Patterns of Prediction

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RogerE
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Re: Patterns of Prediction

Post by RogerE »

Predicting intervals that contain a specified number of special numbers

Where would you find a run of a thousand consecutive integers (also called whole numbers or natural numbers) that contain exactly ten perfect squares? What if you replace "perfect squares" by some other criterion?

There are various ways to go about predicting where to look for such a run (interval). Here's a systematic way that doesn't depend particularly on the nature of perfect squares, but would also work for other kinds of special integers.
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Re: Patterns of Prediction

Post by RogerE »

Predicting intervals having a specified number of special numbers (cont.)


I've been thinking about further predictions along the theme introduced yesterday.
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Screen Shot 2022-07-05 at 1.53.34 am.png
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Footnote: Do you find it surprising that the interval of size one hundred million located for this calculation contains fifty perfect powers, all of which are squares or cubes, but no others that are higher powers? It shows how rapidly far apart consecutive fifth powers grow, not to mention consecutive higher powers.
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Re: Patterns of Prediction

Post by RogerE »

The latest Covid Update for New South Wales:

The increments over last week's numbers give the best sense of current trends.
Screen Shot 2022-07-06 at 3.51.13 pm.png
Changes from 28 June:
Active cases increased by 20515
Hospitalised cases increased by 296
Intensive care cases increased by 24
Ventilated cases increased by 6
Daily deaths decreased by 1
Screen Shot 2022-07-06 at 3.52.30 pm.png
Changes from 28 June:
Cases in last 24 hours increased by 2708
Cases this week increased by 14648
Cases last week increased by 5384
Cases since start of pandemic increased by 75350
Deaths since start of pandemic increased by 79
.
The above two screen shots come from the website:
https://www.health.nsw.gov.au/Infectious/covid-19/Pages/stats-nsw.aspx
.
Since the previous snapshot of 28 June, there have been 79 deaths added to the NSW total. That's an average around 11 deaths due to Covid, per day, in NSW.


Here are the "local numbers" in my neighbourhood of New South Wales [NSW]:
Screen Shot 2022-07-06 at 3.53.33 pm.png
Changes from 28 June:
Cases increased by 876
Daily Tests increased by 1272
Daily Rate per 1000 increased by 7
Screen Shot 2022-07-06 at 3.53.53 pm.png
Changes from 28 June:
Cases increased by 1015
Daily Tests increased by 1130
Daily Rate per 1000 increased by 6
.
I find the visual representation of the data more informative and comprehensive.
The NSW daily and seven-day average case numbers graph:
Screen Shot 2022-07-06 at 3.55.14 pm.png

https://www.google.com/search?q=covid+numbers+in+nsw+today&oq=covid+numbers+in+NSW+today


Deaths from Covid 19 in NSW
Screen Shot 2022-07-06 at 3.56.16 pm.png
https://www.google.com/search?q=Daily+deaths+from+Covid+in+N ... e&ie=UTF-8
.
A snapshot of recent data in this graph:
.
Screen Shot 2022-07-06 at 3.56.45 pm.png
.
________________


Compare the NSW mortality graph with the mortality graph for the whole of Australia:
Whole of Australia — Covid deaths per day, and seven day average
Whole of Australia — Covid deaths per day, and seven day average
https://www.google.com/search?q=Daily+deaths+from+Covid+in+N ... &coasync=0


Some overall data
During the past weekend, the total number of Covid deaths in Australia passed 10,000.
Latest data shows the total at 10,130.
The total number of cases for the whole of Australia has now passed 8.33 million.
The current total population of Australia is 26.07 million.
Consequently, almost 32% of the Australian population has contracted Covid19.


/RogerE :D
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Re: Patterns of Prediction

Post by RogerE »

An extra note about Covid 19 deaths in Australia:

Not only have 10,000 people in Australia died from Covid 19, since 2020,
but
over 7000 of those deaths have occurred in the last six months


Wearing facemasks

Some recent items about using facemasks advocate their use and the benefits of that use.
Early in the pandemic there was strong general advocacy of facemask wearing. In more
recent times many political leaders have minimised that in favour of advocating a return
to full economic activity, while the health professionals have continued to advocate
maskwearing as a simple and effective way of reducing the spread of the virus.

27 June 2022:
Screen Shot 2022-07-06 at 9.11.38 pm.png
https://www.healthdirect.gov.au/covid19-restriction-checker/face-masks/nsw
.
18 Jan 2022:
Screen Shot 2022-07-06 at 9.25.44 pm.png
https://www.abc.net.au/news/2022-01-18/dr-norman-swan-recomm ... s/13709034
.
6 July 2022:
Screen Shot 2022-07-06 at 9.37.59 pm.png
https://www.sbs.com.au/news/article/if-you-can-wear-a-mask-d ... /sflrdlk91
.
/RogerE :D
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Re: Patterns of Prediction

Post by RogerE »

Sign of the Covid times:
Today's news is that New South Wales recorded 22 deaths from Covid 19 in the last 24hrs.

/RogerE
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Re: Patterns of Prediction

Post by RogerE »

Predicting intervals having a specified number of special numbers (cont.)


Let's apply to prime numbers the earlier ideas about predictions relating to special integers in an interval.
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Screen Shot 2022-07-12 at 12.06.21 pm.png
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Footnote: Even though the detailed local distribution of primes is notoriously "irregular", the "average" distribution is smooth enough that the derivative method, combined with "sliding", leads to very satisfactory predictions.
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Re: Patterns of Prediction

Post by RogerE »

In another thread, I asked the small question: How many palindromes are there between 20,000 and 30,000?

Refer to the very last sentences in the post:
https://www.stampboards.com/viewtopic.php?t=314&start=7180
(The answer turns out to be 100. Some subsequent posts are also relevant.)

I pose this follow up question (without an immediate footnote):

How many palidromes between 20,000 and 30,000 are multiples of 3?

/RogerE :D
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Re: Patterns of Prediction

Post by lesbootman »

33 perhaps?

@int(100/3)
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Re: Patterns of Prediction

Post by RogerE »

Thanks lesbootman. Dividing the total by 3 and then rounding to the nearest integer (or rounding down) can be expected to give a good estimate, because numbers with the required property are rather evenly spaced, and the probability that a random palindrome is a multiple of 3 seems likely to be 1/3. But that's a "heuristic" argument. Can we be sure it gives the exact answer?

It's often a good idea to test a proposed method on a smaller example. Consider this run of consecutive 3-digit palindromes:
202, 212, 222, 232, 242, 252, 262
The multiples of 3 are shown in colour.

From this example,
The interval [200, 270] contains 7 palindromes, of which 2 are multiples of 3
The interval [200, 260] contains 6 palindromes, of which 2 are multiples of 3
The interval [210, 260] contains 5 palindromes, of which 2 are multiples of 3
The interval [200, 250] contains 5 palindromes, of which only 1 is a multiple of 3
All but the third instance give correct count using "dividing by 3 and rounding down".
However, the third instance requires "dividing by 3 and rounding up" to get the correct answer.

Conclusion: The "dividing by 3 and rounding down" method gives a good estimate, but not necessarily the exact answer...

/RogerE :D
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Re: Patterns of Prediction

Post by lesbootman »

I was fairly sure that 33 was the correct answer. Not being an advanced mathematical theorist, I had set up a spreadsheet and tested my very basic theory that way. X+100 (X+110 every 10th step) and Y/3 aren't too challenging to copy and paste.
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Re: Patterns of Prediction

Post by RogerE »

The latest Covid Update for New South Wales:

The increments over last week's numbers give the best sense of current trends.
Screen Shot 2022-07-12 at 11.46.04 pm.png
Changes from 5 July:
Active cases increased by 12022
Hospitalised cases increased by 227
Intensive care cases decreased by 6
Ventilated cases increased by 1
Daily deaths increased by 10
Screen Shot 2022-07-13 at 12.29.05 am.png
Changes from 5 July:
Cases in last 24 hours decreased by 2969
Cases this week increased by 1894
Cases last week increased by 12664
Cases since start of pandemic increased by 64314
Deaths since start of pandemic increased by 158
.

The above two screen shots come from the website:
https://www.health.nsw.gov.au/Infectious/covid-19/Pages/stats-nsw.aspx
.


Since the previous snapshot of 5 July, there have been 158 deaths added to the NSW total. That's an average around 22 deaths due to Covid, per day, in NSW.


Here are the "local numbers" in my neighbourhood of New South Wales [NSW]:
Screen Shot 2022-07-12 at 11.47.26 pm.png
Changes from 5 July:
Cases increased by 931
Daily Tests increased by 875
Daily Rate per 1000 increased by 6
Screen Shot 2022-07-12 at 11.48.42 pm.png
Changes from 5 July:
Cases increased by 930
Daily Tests increased by 1107
Daily Rate per 1000 increased by 5
.
I find the visual representation of the data more informative and comprehensive.
The NSW daily and seven-day average case numbers graph:
New South Wales daily new cases, and 7 day average
New South Wales daily new cases, and 7 day average

https://www.google.com/search?q=covid+numbers+in+nsw+today&oq=covid+numbers+in+NSW+today


Deaths from Covid 19 in NSW
New South Wales — Covid deaths per day, and 7 day average
New South Wales — Covid deaths per day, and 7 day average
https://www.google.com/search?q=Daily+deaths+from+Covid+in+N ... e&ie=UTF-8
.
A snapshot of recent data in this graph:
.
Screen Shot 2022-07-13 at 12.01.56 am.png
.
________________


Compare the NSW mortality graph with the mortality graph for the whole of Australia:
Whole of Australia — Covid deaths per day, and 7 day average
Whole of Australia — Covid deaths per day, and 7 day average
https://www.google.com/search?q=Daily+deaths+from+Covid+in+N ... &coasync=0



/RogerE :D
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Re: Patterns of Prediction

Post by RogerE »

lesbootman wrote: 13 Jul 2022 01:12 I was fairly sure that 33 was the correct answer. Not being an advanced mathematical theorist, I had set up a spreadsheet and tested my very basic theory that way. X+100 (X+110 every 10th step) and Y/3 aren't too challenging to copy and paste.
Well done lesbootman.

Here's the calculation I used. It confirms your result. :D

Screen Shot 2022-07-13 at 1.06.53 am.png
Incidentally, the smallest and largest palindromes in the interval that are multiples of 3 are 20202 and 29892.

/RogerE :D
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Re: Patterns of Prediction

Post by RogerE »

Predicting intervals having a specified number of special numbers (cont.)


After looking at the number of primes in an interval, let's now look at the number of pairs of twin primes in an interval. For instance, below 10 there are 8 twin prime pairs:
(3,5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), (149, 151), (179, 181), (191, 193), (197, 199)
The fine scale distribution is notoriously irregular, but like the primes themselves, the large scale distribution of twin prime pairs seems to "behave". The status of the distribution, and the status of the assumption that there are infinitely many such pairs, are still essentially conjectural. (Progress is being made in both these matters.)
Let's proceed with application of a likely distribution function for twin prime pairs — real applications are possible!

Screen Shot 2022-07-14 at 1.07.24 pm.png
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Re: Patterns of Prediction

Post by RogerE »

Typo! Previous post: The list of twin primes covers the pairs below 200, not 10.

Another instance of Muphry's Law ;)
https://www.stampboards.com/viewtopic.php?p=7000314&hilit=Peter+Principle#p7000314


Adding this correction here gives me the opportunity to show that the approximation to the twin prime pair counting function used in the previous post does remarkably well for predicting the number of pairs below 200:

Screen Shot 2022-07-14 at 5.16.41 pm.png
My footnote in the previous post also has a Muphry's Law error: it should remark on the unexpectedly high number of twin prime pairs between 87,500 and 88,000 (but I had a concentration slip. and stated "between 87,000 and 87,500").

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Re: Patterns of Prediction

Post by RogerE »

Predicting intervals having a specified number of special numbers (cont.)


I've been thinking more about counting perfect powers.
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Screen Shot 2022-07-17 at 11.30.29 am.png
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Footnote: Once again, the large spacing between consecutive higher powers means that they scarcely figure in an interval of size one hundred million in the neighbourhood of ten billion (US terminology).
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Re: Patterns of Prediction

Post by RogerE »

Predicting intervals having a specified number of special numbers (cont.)


Here's another instalment about counting perfect powers.
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Screen Shot 2022-07-18 at 1.06.35 am.png
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Re: Patterns of Prediction

Post by RogerE »

The latest Covid Update for New South Wales:

The increments over last week's numbers give the best sense of current trends.
.
Screen Shot 2022-07-21 at 3.22.58 am.png
Changes from 11 July:
Active cases increased by 7179
Hospitalised cases increased by 187
Intensive care cases increased by 5
Ventilated cases decreased by 3
Daily deaths increased by 10
Screen Shot 2022-07-21 at 3.23.53 am.png
Changes from 11 July:
Cases in last 24 hoursincreased by 4546
Cases this week increased by 8389
Cases last week increased by 1230
Cases since start of pandemic increased by 96650
Deaths since start of pandemic increased by 104
.
The above two screen shots come from the website:
https://www.health.nsw.gov.au/Infectious/covid-19/Pages/stats-nsw.aspx
.
Since the previous snapshot of 11 July, there have been 104 deaths added to the NSW total. That's an average around 11 deaths due to Covid, per day, in NSW.


Here are the "local numbers" in my neighbourhood of New South Wales [NSW]:
Screen Shot 2022-07-21 at 3.33.28 am.png
Changes from 11 July:
Cases increased by 970
Daily Tests increased by 372
Daily Rate per 1000 increased by 2
Screen Shot 2022-07-21 at 3.33.54 am.png
Changes from 11 July:
Cases increased by 910
Daily Tests decreased by 386
Daily Rate per 1000 decreased by 2
.
I find the visual representation of the data more informative and comprehensive.
The NSW daily and seven-day average case numbers graph:
Screen Shot 2022-07-21 at 3.26.07 am.png
The current wave is predicted to continue to increase over the next few weeks.
https://www.google.com/search?q=covid+numbers+in+nsw+today&oq=covid+numbers+in+NSW+today


Deaths from Covid 19 in NSW

Screen Shot 2022-07-21 at 3.27.23 am.png
https://www.google.com/search?q=Daily+deaths+from+Covid+in+N ... e&ie=UTF-8
.
A snapshot of recent data in this graph:
.
Screen Shot 2022-07-21 at 3.28.18 am.png
.
________________


Compare the NSW mortality graph with the mortality graph for the whole of Australia:
Screen Shot 2022-07-21 at 3.31.54 am.png
https://www.google.com/search?q=Daily+deaths+from+Covid+in+N ... &coasync=0


/RogerE :D
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Re: Patterns of Prediction

Post by RogerE »

A serious footnote to the previous post:

Yesterday, to 4pm, there were 25 deaths due to Covid 19 in New South Wales.
Screen Shot 2022-07-21 at 11.41.17 am.png
https://www.health.nsw.gov.au/Infectious/covid-19/Pages/stats-nsw.aspx
/RogerE
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Re: Patterns of Prediction

Post by RogerE »

The Uncertainty Principle

Great Britain, 1990: Greetings, 20p Cheshire cat (SG 1486)<br />[The Queen's profile and the face value have almost disappeared]
Great Britain, 1990: Greetings, 20p Cheshire cat (SG 1486)
[The Queen's profile and the face value have almost disappeared]
.
In another thread, almost accidentally, the Cheshire cat and Schrödinger's cat have briefly featured,
most recently at
https://www.stampboards.com/viewtopic.php?p=8904355#p8904355
______________________
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Predictability
.
We like the idea of predictability and certainty. It is reassuring. Its absence or denial are unsettling. Even thinking about unpredictability meets with a high level of resistance.

Politicians are urged to make definitive commitments to give "the market" certainty about what to expect, so investments in infrastructure can be made confidently, with strong expectation of gains in productivity and dividends.

We wish for predictability and certainty in managing the trajectory of the Covid 19 pandemic by adjusting health policy and its implementation. We wish for predictability and certainty in managing global climate change. And in so many other areas of consequence for us all, we wish for predictability and certainty.

But at a deep level it turns out that our very understanding of reality is fundamentally unable to support predictability and certainty.

In another thread I recently quoted from a Q&A article by Eric Powell interviewing quantum physicist Anton Zeilinger in Discover, July-Aug 2011, pp78-84. Zeilinger remarked:

In the history of physics we have always found something deeper.
That deeper thing is usually more counterintuitive than what we had before
and takes awhile to get used to.

https://www.stampboards.com/viewtopic.php?t=96204&start=106
.
That is precisely the case with Heisenberg's deep insight into physical reality.


The Heisenberg Uncertainty Principle

Screen Shot 2022-07-23 at 11.47.32 am.png
https://en.wikipedia.org/wiki/Uncertainty_principle
.
Not just an observer limitation


From the above Wikipedia article:
Screen Shot 2022-07-23 at 1.13.07 pm.png
.
Practical consequences

Screen Shot 2022-07-23 at 1.13.51 pm.png
.
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"Curiouser and curiouser!" cried Alice (she was so much surprised,
that for the moment she quite forgot how to speak good English).

.
/RogerE :D
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Re: Patterns of Prediction

Post by RogerE »

Today's Covid 19 news for Australia:
.
102 deaths from Covid 19 in the last 24hrs
Cumulative total number of deaths from Covid 19 in Australia has now passed 11,000
— in fact, it's now at 11,134
.
The numbers are accelerating, that is, a positive rate of change in the average rate of change.

/RogerE
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Re: Patterns of Prediction

Post by RogerE »

Sudoku revisited.


Back in Nov—Dec 2021, this thread included a sequence of posts about sudoku puzzles.
Some of those posts can be found at, and following after, this link:
https://www.stampboards.com/viewtopic.php?p=7730805#top
.
________________________
.
Without going over that territory again, I would like to discuss a sudoku puzzle that I've been "working".
Here's the starting grid:

Grid A
Grid A
.
After 20 "steps" I produced the following grid.
Invitation 1: check whether you agree with this grid.

Grid B
Grid B


Here is a "near-solution" to grid B.
It is not a solution, because to two cells marked X have no admissible entries.
Invitation 2: check that no repeated entry occurs in any row, any column, or any "field" (3x3 square), and that grid C is missing one entry 6 and one entry 7, but neither can be placed in a cell marked X.
Grid C
Grid C
.
Invitation 3: Prove, or disprove, that if a partially completed sudoku grid has no admissible further entry, then at least two cells are empty and unable to be filled.
Invitation 4: Find a "forced" next step after grid B, and explain why it is forced.

I assure readers that there is a valid completion to grid B, but I will leave it to others to find and show it.

/RogerE :D
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Re: Patterns of Prediction

Post by capetriangle »

RogerE

I completed the puzzle fairly easily.

It only required one guess - in the fourth cell with the numbers 4 and 5 in positions 8 and 9 of the cell respectively. I apparently guessed correctly as it led to the solution.

Your Grid C is riddled with mistakes.

Kindest regards

Richard
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Re: Patterns of Prediction

Post by RogerE »

Hello Richard = capetriangle. Thanks for tackling that sudoku.
capetriangle wrote:I completed the puzzle fairly easily.

It only required one guess - in the fourth cell with the numbers 4 and 5 in positions 8 and 9 of the cell respectively. I apparently guessed correctly as it led to the solution.
• Richard, can you (or others) show an argument for a forced move to follow from the position in Grid B?
I am really interested to see reasoning for the next move, not simply a statement of what the next move is.

• I can't understand your "numbers 4 and 5 in positions 8 and 9 of the cell". Could you please clarify, using the following
Terminology: A "cell" is any one of the 81 components of the grid; a "row" for a horizontal line of 9 cells; a "column" for a vertical line of 9 cells; and a "field" (or "cage") is any one of the nine 3x3 special groups of cells.
The earlier discussions of sudoku puzzles (Nov-Dec 2021) in this thread used several useful notations (depending on the person posting) to explain moves. It would be helpful if you would use one of those notations so we could have a clear understanding of your reasoning. For instance, one notation is used to compactly describe a sequence of steps ("moves" = reasoning plus resulting entry) in
https://www.stampboards.com/viewtopic.php?t=93774&start=270


• The point of Grid C was to display an almost complete continuation from Grid B which only runs into an impasse with the last two unfilled cells.
capetriangle wrote:Your Grid C is riddled with mistakes.
I think your classification of an entry in a particular cell as a "mistake" means you have reasoning which shows a different entry is forced for that cell. That brings me back to a request for reasoning to determine just the next move from Grid B.

Thanks for any reasoning you might share with us.

/RogerE :D
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Re: Patterns of Prediction

Post by capetriangle »

RogerE

• I can't understand your "numbers 4 and 5 in positions 8 and 9 of the cell". Could you please clarify, using the following

My solution was:

261 378 549
739 514 862
458 692 137

196 247 358
327 851 496
845 963 721

572 489 613
914 736 285
683 125 974

So in row 6, the figures 4 and 5 are both exclusive choices for the second and third cells of that row.

In the second cell of row 6 I first attempted to solve using the figure 4 and it worked to completion. Had it not I would have tried the figure 5.

Hope the above helps.

Kindest regards

Richard
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Re: Patterns of Prediction

Post by RogerE »

Thanks Richard = capetriangle. Yes, I now understand.
May I paraphrase?
You noticed that with Grid B as starting configuration,
rows d and e, and column 1, each contain an entry 4 and an entry 5.
Therefore, in field/cage D, the entries 4 and 5 must be in cells f2 and f3, in some order.
You decided to try 4 in f2 and 5 in f3.
You were able to complete the sudoku puzzle from there,
so you were happy, and you concluded that was "the" solution.
I think you also concluded implicitly that assigning 5 to f2 and 4 to f3
would not have led to a solution.


I confirm that the solution I found is the same as the solution that you found.
Like you, I found an open cell in Grid B that has just two possible entries, and I tested what would follow from each of the two possible choices. One of the choices led to the "almost solution" in Grid C; the other led to the full solution we each found. Because I followed the other option as far as the "almost solution", I concluded that there is just one solution.

Open question
The method of solution Richard = capetriangle used for Grid B was a "branch point" strategy: find a cell (such as f2) where there are just two options, and test which option leads to a full solution. (Once f2 is filled, the entry in f3 is forced.) My method was also a "branch point" strategy, though I began with a different cell. (I think I used g2, where 6 and 7 are the only possibilities.)
My question/invitation to fellow Stampboarders: can you find a forced move to continue Grid B? If so, tell us the reasoning. Failing that, must we conclude that a "branch point" strategy is essentially the only way to proceed?

Comment: My preference is for a sudoku puzzle to have a solution that has at least one forced entry at every step. It is clear that some published sudoku puzzles don't have that property, because it turns out that they have more than one solution. My interest here is in the question: if a given sudoku puzzle has a unique solution, must there be a way to avoid using a "branch point" strategy to reach that solution?

/RogerE :D
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Re: Patterns of Prediction

Post by satsuma »

RogerE wrote: 25 Jul 2022 00:46 Sudoku revisited.
After 20 "steps" I produced the following grid.
Invitation 1: check whether you agree with this grid.

Image
/RogerE :D
Hi Roger
I just saw you are looking at sudoku again.
I agree with your above grid.
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Re: Patterns of Prediction

Post by capetriangle »

RogerE wrote: 26 Jul 2022 12:44 Thanks Richard = capetriangle. Yes, I now understand.
May I paraphrase?
You noticed that with Grid B as starting configuration,
rows d and e, and column 1, each contain an entry 4 and an entry 5.
Therefore, in field/cage D, the entries 4 and 5 must be in cells f2 and f3, in some order.
You decided to try 4 in f2 and 5 in f3.
You were able to complete the sudoku puzzle from there,
so you were happy, and you concluded that was "the" solution.
I think you also concluded implicitly that assigning 5 to f2 and 4 to f3
would not have led to a solution.


That is exactly what I did.


Open question
The method of solution Richard = capetriangle used for Grid B was a "branch point" strategy: find a cell (such as f2) where there are just two options, and test which option leads to a full solution. (Once f2 is filled, the entry in f3 is forced.) My method was also a "branch point" strategy, though I began with a different cell. (I think I used g2, where 6 and 7 are the only possibilities.)

I also confirm G2 and G7 as similar "branch points" for the numbers 6 and 7.

My question/invitation to fellow Stampboarders: can you find a forced move to continue Grid B? If so, tell us the reasoning. Failing that, must we conclude that a "branch point" strategy is essentially the only way to proceed?

I believe that is the solution.

Comment: My preference is for a sudoku puzzle to have a solution that has at least one forced entry at every step. It is clear that some published sudoku puzzles don't have that property, because it turns out that they have more than one solution.

At one time I thought that each Sudoku had to have a unique solution but I did, on at least one occasion, find a puzzle with two solutions.

My interest here is in the question: if a given sudoku puzzle has a unique solution, must there be a way to avoid using a "branch point" strategy to reach that solution?

/RogerE :D
My comments in red.

Richard
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Re: Patterns of Prediction

Post by satsuma »

RogerE wrote: 26 Jul 2022 12:44
Open question
The method of solution Richard = capetriangle used for Grid B was a "branch point" strategy: find a cell (such as f2) where there are just two options, and test which option leads to a full solution.

/RogerE :D
I'm not sure whether this statement is implicitly true. I can envisage a scenario where a second branch point is reached before a full solution can be attained.

I have encountered such puzzles, although I accept it may have been due to my not finding a forced move which was available.
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Re: Patterns of Prediction

Post by satsuma »

[quote=RogerE


The point of Grid C was to display an almost complete continuation from Grid B which only runs into an impasse with the last two unfilled cells.

/RogerE :D
[/quote]

Again I'm not sure if this statement is correct. It may seem that way following the sequence you used to get there but you might like to check grid b again.

It follows that cells F8 and F9 are limited to 5 and 6.

Therefore as soon as you needed to place H9 as 6 you were at an impasse, if not before
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Re: Patterns of Prediction

Post by satsuma »

I've spent some time thinking about this puzzle.
From Grid B

Image

You can get to this position which I will call grid k
possibilities filled in and winnowed
possibilities filled in and winnowed
You will notice there are 7 pairs (F1F2, B6C6, A6G6, B7G7, F8F9, C4I4, and G7H9) plus several triples. I believe I have investigated all of them to either the solution or a failure to solve.
Capetriangle was fortunate in his choice of an either or pair leading to a solution.

I believe this puzzle was constructed to not have a logical solution but rather an elimination solution.


Interestingly the simplest method I found to arrive at a successful starting point did not come from a testing of any of these pairs, but from cell B1.

Consider Columns ABC
col abc
col abc

If
1 B1=8, then
2 C3=5, then
3 B3=3, then
4 B2=7, then
5 B7=6, then
6 C2=9, then
7 C6=4, then
8 C8=8, then
9 C4=6, then
10 C5=7, then
11 and A6=8.

which brings us to this:
grid L
grid L
Now if you consider row 6; if A6 = 8, then G6 = 7,
and if you consider row 7; if B7 = 6 then G7 = 7.

As both G6 and G7 cannot equal 7 the starting value of B1 = 8 is incorrect. B1 must equal 6 allowing the puzzle to be solved.
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Re: Patterns of Prediction

Post by RogerE »

The latest Covid Update for New South Wales:

I'm running several days behind my intended update timing this week, but I will still show the increments over the previous weekly post, which had data for 19 July 2022.
The increments over last week's numbers give the best sense of current trends.
We see increases across all measures this week.
.
Screen Shot 2022-07-28 at 10.14.55 pm.png
Changes from 19 July:
Active cases increased by 27307
Hospitalised cases increased by 46
Intensive care cases increased by 2
Ventilated cases increased by 2
Daily deaths increased by 14
Screen Shot 2022-07-28 at 10.15.27 pm.png
Changes from 19 July:
Cases in last 24 hoursincreased by 352
Cases this week increased by 13300
Cases last week increased by 14496
Cases since start of pandemic increased by 116611
Deaths since start of pandemic increased by 186
.
The above two screen shots come from the website:
https://www.health.nsw.gov.au/Infectious/covid-19/Pages/stats-nsw.aspx
.
Since the previous snapshot of 19 July, there have been 186 deaths added to the NSW total. That's an average around 21 deaths due to Covid, per day, in NSW.


Here are the "local numbers" in my neighbourhood of New South Wales [NSW]:
Screen Shot 2022-07-28 at 10.16.22 pm.png
Changes from 19 July:
Cases increased by 822
Daily Tests increased by 958
Daily Rate per 1000 increased by 6
Screen Shot 2022-07-28 at 10.16.44 pm.png
Changes from 19 July:
Cases increased by 1169
Daily Tests increased by 1740
Daily Rate per 1000 increased by 9
.
I find the visual representation of the data more informative and comprehensive.
The NSW daily and seven-day average case numbers graph:
Screen Shot 2022-07-28 at 10.19.25 pm.png
The current wave is predicted to continue to increase for a few weeks.
https://www.google.com/search?q=covid+numbers+in+nsw+today&oq=covid+numbers+in+NSW+today


Deaths from Covid 19 in NSW

Screen Shot 2022-07-28 at 10.21.18 pm.png
https://www.google.com/search?q=Daily+deaths+from+Covid+in+N ... e&ie=UTF-8
.
A snapshot of recent data in this graph:
.
Screen Shot 2022-07-28 at 10.48.07 pm.png
.
________________


Compare the NSW mortality graph with the mortality graph for the whole of Australia:
Screen Shot 2022-07-28 at 10.50.56 pm.png
https://www.google.com/search?q=Daily+deaths+from+Covid+in+N ... &coasync=0


A snapshot of data for the whole of Australia for 27 July 2022.
This is the highest number of deaths recorded in any one day in Australia.
.
Screen Shot 2022-07-28 at 11.47.17 pm.png


/RogerE
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Re: Patterns of Prediction

Post by satsuma »

Hi RogerE
I believe I have discovered a counter-intuitive, but logical proof; assisted by already knowing the answer.

Here is your Grid B:
grid B
grid B
.

Step 1: Add the possible 7s
Possible 7s added
Possible 7s added
Because only one of G6 and G7 can be a seven, one of A6 and B7 must be a 7.

Case 1, A6 = 7
case 1
case 1
The yellow highlit placements follow logically.

Case 2, B7 = 7
case 2
case 2
The green highlit placements follow logically.

This gives us the information that B2 cannot be 7.

Now lets fill in the top left cage with the possible values:
cage 1
cage 1
As cells B2, B3 and C3 are a triple of values 3, 5 and 8; therefore cell B1 = 6; and as B7 can only be 6 or 7, case 2 with B7 = 7 is the correct alternative.

The rest of the puzzle can then be solved.
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Re: Patterns of Prediction

Post by RogerE »

Hello satsuma and Richard =capetriangle.
Thanks to both of you for your participation with me in my latest sudoku discussion.

Terminology
I'm pleased that we seem to have consistency in terminology about the sudoku grid:


cell — the smallest squares in a sudoku grid; the whole grid comprises 81 cells, in a 9x9 array.
row — a horizontal line of 9 cells in a sudoku grid; the full sudoku grid comprises 9 rows.
column — a vertical line of 9 cells in a sudoku grid; the full sudoku grid comprises 9 columns.
cage — a special 3x3 array of 9 cells in a sudoku grid; the full sudoku grid is partitioned into 9 cages.(*)

(*) Originally I wanted to use square, and later field, for the special 3x3 arrays of 9 cells making up the sudoku grid. Eventually, following satsuma's lead, I adopted the term cage. Evidently that term is widely used in connection with sudoku puzzles.

Notation
However, the labelling of the cells, rows, columns and cages has not been uniformly practised in our posts.
Here is the notation I proposed when we previously discussed sudoku puzzles:

Screen Shot 2021-12-07 at 5.53.35 pm.png
https://www.stampboards.com/viewtopic.php?t=93774&start=269


Here's how that notation was proposed:
RogerE wrote: 07 Dec 2021 20:59 ...

Opportunity for sudoku
.
Recording the moves in a "game" of sudoku is a practice which has not become popular and widespread. Typically the only assistance offered to players is the publication of the full configuration. The implication is that players can "go figure" how to arrive at that solution — they are offered no insight into a sequence of moves that would reach the solution.

Many will be familiar with the feeling of frustration that occurs when a mistake becomes evident in the course of an attempted solution to a sudoku, but it's almost impossible to retrace the moves to identify where the error actually occurred.

I suggest that an efficient notation for recording sudoku moves would be a great step forward. It would allow the "solution grid" to be accompanied by a list of moves for reaching that solution, so "players" could learn from working through that sequence (especially if they were not able to complete the solution independently).
Furthermore, recording one's moves provides a way of retracing the moves if a contradiction is encountered, so the source of the error can be identified and understood. In summary, notation recording a sequence of sudoku moves would serve as a learning tool.

Suggested notation for sudoku
.
Learning from the precedents of chess notation, and the grid labelling in some recent posts in this thread, I would like to suggest the following notation as an experiment for others to try out.
.
Image
.
Comments:
• It is helpful to use cells as the name of the 81 smallest elements of the sudoku grid, and squares as the name of the nine (intermediate sized) 3x3 sets of cells. The horizontal lines of 9 cells are rows, and the vertical lines of 9 cells are columns. The symbols to be inserted in the cells are the entries.
• The lower case letters a–j serve to label the rows, while the upper case letters A–J serve to label the squares. The numbers 1–9 serve to label the columns, and also the entries. The notation distinguishes between numbers as column labels and numbers as entries — illustrated by the same solution below.
• The use of J rather than I is a standard way of avoiding confusion between the letter I and the numeral 1.
...
/RogerE :D
By contrast, although satsuma also numbered the columns, he labelled the rows with capital [upper case] letters (and did not assign notation to the cages) in his post
https://www.stampboards.com/viewtopic.php?t=93774&start=259
It seems to me that lower case letters for rows, and upper case letters for cages, has a more natural association — lower case for the one-dimensional sets, upper case for the two-dimensional sets.

Using the notation in practice
.
Specific cells are "addressed" by their row first, column second — following the standard mathematical practice for matrices, and the geographical practice for points on a map. Thus, the top left corner cell of a sudoku grid is a1, the top right corner cell is a9, the bottom left corner cell is j1, and the bottom right corner cell is j9. It's important not to use capital letters for the rows, because capital letters indicate cages. For example, the middle cage in a sudoku grid is E; row e intersects cage E in the cells e4, e5 and e6; the central cell of the whole sudoku grid is e5.

[Most recently, satsuma has been using capital letters for the columns, with numbers for the rows, and addressing cells by their column first, row second. In that notation, the top left corner cell of a sudoku grid is A1, the top right corner cell is J1, the bottom left corner cell is A9, and the bottom right corner cell is J9. That's not the same as his notation in the post
https://www.stampboards.com/viewtopic.php?t=93774&start=259

When there is no introductory explanation of what notation is being used, it takes time and effort to decipher. The chess world has had several quite distinct notations for recording moves and discussing strategies, and that has required serious players to master more than one notation. However, at least within particular large sectors of the chess world, a single notation is the standard followed in publications and records.]


Using the notation to follow satsuma's reasoning
.
For example, consider our recently discussed Grid B:

Grid B
Grid B
Just four cells in Grid B contain the entry 7: they are a5, c9, d6 and h4.
In his latest discussion, satsuma has added to Grid B the possible locations where open cells could contain 7. He identifies the cells
b1, b2, b3
e3, e8
f1, f7
g2, g7
j1, j2, j8
As a check, notice that the letters a, c, d, h are absent, and the numbers 4, 5, 6, 9 are absent.
Equivalently, the five remaining entries 7 must be in rows b, e, f, g, j and in columns 1, 2, 3, 7, 8.
However, why have cells f2 and f3 not been included among the possible locations for 7?

I agree with satsuma's case 1: if f1 contains 7, then 7 must be in b3, e8, f1, g7, j2.
I also agree with case 2: if g2 contains 7, then 7 must be in b1, e3, f7, g2, j8.
Thus, satsuma's discussion (with my two added sites) shows the following possible solutions (monochromatic sets) for cells containing 7 :
b1, b2, b3
e3, e8
f1, f2, f3, f7
g2, g7
j1, j2, j8
Case 1: If 7 is in any one of f1, f2, f3, then row f —> 7 is not in f7, then column 7 —> 7 is in g7, —> 7 is not in g2.
Case 2: If 7 is in g2, then row g —> 7 is not in g7, then cage J —> 7 is in j8, then cage F —> 7 is in f7, —> 7 is not in any one of f1, f2, f3.
Case 3: If we are not in Case 1 or Case 2, then row f —> 7 is in f7; row g —> 7 is in g7: contradiction in column 7.
Hence either Case 1 or Case 2 must apply (assuming the sudoku puzzle has a solution!), so b2 and j1 do not hold 7.
Note that row g requires the cell g2 to contain a number in the set {6, 7}.
From here, satsuma cleverly observes that entries in the cells b2, c2 and c2 together have to comprise the set {2, 3, 8}. Then 8 cannot be in cell a2, so row a, column 2 and cage A require 6 to be in a2.
Then 6 in a2 —> g2 holds 7.
"The rest of the puzzle can then be solved" says satsuma.

A clever piece of reasoning! The fact that Case 1 and Case 2 both have to be explored, and Case 3 briefly used to show that those first two cases are the only viable possibilities, still demonstrates a branch point in this line of reasoning, but the two branches do not have to be pursued so far that one leads to a contradiction; rather, they combine to reach as point where one of them is shown to be "correct".

Thanks satsuma, very interesting indeed!

/RogerE :D
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Re: Patterns of Prediction

Post by RogerE »

Here is a follow-up discussion about sudoku puzzles, including proofs of some formal propositions about the puzzles as a class.

Screen Shot 2022-08-01 at 2.27.13 am.png
Screen Shot 2022-08-01 at 2.45.16 am.png
.
These results are original, so they should not be reported or used without attribution.

/RogerE :D
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Re: Patterns of Prediction

Post by satsuma »

Some interesting work, RogerE.

However, I'm having a little bit of difficulty understanding the utility of the proofs.

Let's say a puzzle solver has filled in much of a particular puzzle and is unable to proceed without guessing.

I'm imagining at that time, the consideration of proofs to classify the puzzle as solvable or not, are investigated.

My difficulty is that being able to prove that the puzzle is without solution, or with manifold solutions, at that time; proves nothing about whether the puzzle as originally set had a unique solution.

The potential to have already entered the value of a cell incorrectly negates this.

Do you see that differently?
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Re: Patterns of Prediction

Post by RogerE »

The latest Covid Update for New South Wales:

This week, we see increases across all measures but one.
.
Screen Shot 2022-08-03 at 5.05.33 pm.png
Changes from 27 July:
Active cases increased by 70
Hospitalised cases increased by 6
Intensive care cases increased by 2
Ventilated cases increased by 4
Daily deaths increased by 5
Screen Shot 2022-08-03 at 5.05.48 pm.png
Changes from 27 July:
Cases in last 24 hoursincreased by 944
Cases this week decrease by 8596
Cases last week increased by 11726
Cases since start of pandemic increased by 74962
Deaths since start of pandemic increased by 166
.
The above two screen shots come from the website:
https://www.health.nsw.gov.au/Infectious/covid-19/Pages/stats-nsw.aspx
.
Since the previous snapshot of 27 July, there have been 166 deaths added to the NSW total. That's an average around 23 deaths due to Covid, per day, in NSW.


Here are the "local numbers" in my neighbourhood of New South Wales [NSW]:
Screen Shot 2022-08-03 at 5.06.34 pm.png
Changes from 27 July:
Cases increased by 11
Daily Tests increased by 466
Daily Rate per 1000 increased by 2
Screen Shot 2022-08-03 at 5.06.55 pm.png
Changes from 27 July:
Cases increased by 438
Daily Tests increased by 1236
Daily Rate per 1000 increased by 6
.
I find the visual representation of the data more informative and comprehensive.
The NSW daily and seven-day average case numbers graph:
Screen Shot 2022-08-03 at 5.09.01 pm.png
The current wave is predicted to continue to increase for a few weeks.
https://www.google.com/search?q=covid+numbers+in+nsw+today&oq=covid+numbers+in+NSW+today


Deaths from Covid 19 in NSW

Screen Shot 2022-08-03 at 5.09.57 pm.png
https://www.google.com/search?q=Daily+deaths+from+Covid+in+N ... e&ie=UTF-8
.
________________


Compare the NSW mortality graph with the mortality graph for the whole of Australia:
Screen Shot 2022-08-03 at 5.10.46 pm.png
https://www.google.com/search?q=Daily+deaths+from+Covid+in+N ... &coasync=0



/RogerE
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Re: Patterns of Prediction

Post by RogerE »

The latest Covid Update for New South Wales:

This week, chief medical commentary reassures us that Australia has passed the peak in the current wave of new Covid19 infections. The current data confirm this.
.
Screen Shot 2022-08-10 at 11.56.24 pm.png
Changes from 2 August:
Active cases decreased by 20937
Hospitalised cases decreased by 76
Intensive care cases decreased by 12
Ventilated cases decreased by 6
Daily deaths decreased by 9
Screen Shot 2022-08-10 at 11.56.56 pm.png
Changes from 2 August:
Cases in last 24 hours decreased by 5292
Cases this week decreased by 13321
Cases last week decreased by 9533
Cases since start of pandemic increased by 78958
Deaths since start of pandemic increased by 166
.
The above two screen shots come from the website:
https://www.health.nsw.gov.au/Infectious/covid-19/Pages/stats-nsw.aspx
.
Since the previous snapshot of 2 August, there have been 166 deaths added to the NSW total. That's an average around 23 deaths due to Covid, per day, in NSW.


Here are the "local numbers" in my neighbourhood of New South Wales [NSW]:
Screen Shot 2022-08-10 at 11.59.32 pm.png
Changes from 2 August:
Cases decreased by 24
Daily Tests increased by 936
Daily Rate per 1000 increased by 6
Screen Shot 2022-08-11 at 12.00.01 am.png
Changes from 2 August:
Cases increased by 345
Daily Tests increased by 1559
Daily Rate per 1000 increased by 7
.
I find the visual representation of the data more informative and comprehensive.
The NSW daily and seven-day average case numbers graph:
Screen Shot 2022-08-11 at 12.05.03 am.png
The current wave is assessed to have passed its peak, and now be definitely decreasing.
https://www.google.com/search?q=covid+numbers+in+nsw+today&oq=covid+numbers+in+NSW+today


Deaths from Covid 19 in NSW

Screen Shot 2022-08-11 at 12.05.54 am.png
https://www.google.com/search?q=Daily+deaths+from+Covid+in+N ... e&ie=UTF-8
.
________________


Compare the NSW mortality graph with the mortality graph for the whole of Australia:
Screen Shot 2022-08-11 at 12.07.31 am.png
https://www.google.com/search?q=Daily+deaths+from+Covid+in+N ... &coasync=0

The seven-day average number of deaths due to Covid per day across Australia is 76, and the daily number yesterday was well above that, at 121. Yet I see a relatively low level of mask-wearing in public — people have become less concerned with the possibility of contracting the virus, and the death rate is no longer prominently reported. It reminds me that the business counter in the supermarket sells cigarettes (very expensive) while prominently displaying the slogan health warning SMOKING KILLS. Evidently, if the warning is not dramatised then people grow used to it and ignore it.


/RogerE
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Re: Patterns of Prediction

Post by RogerE »

Patterns in the prime numbers are a source of surprise and pleasure — a gem is a lovely thing, and the primes include some naturally occurring gems.

In another post I just came across this nice pair of twin primes:

179999, 180001

If N is a multiple of 1000, can N – 1 and N + 1 be twin primes? Yes, but N must be a multiple of 3000
(otherwise one of N – 1 and N + 1 is a multiple of 3).
And N – 1 and N + 1 are not necessarily prime whenever N is a multiple of 3000.
For example, when N = 6000 we have composites N – 1 = 5999 = 7 x 857 and N + 1 = 6001 = 17 x 353.

However, the following are twin primes:
2999, 3001
8999, 9001
101999, 102001
164999, 165001
...
There is of course no known proof that there are infinitely many such twin primes, but it is natural to expect that the list will go on forever. We have seen the first five pairs (provided I haven't made any slips). Can you uncover the next few such gems?

/RogerE :D
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Re: Patterns of Prediction

Post by RogerE »

The study of sudoku puzzles continues to be of interest.

In a recent post I proposed a definition for the term incomplete sudoku puzzle, and distinguished three types of such puzzles: improper (no valid completion); proper (a unique valid completion); deficient (more than one valid completion).
https://www.stampboards.com/viewtopic.php?t=93774&start=433
.
I proved some modest general propositions showing that
improper incomplete sudoku puzzles necessarily include at least two independent open cells.
.
I also showed the complementary result, that
for any two independent cells in two dependent cages of a sudoku grid, there exists an improper incomplete sudoku puzzle in which the only open cells are exactly those two.
.
I did not determine whether the corresponding statement is true when "two independent cages" is substituted for "two dependent cages".

I raised some general questions, including: Among all proper incomplete sudoku puzzles, what is the least number of filled cells? I denoted this number by M_0 ("capital M, subscript 0"). I don't know the numerical value of M_0, but I can offer a relevant example ("Exhibit A") from among some published proper incomplete sudoku puzzles that I've solved recently:

Incomplete sudoku puzzle &quot;Exhibit A&quot;<br />A proper puzzle with just 22 filled cells
Incomplete sudoku puzzle "Exhibit A"
A proper puzzle with just 22 filled cells
.
Exhibit A is a proper incomplete sudoku puzzle, with just 22 filled cells. This shows that M_0 is no larger than 22. Are there any proper incomplete sudoku puzzles with fewer than 22 filled cells?

Notice that Exhibit A has no entry 4 and only one entry 7. Completing the puzzle is interesting, because identification of the cells that must contain 4 or 7 must wait until quite a lot of other cells have been filled.
_______________
.
In a follow-up post (post index 434 in this thread) satsuma commented: "I'm having a little bit of difficulty understanding the utility of the proofs." My response is that we are coming to the study of sudoku puzzles from different viewpoints.

I think satsuma is looking for results that will help with the solution of particular types of sudoku puzzles, whereas I'm looking for results that tell me about properties of the whole class of sudoku puzzles — how few filled cells are sufficient to determine a unique solution; how many filled cells can still lead to multiple solutions; how few or how many filled cells can rule out any valid solution; what types of solution strategy are sufficient, what types are necessary; whether there are sudoku puzzles that unavoidably require a branch-point strategy for their solution, and if so, how many branch-points can occur; and so on.

The general philosophical question might come down to something like this: Utilitarianism is a test which can be applied to assign value to objects or activities, but it need not be the only way we can assign value. Is a field of science or art to be judged solely by its practical usefulness, or can understanding or appreciation or enjoyment or admiration be a valid basis for pursuit of such fields?

/RogerE :D
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Re: Patterns of Prediction

Post by satsuma »

Hi Roger E

Here is your exhibit A
exhibit A
exhibit A
I note with interest that the first move chosen can be any of a 1;2;3;5; or 8.

I was able to find a solution to exhibit A, but not without testing past a branch point. I reached this stage but found no further forced move:
exhibit A progressed
exhibit A progressed
Consequentially I had to use the test an option till it doesn't fail method, which enabled a found solution, but not necessarily a unique solution.

But, your definition of a proper incomplete puzzle is one with a unique valid solution, and you state exhibit A is one such: - How were you able to prove that?

Regarding the least number of filled cells to solve a proper incomplete sudoku puzzles, have you considered for example, testing whether any of the given cells in exhibit A can be deleted without compromising the logic required to solve it?

I would also question how the placement of the values affects M_O.
You might recall our earlier discussion about geometric aspects of sudoku.
equal values
equal values
I raise this because exhibit A has 6 different values provided in the boxes and 2 more different values provided in the rectangular ring.

I wonder if eight different values somewhere in those 32 cells is a part of the minimum M_O?
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Re: Patterns of Prediction

Post by RogerE »

Ooohh! My apologies — you will notice that my previous post of Exhibit A did not have 22 filled cells — it only had 21, because I accidentally omitted 5e1. Here's the correct Exhibit A:

Exhibit A (corrected!)<br />Proper partial sudoku puzzle with just 22 filled cells
Exhibit A (corrected!)
Proper partial sudoku puzzle with just 22 filled cells
.
I'm sorry for not noticing that my earlier version of Exhibit A was missing 5 in cell e1.

I will need to think more about some of the points raised by satsuma about the defective version of "Exhibit A". It appears that by using a branch-point strategy for the defective "Exhibit A", satsuma was able to find one valid completion, but whether or not it included 5 in the cell e1 was not disclosed. I haven't yet tested whether there is a valid completion of the defective version of "Exhibit A" with an entry different from 5 in e1.

/RogerE :D
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Re: Patterns of Prediction

Post by satsuma »

RogerE wrote: 15 Aug 2022 15:56

I will need to think more about some of the points raised by satsuma about the defective version of "Exhibit A". It appears that by using a branch-point strategy for the defective "Exhibit A", satsuma was able to find one valid completion, but whether or not it included 5 in the cell e1 was not disclosed.

/RogerE :D
It did not. The cell value was 9.
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Re: Patterns of Prediction

Post by RogerE »

A different Patterns of Prediction topic caught my attention earlier this afternoon. This is Science Week in Australia, and a radio conversation I heard raised the question: Can we understand quantum physics?
.
Quora contains this interesting post, by physicist Mark John Fernee:
.
Screen Shot 2022-08-15 at 2.35.01 pm.png
Screen Shot 2022-08-15 at 2.37.47 pm.png
https://www.quora.com/Was-Feynman-right-when-he-said-If-you- ... -mechanics
/RogerE :D
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Re: Patterns of Prediction

Post by satsuma »

RogerE wrote: 15 Aug 2022 15:56
Image
Exhibit A (corrected!)
Further to exhibit A corrected:
You will note that 19 of the cells are given values 1, 2, 3, 5, 6, or 8.
There are two 9s and one 7.

Simply knowing that cells c9 and j8 are equal and that cell c4 is different, the puzzle can still be solved completely.

So the two 9s and the 7 are only required to enable the placement of all the 1s, 2s, 3s, 5s, 6s, and 8s.
This demonstrates that the 4s, 7s, and 9s are independent of the rest of the solution.
ABC exhibit A
ABC exhibit A
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Re: Patterns of Prediction

Post by RogerE »

More about sudoku.

Thanks satsuma. I've now had time to do some more experimenting along the lines of your most recent sudoku post.

I looked at Exhibit A*, which is the original Exhibit A partial sudoku puzzle (with 22 entries), modified so that the explicit entries 1, 2, 3, 6, 8 are present, together with literal entries w, x, y (occupying the positions formerly occupied by 5, 9 and 7 respectively). Recall that there is just one entry y, and no entry for the remaining symbol, which I denote by the literal z. (I avoided A, B, C for the literals, because those are already used to denote cages.)

Screen Shot 2022-08-18 at 1.24.34 am.png
.
What I found is that Exhibit A* has a unique solution, with no branch point strategies required. That is, Exhibit A* is a proper incomplete sudoku puzzle with just 22 entries.
Here is the reasoning that I used to complete the puzzle:

Screen Shot 2022-08-18 at 1.59.50 am.png
.
The completed puzzle:

Screen Shot 2022-08-18 at 1.26.17 am.png
.
Notice that the numbers corresponding to the literals w, x, y, z can be 4, 5, 7, 9 in any order. All that is required is that they be four markers different among themselves, and different from the five numbers used to fill 19 cells in the original puzzle. The fact that we can assign 4, 5, 7, 9 in any order does not show that they are "independent". The uniqueness of the solution is what is important, and the choice of symbols for the entries is incidental and arbitrary. Indeed, the eight different symbols used to specify Exhibit A* could be replaced by any set of 8 symbols, and the puzzle would still be proper (that is, its solution would be the unique array of the corresponding 9 symbols).

Can any Stampboarder show us a proper incomplete sudoku puzzle with fewer than 22 filled cells?

/RogerE :D
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Re: Patterns of Prediction

Post by RogerE »

The latest Covid Update for New South Wales:

This week, medical commentary continues to affirm that Australia has passed the peak in the current wave of new Covid19 infections, but that full vaccination is highly recommended, and mask wearing in public is still encouraged.
.
Screen Shot 2022-08-18 at 2.26.40 am.png
Changes from 9 August:
Active cases decreased by 33184
Hospitalised cases decreased by 97
Intensive care cases decreased by 3
Ventilated cases increased by 3
Daily deaths decreased by 4
Screen Shot 2022-08-18 at 2.27.07 am.png
Changes from 9 August:
Cases in last 24 hours decreased by 3292
Cases this week decreased by 21068
Cases last week decreased by 14481
Cases since start of pandemic increased by 55729
Deaths since start of pandemic increased by 164
.
The above two screen shots come from the website:
https://www.health.nsw.gov.au/Infectious/covid-19/Pages/stats-nsw.aspx
.
Since the previous snapshot of 9 August, there have been 164 deaths added to the NSW total. That's an average around 23 deaths due to Covid, per day, in NSW.


Here are the "local numbers" in my neighbourhood of New South Wales [NSW]:
Screen Shot 2022-08-18 at 2.27.47 am.png
Changes from 9 August:
Cases decreased by 859
Daily Tests increased by 621
Daily Rate per 1000 increased by 4
Screen Shot 2022-08-18 at 2.28.12 am.png
Changes from 9 August:
Cases increased by 695
Daily Tests increased by 215
Daily Rate per 1000 increased by 1
.
I find the visual representation of the data more informative and comprehensive.
The NSW daily and seven-day average case numbers graph:
Screen Shot 2022-08-18 at 2.29.34 am.png
The current wave is assessed to have passed its peak, and now be definitely decreasing.
https://www.google.com/search?q=covid+numbers+in+nsw+today&oq=covid+numbers+in+NSW+today


Deaths from Covid 19 in NSW

Screen Shot 2022-08-18 at 2.30.47 am.png
https://www.google.com/search?q=Daily+deaths+from+Covid+in+N ... e&ie=UTF-8
.
________________


Compare the NSW mortality graph with the mortality graph for the whole of Australia:
Screen Shot 2022-08-18 at 2.31.45 am.png
https://www.google.com/search?q=Daily+deaths+from+Covid+in+N ... &coasync=0



/RogerE
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Re: Patterns of Prediction

Post by satsuma »

RogerE wrote: 18 Aug 2022 03:19 More about sudoku.

Thanks satsuma. I've now had time to do some more experimenting along the lines of your most recent sudoku post.

I looked at Exhibit A*, which is the original Exhibit A partial sudoku puzzle (with 22 entries), modified so that the explicit entries 1, 2, 3, 6, 8 are present, together with literal entries w, x, y (occupying the positions formerly occupied by 5, 9 and 7 respectively). Recall that there is just one entry y, and no entry for the remaining symbol, which I denote by the literal z. (I avoided A, B, C for the literals, because those are already used to denote cages.)

Image
.
What I found is that Exhibit A* has a unique solution, with no branch point strategies required. That is, Exhibit A* is a proper incomplete sudoku puzzle with just 22 entries.
Here is the reasoning that I used to complete the puzzle:

Image
.
The completed puzzle:

Image
.
Hi RogerE

I think you'll find there is a typo in your proof in line 29:
29. z(C)>zc7 should read:
29. z(C)>zb7.
Apart from that I can see no issue with the proof.


I would now like to draw your attention to some comparisons between Exhibit A (incomplete) and Exhibit A (complete). Below is the solution I found (which may not be unique) for the incomplete puzzle:
incomplete exhibit A, my solution
incomplete exhibit A, my solution
(The change in fill colours only represent placements after I had to make an either or decision)


You will remember that the missing given was cell E1 = 5 and as I advised my solution required cell e1 = 9.
I decided to try to assess how critical that omission was and tested the complete vs the incomplete exhibits solutions for variances. Here is a summary:
complete vs incomplete
complete vs incomplete
What I find interesting is that although almost all the changes relate to the values 4,5,7 and 9; which you identified as interchangeable as a group amongst themselves; some 1s and 2s have also appeared. I also note that although there are some variances in the placements of the value 7; none of those placements relate to the 1s and 2s.

So I'm still thinking about which cells were critical to a unique solution and which functioned as placeholders only.
Could any of the 22 givens be omitted, yet still enable a unique solution to be found?
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Re: Patterns of Prediction

Post by capetriangle »

satsuma, RogerE

My brain has glazed over on these.

However, satsuma, I do get a different solution for the complete (22 numbers) Exhibit A. I found that the solution for the 22 numbers puzzle was quite easy.

The difference is in Cage G.

I get the solution as:

4,3,6
1,9,2
8,7,5

Kindest regards

Richard
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Re: Patterns of Prediction

Post by RogerE »

• Hello Richard = capetriangle. Your solution to the intended 22-entry version of "Exhibit A" exactly agrees with the unique solution I showed for "Exhibit A*". The correspondence is fully described by setting
w = 5, x = 9, y = 7, z = 4.


• Hello satsuma. You are correct about the typo you detected in my "game play" for my solution to "Exhibit A*". The solution I displayed in the completed sudoku grid shows that I did indeed enter z in b7, not c7, so the "zc7" in step 29 of my "game play" was just a typo.

Your investigation of the consequences of 5v9 in cell e1 of Exhibit A is indeed interesting. I think it has some "deep" consequences, not yet clear to me — meaning, I will try to keep it in mind, and consider it further, to try to discern some of those consequences... You have definitely contributed to the structural thinking about sudokus that I find very interesting, and also the analysis of strategies.

Now let's comment on the very interesting question: Is there any one entry in the 22 givens of Exhibit A that can be omitted, with the remaining 21 givens determining a unique solution? You have shown that 5e1 is essential for Exhibit A to have a unique solution, and you have reported that replacing it with 9e1 also yields a unique solution. I think we have not yet decided definitively whether any other entry in e1 (while keeping the other 21 givens unchanged) gives a proper sudoku. The question about whether Exhibit A contains a 21-given subset that is a proper sudoku is very interesting, and I conjecture that the answer is "no". Equivalently, I conjecture that deleting any one given from Exhibit A produces a deficient puzzle (you have shown that is the case for cell e1 — "only" 21 more cells to check!)

Thanks for your interest, and contributions so far :D

/RogerE :D
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Re: Patterns of Prediction

Post by RogerE »

More on the incomplete sudoku puzzle with 22 givens, referred to as Exhibit A.

Here is a modified version of Exhibit A, in which I have replaced the 5 in e1 with the literal x (denoting a potentially given entry that we don't yet know explicitly). I will call this Exhibit Ax(e1), so we have some label by which to refer to this incomplete sudoku puzzle.

Exhibit Ax(e1)
Exhibit Ax(e1)


Available choices for x
We know from earlier discussion in this thread that when we assign x = 5 the puzzle is proper — that is, it has a unique completion.
Here I am interested in what is the status of the incomplete puzzle for other choices of x.
Clearly x cannot take the values 1, 3 or 6, because any of those choices violate the definition of an incomplete sudoku puzzle.
Results reported by satsuma apparently show that x = 9 yields a proper puzzle, though I would like to see the full details — the claim is that there is a valid completion, and that completion is unique.


Analysis while retaining x as a literal
Consider this partial completion of Exhibit Ax(e1), after 24 moves (specified below the displayed grid):

Partial completion of Exhibit Ax(e1) — after 24 moves.
Partial completion of Exhibit Ax(e1) — after 24 moves.
Screen Shot 2022-08-22 at 12.07.41 am.png


Note that the 24 red entries are all determined uniquely by the 21 givens (in black) together with the fact that e1 is occupied, but we don't know the numerical value of its entry.
Notice that 2 is now explicit in row e. If we had initially assigned the value x = 2 in Exhibit Ax(e1), we now deduce that this results in an improper puzzle.


After 24 moves followed by a particular numerical assignment for x
I didn't find a forced next move after the above 24 moves, so my approach was to treat this stage as a Branch Point in my strategy. I decided to try the option x = 4. I made all new entries in green.

I was able to continue quite far before discovering (perhaps around move 50) a violation of the sudoku constraints — namely, a cell with no possible entry. For interest, I found that I could validly complete all but two cells, while the remaining two have no possible entry. Let's call this Exhibit A4(e1)*.
Conclusion: Exhibit A4(e1) is an improper incomplete sudoku, and it has an extension to an improper incomplete sudoku Exhibit A4(e1)* with only two open cells, d9 and f7:

Exhibit A4(e1)* = improper partial sudoku puzzle containing Exhibit A4(e1), <br />with d9 and f7 the only open cells
Exhibit A4(e1)* = improper partial sudoku puzzle containing Exhibit A4(e1),
with d9 and f7 the only open cells


This is a counter-example to an earlier claimed "proof"
In an earlier post in this thread I offered proofs of several claims about incomplete sudoku puzzles.
https://www.stampboards.com/viewtopic.php?t=93774&start=433
Now Exhibit A4(e1) is a counter-example to one of the claims, so there must be an error in its "proof":
.
Screen Shot 2022-08-22 at 12.51.10 am.png
.
Let's apply the "proof" to Exhibit A4(e1)*.
Cage F is the only incomplete cage.
The two missing entries are 4 and 8. Since only one 4 is missing from the whole grid, two of the three rows intersecting cage F must contain 4 — they are rows d and e, so f is the row that is missing 4.
Two of the three columns intersecting cage F must contain 4 — they are columns 7 and 8, so 9 is the column that is missing 4.
The intersection of f and 9 is the cell f9, which can legitimately contain the entry 4 without causing any row or column to contain two 4s.
The trouble is, f9 is not open, it already contains 6. The error in the "proof" was to assume that the intersection of the row and column lacking 4 would be an empty cell that can be assigned 4.
(Similarly, row d and column 7 both lack 8, so cell d7 can legitimately contain the entry 8, but once again it is not empty, it already contains 1.)
So, we see that "Corollary 2" must be discarded as incorrect. The proposed "proof" overlooked the fatal fact that the cells which can legitimately be filled with the missing entries might already be occupied by different entries.


An apt quotation
I am currently reading a relevant book:
John and Mary Gribbin, Science — a History in 100 Experiments, William Collins Books (2016)
My quote comes from William Gilbert (1544-1603), who experimented extensively with magnetic objects, and wrote about his experiments and conclusions in the book De Magnete (Concerning Magnetism, Magnetic Bodies, and the Great Magnet Earth).
William Gilbert wrote: In the discovery of secret things, and in the investigation of hidden causes, stronger reasons are obtained from sure experiments and demonstrated arguments than from probable conjectures and the opinions of philosophical speculators.
.
So there you have the scientific method in a nutshell. My claimed "proof" turned out to be a probable conjecture, and the experiment of closely examining Exhibit A4(e1) turned out to yield a different outcome than my conjecture anticipated. ;)
.
/RogerE :D
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