Coin rotation paradox

The coin rotation paradox described here has existed for a long time. Before February 21, 2020 the Wikipedia page gave no explanation of why the moving coin rotates twice as it makes one revolution around the fixed coin. I did not find a good explanation anywhere else on the Internet either. I constructed my own solution. Then I edited the page to explain why, adding the following new section. 

Analysis and Solution

From start to end the center of the moving coin travels a circular path. The edge of the stationary coin and said path form two concentric circles. The radius of the path is twice either coin's radius. Hence, the circumference of the path is twice either coin's circumference. To go all the way around the stationary coin, the center of the moving coin must travel twice the coin's circumference. How much the moving coin rotates around its own center en route, if any, or in what direction – clockwise, counterclockwise, or some of both – has no effect on the length of the path. That the coin rotates twice as described above and focusing on the edge of the moving coin as it touches the stationary coin are distractions.

At the time of writing this, there is a warning that says the page lacks citations and has unsourced material. Like I said above, I did not find anything to cite or use as a source. I solved it on my own. 

Wikipedia can be edited at any time by almost anybody. I hope nobody removes or ruins the above section. Anyway, I hereby document it in my blog, which is far less subject to being altered by others. 

Simple Sukodu Maker #3

The following shows another method of making a completed Sudoku puzzle. This method involves no arithmetic, using copy and paste after filling in one 3x3 part of the puzzle with the numbers 1-9.

Start by putting the numbers 1-9 in any order in the top left 3x3 array (or block) of the 9x9 array. In the following B2-B9 are used to designate the other 3x3 arrays for ease of reference. B1 will refer to the top left 3x3 block of numbers.

7	1	2
3	5	6		B2			B3
8	9	4
	B4			B5			B6
	B7			B8			B9

Next copy the first row of numbers in B1 to the 2nd row of block B2 and the 3rd row of B3. Then copy the 2nd row to the 3rd row of B2 and the 1st row of B3. The different colors in the following highlight the steps. Similarly but not shown, one could copy the 2nd row of B1 to the 1st row of B2 and the 3rd row of B3. Lastly copy the 3rd row of B1 to the remaining open row of B2 & B3. 

7	1	2	8	9	4	3	5	6
3	5	6	7	1	2	8	9	4
8	9	4	3	5	6	7	1	2
	B4			B5			B6
	B7			B8			B9

Next copy the first column of numbers in B1 to the 2nd column of block B4 and the 3rd column of B7. Then copy the 2nd column of B1 to the 3rd column of B4 and the 1st column of B7. The different colors in the following highlight the steps. Similarly but not shown, one could copy the 2nd column of B1 to the 1st column of B4 and the 3rd column of B7. Lastly copy the 3rd column of B1 to the remaining open column of B4 & B7.

7	1	2	8	9	4	3	5	6
3	5	6	7	1	2	8	9	4
8	9	4	3	5	6	7	1	2
2	7	1
6	3	5		B5			B6
4	8	9
1	2	7
5	6	3		B8			B9
9	4	8

Complete the remaining blocks in similar fashion. One could copy (1) rows from B4 to B5 & B6 and rows from B7 to B8 & B9, or (2) columns from B2 to B5 & B8 and columns from B3 to B6 & B9.

One could also start by first putting the numbers 1-9 in any other 3x3 block and fill the remaining blocks following a method similar to the one used above. Whichever block is first filled, fill the other two blocks in the same 3x9 band and the other two blocks in the same 9x3 stack, or vice-versa.

Once the entire 9x9 array is filled, one could used the ranked random number process described in my May 17 post to make numerous more completed Sukodu puzzles. 

Simple Sukodu Maker #2

In my last post I showed how to make two simple Sudoku base puzzles. One started with entering the numbers 1 4 7 2 5 8 3 6 9 in the first row. It was completed by filling each column adding 1 to the entry above it, with the exception that 1 follows 9. The other started with entering the same numbers in the first column. It was completed by filling each row adding 1 to the entry to the left of it, with the exception that 1 follows 9.

Similarly, ten more almost as simple base puzzles can be made by adding 2, 4, 5, 7, and 8 instead of 1, with an exception -- if adding yields a number more than 9, then subtract 9 from the result.  Adding 3 or 6 doesn't work since they violate the Sudoku rule that each of digits 1-9 appears exactly once in each row, column, and 3x3 block. 

The following shows the puzzle made by filling the first row and then adding 4 in columns.

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

The following shows the puzzle made by filling the first column and then adding 4 in rows. It is also the above array transposed.

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

Like explained in my prior post, numerous more puzzles can be made from any of these 10 base puzzles using ranks of random numbers. 

More base puzzles could be made by swapping entire columns within the groups of columns 1-3, 4-6, and 7-9 and/or reordering the three 9x3 arrays. Similarly, more base puzzles could be made by swapping entire rows within the groups of rows 1-3, 4-6, and 7-9 and/or reordering the three 3x9 arrays. 

Simple Sudoku Maker

There is much on the Internet about how to solve Sudoku puzzles. However, there is little about how to make a Soduku puzzle. The following shows a simple non-trial-and-error method to make a Sudoku puzzle that satisfies the requirements. That is, each of the rows, columns, and nine 3x3 blocks of cells contain each number 1-9 exactly once. Also, many alternatives can be made from this result. 

Make the Base Puzzle

First enter the numbers shown below in red and blue backgrounds. Note that the numbers in blue are copied from the 2nd and 3rd rows of numbers in red. Then going down each column, add 1 to get the number in the next row until the puzzle is filled, with one exception -- 1 follows 9. 

There may be, but I doubt there is a simpler method. I have seen only one other method that is about as simple (link). 

Many alternatives can be made from this Base Puzzle. For each of numbers 1-9, generate a random number, then rank the random numbers. The random numbers and ranks are easily produced in a spreadsheet. Then pair each number 1-9 with its random number's rank among all the random numbers, and create the next puzzle from the ranks as shown below. In effect, the numbers in the Base Puzzle are shuffled while still meeting the requirements. For example, a 1 in the Base Puzzle yields a 6 in the Output Puzzle, a 2 in the Base Puzzle yields a 4 in the Output Puzzle, and so forth. 

Base Puzzle

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

Number

random

rank

1	0.47460201	6
2	0.66240441	4
3	0.30831936	7
4	0.71401711	2
5	0.28997433	8
6	0.68553150	3
7	0.05804816	9
8	0.73011822	1
9	0.47537093	5

Output Puzzle

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

Equally simple is to create the Base Puzzle by transposing the numbers in red and blue and then adding 1 in rows rather than in columns. The result would be the above Base Puzzle fully transposed. It would match the one at the URL linked above, but its author fills the puzzle by shifting rather than adding. The link also uses a different shuffling method to make alternatives.

Finishing is easy. Delete as many numbers from the Output Puzzle as desired. I leave allowing only one solution to others.

Coronavirus - herd immunity

Herd immunity "is a form of indirect protection from infectious disease that occurs when a large percentage of a population has become immune to an infection, whether through previous infections or vaccination, thereby providing a measure of protection for individuals who are not immune. In a population in which a large proportion of individuals possess immunity, such people being unlikely to contribute to disease transmission, chains of infection are more likely to be disrupted, which either stops or slows the spread of disease. The greater the proportion of immune individuals in a community, the smaller the probability that non-immune individuals will come into contact with an infectious individual, helping to shield non-immune individuals from infection" (Wikipedia).

It's clear that herd immunity is effective when the already-immune portion of the herd is high, but has little effect when the already-immune portion of the herd is very low.  Amid all the panic, herd immunity has received little attention during the ongoing coronavirus pandemic. Given the newness of the virus, SARS-CoV-2, and no vaccine for it, the already-immune portion of any country's population is very low.

In this interview Professor Knut Wittkowski relies heavily on the idea of herd immunity to combat the coronavirus. He may be correct, but I didn't share his confidence. First, the already-immune portion of people is very low. Without vaccination, many, many people would have to become infected -- in a non-controlled way unlike vaccination -- to reach the critical mass for herd immunity to have much effect. Second, it seems he relied too much on mortality rates from the flu. The range of annual flu deaths is wide. He predicted the number of deaths in the USA will be in the low end of that range. Deaths from the SARS-CoV-2 being one cause already are near the high end of that range and will exceed it. Wittkowski emphasizes children becoming immune via exposure to SARS-CoV-2 without it being fatal (or even sickening). However, children ages 0-14 are only about 19% of the USA's population, with ages 0-18 about 24%. In summary, the downside of non-immune people being exposed to SARS-CoV-2 seems risky. He also relies on isolating the elderly, but there are many elderly living among the general population, not in nursing homes or assisted care residences. Of course, non-immune people avoiding exposure to the virus does not require the sort of massive lock-downs that have been implemented by governments.

These two articles #1, #2 reach the same conclusion about how lethal the virus is. The lack of a vaccine is likewise a weighty factor.

One country, Sweden, has relied on a herd immunity approach and no strict lock-down. Its neighbors Denmark, Finland, and Norway have kept reported infections and deaths lower, especially deaths (link). This article addresses Sweden's case somewhat favorably. This article addresses Sweden's case more negatively.

Coronaviris - innovation

A Wall Street Journal article is about innovation during the coronavirus pandemic. Shoes to Masks: Corporate Innovation Flourishes in Coronavirus Fight. It features several cases of how private companies have changed what they do in response to economic demand. Since the article is behind a paywall, the following -- until the asterisks -- are some excerpts.

The innovations "run the gamut from individuals using 3-D printers to turn out N95 masks and academic labs repurposing themselves into coronavirus test centers to General Motors Co. teaming up with Ventec Life Systems to make ventilators."

The "shoes to masks" in the title refers to a small Pennsylvania company that switched from making children's shoes to making face masks for protection against the coronavirus.

The True Value Co. switched some of its paint production lines to make hand sanitizer.

During World War II, government agencies "such as the War Production Board and the U.S. Maritime Commission decided what should be produced and which companies would get contracts, which generally had a guaranteed profit margin."

"Yet compared with wartime, the federal government has thus far mostly played a supporting role in mobilizing economic resources to combat the pandemic. Most of the acquisition and production of vital supplies has fallen to states and health-care providers working with private suppliers and intermediaries."

*******

The above innovations occurred in a partly free market economy, where production and decision-making are not centralized. Any market participants may make decisions and take initiative. They do not need authorization from government officials (with exceptions). Socialism is usually the idea that government should own and control all production. (Even socialists who advocate worker-owned firms typically desire extensive government planning at a macro level, especially for finance and capital allocation.) Production, capital allocation, and decision-making are centralized, subject to laws, regulations, and approval by government officials. The spontaneous order of markets is more conducive to innovation and more adaptive to changing circumstances than the command or planned order of politicians and government bureaucrats.

Net neutrality - USA vs Europe

Coronavirus Crisis Vindicates the FCC’s ‘Net Neutrality’ Rollback is the title of a Wall Street Journal op-ed. The subtitle is: In Europe, meanwhile, heavy-handed regulation is forcing internet providers to throttle video speeds.

The article is behind a paywall. The title and subtitle summarize it nicely. The following excerpts tell more.

The widespread imposition of stay-at-home orders has underscored the critical role that access to the internet plays in modern society. In Europe, networks have struggled to meet bandwidth demand. U.S. networks have faced fewer problems adjusting to the increase in demand. The European Union has embraced a heavy-handed regulatory scheme designed to allocate access to the existing network, while the U.S. has emphasized private investment to expand network capacity.

European regulators were guided by the legal system developed to govern traditional telephone service largely built with taxpayer funds.  Rather than fold the internet into an outdated legal regime developed for a different era, the American vision concentrates on encouraging telephone and cable companies to compete by investing to increase their bandwidth. The only major deviation from this pattern occurred in 2015, when the Federal Communications Commission adopted a “net neutrality” rule applying legacy telephone regulation to the internet for the first time. The agency returned to its longstanding investment-oriented policy in 2018.

The U.S. and EU have seen dramatically different investment and utilization. Between 2010 and 2016, American providers invested on average annually 2.35 times as much per household as their European counterparts. This allowed the average U.S. household to consume more than three times as much data as the average European household in 2017, according to Cisco. [End].

The article doesn't say what net neutrality is. Wikipedia and Investopedia say more.