...|   |...|8.5|...|
|6.4|...|1.9|   |5.6|...|8.3|   |9.3|...|5.1|   |9.4|...|5.8|   |5.9|...|6.8|   |7.8|...|9.6|
|...|4.3|...|   |...|7.3|...|   |...|3.2|...|   |...|2.9|...|   |...|5.6|...|   |...|3.7|...|
+---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
|5.7|...|3.2|   |1.5|...|7.2|   |1.4|...|2.7|   |8.2|...|1.5|   |1.3|...|9.4|   |2.5|...|3.4|
|...|2.8|...|   |...|4.7|...|   |...|5.7|...|   |...|8.5|...|   |...|6.8|...|   |...|6.9|...|
|3.8|...|4.1|   |2.3|...|4.8|   |2.5|...|8.9|   |7.5|...|4.3|   |4.6|...|7.5|   |1.7|...|6.2|
+---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
|...|3.6|...|   |...|1.5|...|   |...|4.1|...|   |...|1.6|...|   |...|2.1|...|   |...|9.3|...|
|1.9|...|7.3|   |9.2|...|5.6|   |5.7|...|4.8|   |6.7|...|9.4|   |6.8|...|5.2|   |9.1|...|7.8|
|...|9.1|...|   |...|3.2|...|   |...|2.5|...|   |...|4.2|...|   |...|8.3|...|   |...|2.1|...|
+---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
10              11              12              13              14              15

Can we have more than 7 boxes with the same value?
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Re: More Homework 6

Postby blue » Sat Nov 26, 2022 3:05 pm

JPF wrote:Can we have more than 7 boxes with the same value?

After exhaustive testing, the answer is "no".
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