Ngaio Marsh and Chekhov

I recently watched a new version of Anton Chekhov’s “The Seagull” at Soulpepper Theatre in Toronto.  A good presentation of a complex play.  The director [Daniel Brooks] wrote in his programme notes, “I love [Chekhov’s] cheekiness and wit.  And his tenderness, especially for those who yearn.  I love him because he blames no one.”  Those remarks by Brooks helped me to a new appreciation of “The Seagull”.

A bit earlier, I had read Ngaio Marsh’s 1980 mystery “Photo Finish”.  One realizes that Marsh patterned her confused, erratic young opera composer after Chekov’s playwright in “The Seagull”.  Thus another way to appreciate the two works.  It is no coincidence: Marsh was very engaged with plays and performances in her books.

Happy reading, and Best Wishes, from Ken Roberts, 17-April-2023.

What is Bode’s law ?

Bode’s law is an observation that the distances of the planets from the Sun are in an offset geometric progression.  It is not really a “law” of nature, in the same way we speak of the “law of gravitation”.  But it is an intriguing idea.  Some people call it “Bode’s rule”.  I would even describe it as “Bode’s relationship” between planetary position and orbital radius.  And, just to complexify the name further, the modern practice is to speak of the “Titius-Bode” law, rule, or relationship, because Bode took his inspiration from Titius (and cited Titius). My authority for this information is the excellent Wikipedia article https://en.wikipedia.org/wiki/Titius%E2%80%93Bode_law

Most scientific ideas have a similarly complicated origin, and names are flexible.  In fact, sometimes the opinion is expressed that the person named as the author of some well-known property of nature, is usually not the originator.  That’s glib.  But, certainly, science is a collaborative endeavour.  Anyway, I’m going to stick with “Bode’s law” for simplicity.

So, let’s state Bode’s law.  At the time (1772) , the known planets were Mercury through Saturn.  (Uranus was discovered in 1781, and the asteroid Ceres was discovered in 1801).  Bode assigns Saturn a distance from the Sun of 100 units, and observes that, using that scale, Mercury is at a distance of 4 units, Venus at 7 (=4+3) units, Earth at 10 (=4+6) units, Mars at 16 (=4+12) units, an “empty” position (possible planet to be found) at 28 (=4+24) units, and Jupiter at 52 (=4+48) units.  Moreover, Saturn’s distance at 100 units can be written as 100=4+96 units.

Thus Bode’s law is that, the planets which occupy positions N = 2,3,4,6,7,8 (with position 5 being skipped) are located at distances 4+3*2^(N-1) from the Sun.  This really is an astonishing relationship.

Even more astonishing is that the later discovery of Uranus found it to be at position 9, and the later discovery of Ceres, the largest asteroid, found it to be at position 5.  Neptune is not at position 10, but Pluto is found at that distance.

The Wikipedia article referenced above has quite a lot of further material, and I want to discuss some of that in future posts.  Some questions which I hope to cover include

  • What if we state Bode’s law with periods instead of distance?
  • Is there a physical rationale for planets to be spaced per Bode’s law?
  • Can Bode’s law be observed in exoplanet systems?

Best wishes, from Ken Roberts, 03-Jan-2023

What is the Nerdle logic game ?

The Nerdle logic game involves guessing an 8-symbol arithmetic equation, which uses only integers, an equal sign, and the four arithmetic operations.  For instance, the equation might be 9 / 3 * 4 = 12.  That was my first guess in the example below, which is from the puzzle posted at the gamesite on 01-Jan-2023:

The green colour of the digit “4” means that the solution contains “4” at that position.  The purple colour of the digits “9”, “3”, “1”, the operator “/” and the equal sign “=” means that the solution contains those symbols but at some other positions.  The white colour of the symbol “*” and the digit “2” means that the solution does not utilize those symbols.

Now one can make a second guess.  It is time for some reasoning.  Since the equal sign is not in position 6, it is likely in position 7, so hypothesize that position 8 is a single digit.  The possibilities for position 8 are anything but “2”.

Let’s try putting the division operator in position 4, so that we will be dividing “1xx” or “3xx” or “4xx” by “4x” to get a single digit result, some result between 3 and 9.  For instance, 9 times 43 equals 387.  But that trial solution would not contain a “1”, so cannot be correct.  Also not correct would be 9 times 40, 42, 44, 45, 46, 47, 48, 49 because such a solution would either contain a digit “2” or would be missing a digit “1” or “3”.  However, we can try 9 times 41 which equals 369.  So, our second guess will be 369 / 41 = 9.  Here is the result:

We have some additional information.  The solution does not contain a “6”.  We know that the division operator is in position 4 and have confirmed that the equal sign is in position 7.  As well, we know there is only one “9” digit in the solution, and have excluded “3” and “9” from being in certain positions.

Let’s continue with the hypothesis that the only operator in the solution is a single division.  We will abandon that hypothesis if we cannot find a solution compatible with it.  This means that we have a single digit value (not 2 or 9, and also not 1, hence something between 3 and 8) following the equal sign, and that value times “4x” must be “1xx”.  So we can conclude the value following the equal sign is 3 or 4.  Let’s check some possibilities.  We can calculate the products of 3 or 4 times “4x”.  There must be a “9” in either position 2 or position 6, and there can be only one “9” in the solution.

Considering the case of a “9” in position 6, we look at 3*49 = 147 (which is ok) and 4*49 = 196 (which is not ok, as there would be two “9” digits in the solution).   This is enough to give us a third guess.  We will not bother to consider possible solutions of the form “19x” divided by “4x” where the latter value is not 49.  Here is the result from our third guess at a solution, 147 / 49 = 3:

And that is the solution to today’s Nerdle puzzle.

Some remarks:  1) This post will be scheduled to be posted at 2100 hours Eastern time today 01-Jan-2023, so that the solution to today’s Nerdle puzzle will not be a spoiler.  The Nerdle website rolls over to a new puzzle at 0000 hours UTC daily, so by the time this post appears the current puzzle with be the 02-Jan-2023 puzzle.  (All the times on my LASI site are Eastern North American time, aka “Toronto” time, which is UTC-5 time zone.)

2) The Nerdle website is at https://nerdlegame.com/

3) I usually run Nerdle with an ad-blocker.  The site displays dynamically changing advertisements and that interferes with the pleasure of figuring out the puzzle.

4) My customary initial guess is 9/3*4=12.  That gives a reasonable amount of information as it tests a string of digits (1,2,3,4) and two operators.  That is not the optimal first guess, based upon some calculations done by a friend.  But it is satisfactory for my enjoyment.  There are undoubtedly other web resources which assist with analysis of Nerdle puzzles.

Happy puzzle solving, for a Happy New Year!

Best wishes, from Ken Roberts, 01-Jan-2023

Should you do everything forever ?

“You don’t have to do something forever, for it to have been worth doing.”

That was the advice I received long ago from a friend, and it helps tremendously.

Here, for instance, is a display of santas on the mantle of our house.  The photo was taken today, 31-Dec-2022.  A person who is familiar with our household traditions may notice that the tall santa on the left is new this year and thus be able to date the photo.  These santas occupy the mantle during the month of December, and come the new year they will be carefully removed, packed away to the next December, to be enjoyed again.

So, we have a mix of the temporary (the presence of santas on the mantle), the periodic (every December), and the permanent intention (bring out the santas every Christmas).  It turns out that the question “how long should the santas be displayed on the mantle” has a complicated answer.  Life is like that.

If something is worth doing, do it if and when the circumstances seem appropriate.  You do not have to do it forever.  You do not have to be correct, either.  Life is full of learning opportunities.

Best wishes for the New Year, from Ken Roberts, 31-Dec-2022

What does LASI mean ?

LASI stands for Logic, Astronomy, Science and Ideas. This website has my thoughts about topics in math, physics, other science and astronomy, puzzles, reading, and other ideas — including a few stories and poems. Perhaps some of this will interest you.

Comments are moderated and may take a long time to appear, depending upon my activities.  If you want to get in touch, please feel free to email me at krobe8@outlook.com

Best wishes, from Ken Roberts