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.