Diurnal Age Meets Conway's Game Of Life

 I've written about Conway's Game of Life in two recent posts:

• The Game of Life on Saturday, the 16th of December 2023 and

• 108 Meets Conway's Game Of Life on Friday, the 9th February 2024

I've been playing around with an app called "Life" on my iPhone that allows the game to be run but I prefer on browser-based app that I can access from my laptop. To that end, I've been playing around with one of three websites recommended by Gemini:

• Conway's Game of Life (Golly edition): https://conwaylife.com/

This website utilizes the popular "Golly" simulation software, offering advanced features like pattern libraries, scripting, and different grid geometries. You can save and export your simulations in various formats.

• Game of Life Online: https://playgameoflife.com/

This website allows you to draw patterns directly on the grid with an intuitive interface. While it lacks advanced features, it's great for quick visualizations and sharing creations.

So in this post, I'm looking at the first of the recommendations and playing around with a new idea. I want to investigate how the number associated with my diurnal age behaves under the Game of Life rules. The number for today, 27346, is shown in Figure 1. All the digits from 0 to 9 can be created using a 3 x 5 pixel grid, the smallest possible size.

Figure 1

The rules lead, after 125 steps, to the image shown in Figure 2:

Figure 2

What would be interesting to keep track of are the number of steps required to reach a stable state. It's clear that the stable states arising from numbers are not unique. For example 16161 will end up the same as 19191 if we don't regard mirror images, rotations and reflections as different. However, most numbers should result in stable states that are different from one another. I can attach images of these stable states to my Airtable database. 

I'll explore the other two Gemini recommendations later. Any particularly interesting stable states or record number of steps arising from these diurnal age investigations can be the subject of future posts. In the case of 27346, we can say that the stable state consists of five blocks (the simplest still life) and one hive or beehive (the second most common still life).

 

This ongoing, daily exercise is a great way to deepen ones understanding of a topic. It was only through my adherence to the investigation of the number associated with my diurnal age that I widened and deepened my understanding of number theory. It's a great maxim: once a day but everyday and can be and should be applied to more aspects of my daily life.

Special Numbers, Special Dates

On the 26th January 2023, I posted about Turning Dates Into Numbers and made the observation that the 10th February 2023 was a special day because the number associated with this date (20230210) had the interesting property that it was the only date in the year that produced a number that had seven distinct prime factors:

20230210 = 2 x 5 x 7 x 11 x 13 x 43 x 47

It turns out that this number has another interesting property in that the sum of its digits equals the last two digits (representing the day of the month). In fact, it begins a run of ten consecutive numbers that end with 20230219 (the 19th February 2023) ... permalink:

20230210, 20230211, 20230212, 20230213, 20230214, 20230215, 20230216, 20230217, 20230218, 20230219

Another run of ten similar numbers occurs in November, starting on the 10th and ending on the 19th.

20231110, 20231111, 20231112, 20231113, 20231114, 20231115, 20231116, 20231117, 20231118, 20231119

For many, many years to come, the sum of the digits of the numbers associated with the dates will always contain two digits. Today is the 15th February 2023 and so we are into the second half of the first-mentioned run.

This number can be expressed as a sum of three cubes
See Figure 2

Similar runs will occur every year but the dates will differ naturally. For example, in 2024, the first run of numbers will begin on 11th January 2024 with 20240111 and end on 19th January 2024 with 20240119. 

In my previous post Turning Dates Into Numbers I looked at what numbers were prime, semiprime and so on. In this post I'll turn my attention to cubes and see what numbers can be expressed as the sum of two cubes and as a sum of three cubes. 

It turns out that there is only one number in the entire year that can be expressed as a sum of two cubes and that is \( 20231020 = 69^3 + 271^3\). This corresponds to the 20th October 2023.

The sums of three cubes are more common. The complete list is shown in Figure 2, arranged in chronological order with 20th January being the first date and 26th December the last.

Figure 2: permalink

It can be noted that some dates produce numbers that can be expressed as sums of three cubes in more than one way. Specifically, 20230426 (two ways), 20230523 (two ways), 20230722 (two ways), 20230904 (two ways), 20230911 (three ways), 20231028 (two ways) and 20231226 (two ways). Thus the 11th September (the famous 911 in United States date format) produces the only number (20230911) that can be expressed as a sum of three cubes in three ways.

A History of Tweets

Each day I tweet about the number associated with my diurnal age but I rarely look back at any of these tweets, especially now that they just represent a summary of a far more detailed analysis that I have stored on my AirTable databases (Diurnal Age Part 1 and Diurnal Age Part 2). 

However, this was not always the case and intially my daily tweet was the only record I retained of my analysis. I discovered a site (https://www.allmytweets.net/) that allows you to easily look back at all your previous tweets and it was interesting to look back at my very first tweet on April 30th 2015. See Figure 1.

Figure 1

Short and sweet but this was back in the days where only 140 characters were allowed. It was only on November 8th 2017 that Twitter increased the limit to 280 characters. This is the OEIS sequence to which the tweet refers:

A109724

Sum of the first \(n^2\) primes.

The initial members of the sequence are:

0, 2, 17, 100, 381, 1060, 2427, 4888, 8893, 15116, 24133, 36888, 54169, 77136, 106733, 144526, 191755, 249748, 320705, 406048, 507825, 627294, 768373, 931686, 1119887, 1336090, 1583293, 1864190, 2180741, 2536646, 2935471, 3380980

In the case of 24133, \(n\) has the value 10. I didn't tweet the next day but did for 24135 and then I missed another day and tweeted for 24137. After that I kept tweeting for every day without fail until the day of this post. The tweets were not exclusively mathematical in those days. It was only on August 6th 2016 (24588) that the tweets became solely mathematical.

I'll still keep tweeting, old habits die hard, even though my AirTable database and my mathematical blog (Mathematical Meanderings) are the foci of my mathematical analysis these days.

Of course I was counting days long before I started tweeting about them. Here is a graphic I created to celebrate being 22222 days old. This occurred on Thursday, February 4th 2010!

More Wordle Statistics

My last post titled Wordle Statistics was long enough so I didn't want to add more newly found information to that and hence I'm making a fresh post. 3Blue1Brown has just created a YouTube video that involves a statistical analysis of Wordle.

There's a lot to digest in this video but my main takeaway after first viewing it was that CRANE was a good starting word! I clearly need to watch it again and again to fully absorb what he's saying. However, for today's Wordle I started with CRANE and the results were almost disastrous as can be seen in Figure 1.

Figure 1

Looking at Figure 1, it can be seen that I had a spectacular start with three letters in the correct positions. There were only two remaining letters to guess. However, I nearly failed because there were just so many possible words that could be made from *RA*E. 

Referring to kaggle, a database of English word frequencies, we can see that TRADE was a good second choice because it has by far the highest frequency. Had I known about word frequencies, my third choice would have been FRAME and I would have solved the puzzle in a mere three attempts.

CRANE: 4,888,961 FIFTH

                                        TRADE: 110,086,585 FIRST

                                        ERASE: 3,086,642 SIXTH

                                        GRACE: 17,642,126 THIRD

BRAKE: 9,321,885 FOURTH

                                        FRAME: 46,079,991 SECOND 

Using Google search with quotes e.g. "trade" yields the following statistics:

CRANE: 166,000,000 SIXTH

                                         TRADE: 1,930,000,000 SECOND

                                         ERASE:  242,000,000 FIFTH

                                         GRACE:  918,000,000 THIRD

 BRAKE:  503,000,000 FOURTH

                                         FRAME:   2,350,000,00 FIRST

Interestingly, using the Google search, FRAME and TRADE swap places with the former being markedly more frequent (in searches at least). CRANE and ERASE also swap positions in fifth and sixth places.

I downloaded the CSV file of word frequencies from kaggle (it's only 5MB) and filtered out words that were not five letters in length. Here are the initial five letter words with the highest frequencies:

about 1,226,734,006

other 978,481,319

which 810,514,085

their         782,849,411

there 701,170,205

first         578,161,543

would 572,644,147

these 541,003,982

click         536,746,424

price         501,651,226

state         453,104,133

email 443,949,646

world 431,934,249

music 414,028,837

after         372,948,094

video 365,410,017

where 360,468,339

books 347,710,184

links         339,926,541

years 337,841,309

As can be seen, ABOUT comes out clearly on top with a frequency of over 1.2 billion! This might not be a bad starting word. Anyway, more food for thought went tackling Wordle.