## More Homework 6

Everything about Sudoku that doesn't fit in one of the other sections

### More Homework 6

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

JPF
JPF
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### Re: More Homework 6

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  . |+---–-----+---------+---------+`

Marek
marek stefanik

Posts: 298
Joined: 05 May 2021

### Re: More Homework 6

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 .`

Mike

m_b_metcalf
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Location: Berlin

### Re: More Homework 6

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 | . . . |+-------+-------+-------+`

JPF
JPF
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Posts: 6017
Joined: 06 December 2005
Location: Paris, France

### Re: More Homework 6

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`

m_b_metcalf
2017 Supporter

Posts: 13322
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Location: Berlin

### Re: More Homework 6

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`

JPF
JPF
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Posts: 6017
Joined: 06 December 2005
Location: Paris, France

### Re: More Homework 6

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`

JPF
JPF
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### Re: More Homework 6

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.`

jovi_al01

Posts: 98
Joined: 26 July 2021

### Re: More Homework 6

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

shye

Posts: 241
Joined: 12 June 2021

### Re: More Homework 6

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 !!`
coloin

Posts: 2283
Joined: 05 May 2005
Location: Devon

### Re: More Homework 6

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+-------+-------+-------+`
blue

Posts: 932
Joined: 11 March 2013

### Re: More Homework 6

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)

jovi_al01

Posts: 98
Joined: 26 July 2021

### Re: More Homework 6

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`
coloin

Posts: 2283
Joined: 05 May 2005
Location: Devon

### Re: More Homework 6

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?
JPF
JPF
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### Re: More Homework 6

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

After exhaustive testing, the answer is "no".
blue

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