Cut-the-Knot

In my previous post titled Four Fours Representation of 32, I made reference to a site called Cut-the-Knot. In this post, I'll look at the site more closely and I feel it certainly deserves closer attention. Here is what one reviewer wrote about the site.

The Web may contain almost every possible problem, puzzle, and article imaginable, but it’s decentralized nature makes it’s hard to locate good content in a sea of endless tutorials, amusing pictures, and commercial promotions. If you’re trying to find extracurricular mathematical materials you need to know where to look, but more importantly, what to look for. Knowing an erudite guide makes life much easier. Alexander Bogomolny, a professional mathematician and curator of mathematical recreations and other topics, is that guide. His site Cut-the-Knot is an enormous collection of fascinating articles, illustrations, and animations covering a wide range of mostly non-advanced mathematics. One of the defining features of his articles are the interactive Java applets that illustrate a problem or principle. The site has been continuously updated since 1997, which makes it among the most comprehensive such repositories online. Unfortunately, because it was created more than fifteen years ago, its age shows in the design and technology used (Java applets are no longer the preferred delivery mechanism for interactive media). Although Cut-the-Knot has garnered over twenty awards, including one from Scientific American, it is not as well known as it should be. If you’re looking for a source of enrichment for regular math classes this is one of the best places to start.

In this post, I'll just look at the first problem that occurs under the subject of Arithmetic and this is 100 Grasshoppers on a Triangular Board. Here is the problem statement:

A triangular board has been cut into 100 small triangular cells by the lines parallel to its sides. Two cells that share a side are said to be neighbours. In each cell there is a grasshopper. All at once, the grasshoppers hop from their cells to neighboring cells. This happens 9 times. Prove that at least 10 cells are now empty.

The solution is as follows:

The board is naturally coloured into two colours so that the neighbouring cells are coloured differently. Let the colours be red and brown and label the grasshoppers by the colour of their cells. Convince yourself that there are 55 red and 45 brown cells. In one hop, the brown grasshoppers move to the red cells thus emptying 55 cells. On the same hop, the brown grasshoppers move to the red cells thus filling at most 45 red cells. Inevitably, at least 10 red cells will remain empty.

The next time the grasshoppers move, they can move into their original cells, thus filling all of them. The situation will return to the original one, while the argument will apply to every odd hop. Perhaps there may happen a greater mix-up of the red and brown hoppers, but the best that may be claimed is that after every odd hop, there are at least 10 empty cells.

In general, there are \(n(n+1)/2\) red cells and \(n(n-1)/2\) brown cells. The difference being \(n(n+1)/2 - n(n-1)/2 = n\), there are always at least \(n\) empty cells after an odd number of hops.

The specific number of hops (9) is nothing but a red herring.

So, an interesting little problem, and of course this site is replete with many more, 244 in fact under the subject heading of Arithmetic. The subject headings are:

• Arithmetic with 244 articles
• Algebra with 1059 articles
• Geometry with 1824 articles
• Probability with 204 articles
• Trigonometry with 36 (unnumbered) articles
• Visual Illusions with 51 (unnumbered) articles

There is a glossary with more articles and links to many more. Unfortunately, the author of the site, Alex Bogomolny, died in 2018 at the age of 70 but he has left behind a treasure trove of mathematical information. 

Dr. Alexander Bogomolny, May 2017
(In Costa Rica, holding a sloth)

Here is what his Wikipedia entry had to say:

Alexander Bogomolny (January 4, 1948  – July 7, 2018) was a Soviet-born Israeli American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Mathematics, senior instructor at Hebrew University and software consultant at Ben Gurion University. He wrote extensively about arithmetic, probability, algebra, geometry, trigonometry and mathematical games.

He was known for his contribution to heuristics and mathematics education, creating and maintaining the mathematically themed educational website Cut-the-Knot for the Mathematical Association of America (MAA) Online. He was a pioneer in mathematical education on the internet, having started Cut-the-Knot in October 1996.

Bogomolny attended Moscow school No. 444, for gifted children, then entered Moscow State University, where he graduated with a master's degree in mathematics in 1971.[3] From 1971 to 1974 he was a junior research fellow at the Moscow Institute of Electronics and Mathematics. He emigrated to Israel and became a senior programmer at Lake Kinneret Research Laboratory in Tiberias, Israel (1974 – 1977) and a software consultant at Ben Gurion University in Negev, Be’er Sheva, Israel (1976 – 1977). From 1976 to 1983 he was a Senior Instructor and researcher at Hebrew University in Jerusalem. He received his Ph.D. in mathematics at Hebrew University in 1981. His dissertation is titled, A New Numerical Solution for the Stamp Problem and his thesis advisor was Gregory I. Eskin. From 1981 to 1982 he was also a Visiting Professor at Ohio State University where he taught mathematics.

From 1982 to 1987 he was Professor of Mathematics at the University of Iowa. From August 1987 to August 1991 he was Vice President of Software Development at CompuDoc, Inc.

Cut-the-knot (CTK) was a free, advertisement-funded educational website which Bogomolny maintained from 1996 to 2018. It was devoted to popular exposition of various topics in mathematics. The site was designed for teachers, children and parents, and anyone else curious about mathematics, with an eye to educating, encouraging interest, and provoking curiosity. Its name is a reference to the legend of Alexander the Great's solution to the Gordian knot.

CTK won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003, the Encyclopædia Britannica's Internet Guide Award, and Science's NetWatch award.

The site was remarkably prolific and contained extensive analysis of many of the classic problems in recreational mathematics including the Apollonian gasket, Napoleon's theorem, logarithmic spirals, The Prisoner of Benda, the Pitot theorem, and the monkey and the coconuts problem. Once, in a remarkable tour de force, CTK published 122 proofs of the Pythagorean theorem.

Bogomolny did indeed entertain but his deeper goal was to educate. He wrote a manifesto for CTK in which he said that "Judging Mathematics by its pragmatic value is like judging symphony by the weight of its score."[11] He describes the site as "a resource that would help learn, if not math itself, then, at least, ways to appreciate its beauty." And he wonders why it is acceptable among otherwise well-educated people "to confess a dislike and misunderstanding of Mathematics as a whole."

Many mathematical ideas are illustrated by applets. CTK wiki (powered by PmWiki) extends the main site with additional mathematical content, especially that with more complicated formulae than available on the main site.

Bogomolny had to leave academia because he had an uncorrectable hearing problem and was practically deaf in latter years. He is survived by his wife Svetlana Bogomolny, two sons David (Israel) and Eli (USA) Bogomolny, and granddaughter Liorah Shaindel Bogomolny.

Four Fours Representation of 32

Let's suppose that \(n\), \(x\) and \(y\) are positive integers such that:$$n^{\, x+y}=x||y$$where \(x||y\) represents the concatenation of \(x\) and \(y\). What values of \(n\), \(x\) and \(y\) satisfy this equation? Well it seems that there is only one set of values, namely \(n=2\), \(x=3\) and \(y=2\) where$$2^{3+2}=32$$This curious fact struck me when looking at the periodicity of conjunctions of Mars and Venus, that occur in almost the same location in the tropical zodiac at intervals 32 years (+0.5 to 4 days). See my post Periodicity of Mars-Venus Conjunctions. 

This got me thinking about what other interesting properties the number 32 might possess. Naturally I turned to Numbers Aplenty to find out.

32 can be written using four fours as in \( (4+4)^{\! ^{\frac{\sqrt{\overline{.4}}}{.4}}} \) where:$$\sqrt{\overline{.4}}=\sqrt{\frac{4}{9}}=\frac{2}{3}\\\ \frac{\frac{2}{3}}{.4}=\frac{\frac{2}{3}}{\frac{2}{5}}=\frac{5}{3}\\(4+4)^{\! ^{\frac{5}{3}}}=8^{\! ^{\frac{5}{3}}}=\left ( 2^3 \right )^{ \!\frac{5}{3}}=2^5=32$$Actually this is not a property unique to 32. In fact, all the numbers up 112 can be represented thus and 113 is the smallest natural number that cannot be obtained using four fours, the common arithmetic operations, factorial, roots, and the notations$$.4=0.4=\frac{2}{5} \text{ and }\overline{.4}=0.4444\dots=\frac{4}{9}$$The use of the vinculum or overline is potentially confusing because the symbol is also used for the rising factorial as in:$$4^{\overline{4}}=4 \times 5 \times 6 \times 7=840$$I've always used the overdot as in \( 0.\dot{4}=0.4444 \dots\) to represent the repeating decimal. 

There are other ways to represent 32 using four fours:$$4!! \times 4 + 4 -4 = 32\\4^{\sqrt{4}}+4^{\sqrt{4}}=32\\4 \times 4+4 \times 4=32$$The use of the double factorial can be noted in the previous example, viz. 4!!= 4 x 2 = 8. When attempting the four fours representation, it needs to be made clear what's allowable. If the simple factorial is allowed (4!= 4 x 3 x 2 = 24), then it needs to be determined whether the double factorial (4!!=4 x2) and subfactorial (!4 = 9) are also permitted. I've written about the various types of factorials in my post Subfactorials, Semifactorials and Others. It's also in this post that I address the problem of representing 10 using three threes, which involves the use of the subfactorial (!3=2).

The key to solving these puzzles is the collect various building blocks. These might include (depending on what's allowed):

• \(\sqrt{4}=2\)
• \(4!=24\)
• \(4!!=12\)
• \(4^{\overline{4}}=840\)
• \(!4=9 \)
• \(.\dot{4}=0.4444 \dots =4/9 \)

Four fours is just one of a variety of mathematical puzzles that attempt to represent the counting numbers in terms of a fixed number of identical digits, according to certain prescribed rules. Here are links to a variety of representations:

• Representation of numbers with three 3's
• Representation of numbers with three 4's
• Representation of numbers with three 5's
• Representation of numbers with four 3's
• Representation of numbers with four 4's
• Representation of numbers with four 5's
• A problem of representing 6 by three identical digits

Digital Roots and Additive Persistence

To quote from my recent post titled SOD ET AL (Sum Of Digits And Other Things) on June 29th 2021:

DIGITAL ROOT 

While I've not made a specific post about digital roots, I've nonetheless mentioned them in the following posts:

• Heptagonal Numbers
• Star Numbers
• The Goldbach Conjecture and Lucky Numbers
• An Unhappy Family

To quote from Wikipedia:

The digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum. The process continues until a single-digit number is reached. In base 10, this is equivalent to taking the remainder upon division by 9 (except when the digital root is 9, where the remainder upon division by 9 will be 0).

Associated with the digital root is the concept of additive persistence defined as:

The additive persistence counts how many times we must sum its digits to arrive at its digital root. For example, the additive persistence of 2718 in base 10 is 2: first we find that 2 + 7 + 1 + 8 = 18, then that 1 + 8 = 9. 

Recently I had cause to visit digital roots again in the context of a new sequence that I devised in response to finding a meaningful OEIS sequence associated with the number 26396 which factorises to 2 * 2 * 6599 and that has prime factors of 2 and 6599. I noticed that both of these prime factors have a digital root of 2. This gave me the idea for the following sequence:

S003: Numbers with more than one prime factor such that the digital root of all prime factors is the same.

I developed an algorithm (permalink) to determine all such numbers up to 30,000 and it turned out that there are 1658 numbers in that range, constituting 6.29%. The members of the sequence below 1000 are:

22, 44, 58, 88, 94, 115, 116, 166, 176, 188, 202, 205, 232, 242, 274, 295, 301, 319, 332, 346, 352, 376, 382, 403, 404, 427, 454, 464, 484, 517, 526, 548, 553, 562, 565, 575, 634, 638, 655, 664, 679, 692, 703, 704, 706, 745, 752, 764, 778, 808, 835, 871, 886, 901, 908, 913, 922, 928, 943, 958, 968

It was a short step then to my next sequence where membership is a little more exclusive:

S004: Numbers with more than one prime factor such that the digital root of all prime factors is the same as the digital root of the number itself.

Here is the permalink to the algorithm that I developed. It turns out that there are 106 numbers in the range up to 30,000 and these constitute 0.402 % of the range. Below 1000, there are only two numbers that satisfy this criterion and interestingly they form a pair.

703 = 19 * 37 where 19, 37 and 703 all have a digital root of 1 and 704 = 2^6 * 11 where 2, 11 and 704 all have a digital root of 2. In the range up to 30,000, there are two other pairs:

14527 and 14528

29503 and 29504

The 106 members of this sequence in the range up to 30,000 are:

[703, 704, 1387, 1856, 2071, 2413, 2701, 3008, 3097, 3439, 3781, 3872, 4033, 4699, 5149, 5312, 5833, 6031, 6464, 6697, 7201, 7363, 7543, 7957, 8227, 8768, 9253, 9271, 9937, 10027, 10208, 10279, 10963, 11072, 11359, 11647, 11899, 11989, 12224, 13213, 13357, 13843, 14023, 14041, 14383, 14527, 14528, 14689, 14749, 15317, 15409, 15751, 16021, 16544, 16777, 16832, 17461, 17767, 17803, 17984, 18019, 18829, 19171, 19351, 19729, 19783, 20017, 20197, 20288, 20519, 20701, 20923, 21223, 21296, 21349, 21691, 21907, 22249, 22411, 22592, 22681, 22987, 23347, 24301, 24643, 24896, 25273, 25721, 26011, 26353, 26912, 27037, 27097, 27343, 27667, 27721, 28009, 28352, 28981, 29089, 29216, 29431, 29503, 29504, 29539, 29773]

 Figure 1 shows a plot of these numbers:

Figure 1

I'll make a note about additive persistence. The distribution from 0 to 30,000 is:

• 10 numbers have a digital persistence of 0 (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
• 1531 have a digital persistence of 1
• 25292 have a digital persistence of 2
• 3168 have a digital persistence of 3

The smallest numbers to have persistences of 0, 1, 2 and 3 are 0, 10, 19 and 199 respectively. The first number with an additive persistence of 4 is 19999999999999999999999. Beyond that, the numbers are ridiculously large. Figure 2 shows the relative proportions:

Figure 2

Beyond the OEIS

OEIS stands for the Online Encyclopaedia of Integer Sequences and, in my study of the integers associated with my diurnal age, I've found it to be an invaluable resource. Any registered person can propose a new sequence and in my initial enthusiasm, I decided to do just that. However, I was met with a firm rebuff and accused of creating a vanity sequence based on my diurnal age. It became quickly apparent that the OEIS is zealously guarded by a small group of individuals who must be approached in a certain manner.

I registered again under a new name and approached the arbiters in a less open and more guarded manner and succeeded in having my future submissions approved. It still rankled that I had to adopt such an approach and for some time now I haven't bothered to make any further submissions. Just lately, it occurred to me that I could concoct my own sequences, store them on Google Docs and even make them shareable and discoverable by others with no approval from the OEIS required.

The OEIS is a great resource and Neil Sloane is to be congratulated on being the inspiration behind its inception but he is quite old now and the duty of care for his creation has passed on to younger people. Unfortunately, the clique who now control the OEIS has created an unwelcoming atmosphere that discourages submissions by new contributors. Given my own advanced age and consequent life experiences, I was quickly able to discern what approach was needed in order to get these little Napoleons on side. This is why I registered again under a new name.

Such accommodations shouldn't be necessary and many younger contributors would be put off after their first rebuff. It's probably time for a new database of sequences to be created. Legally, it shouldn't be possible to own a sequence of numbers and so anybody or any group should be able to extract whatever sequences they want and start afresh. So far the sequences that I've created are not currently in the OEIS and I developed them because the existing OEIS entries held no interest for me.

Here is the link to the Google document that I've created:

https://docs.google.com/document/d/1pFqKbLCbhNCZ6euoUavt5QucPHfLRdiicWH_OHBw1L4/edit?usp=sharing

I'm not suggesting that there's anything amazing about these sequences but they're just as meaningful as a lot of the rubbish sequences that clutter up the OEIS today. To get an idea of the flavour of what I've created, here is a link to my latest sequence. Follow the link to see all five of them. These are early days yet and I'm likely to modify the layout in the future but for now:

S005: Composite numbers for which the set of digits associated with their prime factors is equal to the set of digits associated with the numbers themselves.

This sequence is a subset of S001.  

The first member of this sequence is \(132 = 2^2 \times 3 \times 11\) with the set of digits associated with 132 being {1, 2, 3} and the set of digits associated with its prime factors of 2, 3 and 11 also being {1, 2, 3}. 

 

The second member of the sequence, 312, has digits that are a permutation of the first member and that thus form the same set {1, 2, 3}. \(132 = 2^3 \times 3 \times 13\) and so its prime factors are 2, 3 and 13 with the associated set of digits being {1, 2, 3}. 

 

An algorithm (permalink) to generate the members of the sequence in the range from 1 to 30,000 is: 

 

L=[]
lower,upper=1,30000
for n in [lower..upper]:
    if is_prime(n)==0:
        D=Set(n.digits())
        DS=D.subsets()
        P=prime_factors(n)

        PS=[]

        for p in P:

            PS+=p.digits()

            if Set(PS)== D:

                L.append(n)

print("There are",len(L),"eligible numbers in the range from",lower,"to",upper,":")

print("This constitutes",numerical_approx(len(L)/(upper-lower)*100,digits=3),"percent of the range")

print()

print(L)

There are 61 eligible numbers in the range from 1 to 30000 :

This constitutes 0.203 percent of the range.

[132, 312, 735, 1255, 1377, 1775, 1972, 3792, 4371, 4773, 5192, 6769, 7112, 7236, 7371, 7539, 9321, 11009, 11099, 11132, 11163, 11232, 11255, 11375, 11913, 12176, 12326, 12595, 12955, 13092, 13175, 13312, 13377, 13491, 13755, 14842, 15033, 15303, 15317, 15532, 16332, 17272, 17276, 17343, 17482, 17973, 17975, 19075, 19276, 20530, 21345, 21372, 22413, 22714, 23535, 24338, 25030, 25105, 27232, 27393, 27944]

ADDENDUM September 2nd 2021

It seems I'll never learn. I made the mistake of proposing another sequence to the OEIS, throwing myself at the mercy of the little Napoleons guard its gates. Admittedly I'd made a couple of errors and had fixed them but then somebody asked why I thought the sequence was interesting. Hmmm. I saw where it was going so I bailed out, never to return again. I've learned my lesson.

ADDENDUM October 5th 2025

I just noticed that I made a post titled One of a Kind? on the 21st January 2024 in which I note that the sequence is actually in the OEIS database. This is ironic because I was touting this sequence as my own discovery and yet it had already been discovered! Not to worry, it's the same sequence whether in my own database or the OEIS's.

Euler–Mascheroni constant and the Meissel–Mertens constant

I've not written explicitly about either the Euler–Mascheroni constant or the Meissel–Mertens constant before, although the former is made mention of in a Numberphile video that I referenced in a post titled The Harmonic Series on October 12th 2016. 

Let's recount that the harmonic series is simply \(\zeta(1)\) and so:$$\zeta(1)=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+ \dots =\sum_{n=1}^{\infty}\frac{1}{n}$$While this sequence does diverge it does so very slowly and that's what my post The Harmonic Series was all about. The Euler-Mascheroni constant can be defined as:$$\begin{align}

\gamma &= \lim_{n\to\infty}\left(-\log n + \sum_{k=1}^n \frac1{k}\right)\\

&=\int_1^\infty\left(-\frac1x+\frac1{\lfloor x\rfloor}\right)\,dx.

\end{align}$$Here, \(\lfloor x\rfloor\) represents the floor function. The numerical value of the Euler–Mascheroni constant, to 50 decimal places, is:

0.57721566490153286060651209008240243104215933593992... 

Below I've embedded the Numberphile video referred to earlier as it's really quite informative.

Like the harmonic series, the sum of the reciprocals of the prime numbers diverges also and even more slowly. The Meissel-Mertens constant is defined as:$$M = \lim_{n \rightarrow \infty } \left( \sum_{p \leq n} \frac{1}{p} - \ln(\ln n) \right)=\gamma + \sum_{p} \left[ \ln\! \left( 1 - \frac{1}{p} \right) + \frac{1}{p} \right]$$where \( \gamma \) is the Euler-Mascheroni constant. The value of M is approximately:

M ≈ 0.2614972128476427837554268386086958590516... 

Figure 1: source

The two constants are thus intimately linked. It's easy to generate approximations of these functions using SageMathCell. See Figure 2.

Figure 2: permalink

Looking at the results in Figure 2, it can be seen that:

Approximation of Euler-Mascheroni constant up to 100000 is 0.577220664893197

Approximation of Miessel-Mertens constant up to 100000 is 0.261801821365208

The light grey digits do not correspond to the known digits for these constants. It can be seen that the approximation to the Miessel-Mertens constant is less accurate than for the Euler-Mascheroni constant, reflecting the log(log) computation for the former versus the log computation for the latter.

For a post that shows how to determine the sum of the alternating harmonic series, see my post titled Alternating Series Test from April 23rd 2021. The alternating harmonic series converges thus:$$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4} \dots=\ln(2) \approx 0.693147180559945 \dots$$See also:

• Biography of Ernet Meissel
• Biography of Franz Mertens
• Biography of Lorenzo Mascheroni
• Biography of Leonhard Euler

Time Signatures and Staves

Part 2 of Mathematics and Music

In my post of June 29th 2021 titled Equal Temperament Tuning, we saw that ratios were at the heart of heptatonic scales like the major scale (see my post of July 2nd 2021 titled Concrete, Laplace Resonance and Heptatonic Musical Scales). Figure 1 is taken from the former post.

Figure 1

As can be seen from Figure 1, the ratios 9:8, 5:4, 4:3, 3:2, 5:3 and 15:8 are most prominent. It might seem that ratio/fractions appear in musical time signatures. To quote from Wikipedia:

Simple time signatures consist of two numerals, one stacked above the other. The lower numeral indicates the note value that represents one beat (the beat unit). This number is typically a power of 2. The upper numeral indicates how many such beats constitute a bar.

For instance:

• \( \frac{2}{4} \) means two quarter-note (crotchet) beats per bar
• \( \frac{3}{4} \) means three quarter-note (crotchet) beats per bar
• \( \frac{4}{4} \) means four quarter-note (crotchet) beats per bar
• \( \frac{3}{8} \) means three eighth-notes (quavers) per bar

Figure 2 gives an example of the use 3/4 time:

Figure 2

However, ratios compare two quantities of the same type, and here that's not the case. Time signatures are really musical shorthand combining two numbers representing different types of things (beat unit or length of beat on the bottom versus beats to the bar on the top). It looks like a fraction but it's not really.

The stave (a set of five parallel lines on any one or between any adjacent two of which a note is written to indicate its pitch) shown in Figure 2 has a Cartesian coordinate aspect to it where the vertical axis represents pitch and the horizontal axis represents time. See Figure 3.

Figure 3

Figure 4 shows the precise changes in pitch as we move along the time line in whole notes or semibreves:

Figure 4

The notes of course encode additional time-related information, namely how long the note is sounded. The position of the note on the time line simply denotes its position in the sequence of notes but tells us nothing about its duration. This is equivalent to marking a point on the 2-D Cartesian or (\(x-y\) plane with a specific shape that represents its position on the \(z\)-axis. Figure 5 shows the different types of notes:

Figure 5: source

Thus the ratio of semibreve to minim to crotchet to quaver to semiquaver is 16 : 8 : 4 : 2 : 1 or \(2^4 : 2^3 : 2^2 : 2^1 : 2^0\). As with the differences in frequencies between notes, the powers of 2 predominate here also.

Figure 6 shows a table of simple time signatures while Figure 7 shows a table of compound time signatures.

Figure 6: source

Figure 7: source

There's great complexity to be found as we delve deeper into musical theory but that's probably sufficient detail for this post.

Concrete, Laplace Resonance and Heptatonic Musical Scales

My son turned 36 today while my granddaughter is aged 18 and I'm aged 72. I couldn't help noticing that the ratio of these ages 18 : 36 : 72 reduces to 1 : 2 : 4 in its simplest form.

This set me on a course to find out what was interesting, mathematically and otherwise, about this ratio. If you search for these proportions, the first thing that pops up is, oddly enough, concrete. See Figure 1.

Figure 1: source

Yes, it turns out that 1 : 2 : 4 is the ratio of cement : sand : aggregrate, by volume, required to make M15 concrete. It is also known as PCC (Plain Cement Concrete) and can be used in construction of Levelling course, bedding for footing, concrete roads, etc. The site from which the table in Figure 1 was taken has lots of calculations for figuring out weights etc. but I'm not going to go into here as it's rather boring.

When I searched for 1 : 2 : 4 planetary motion, I came up with something more interesting: 

A Laplace resonance is a three-body resonance with a 1 : 2 : 4 orbital period ratio (equivalent to a 4 : 2 : 1 ratio of orbits). The term arose because Pierre-Simon Laplace discovered that such a resonance governed the motions of Jupiter's moons Io, Europa, and Ganymede. It is now also often applied to other 3-body resonances with the same ratios, such as that between the extrasolar planets Gliese 876 c, b, and e. Three-body resonances involving other simple integer ratios have been termed "Laplace-like" or "Laplace-type".

Figure 2 shows a simulation of the Jovian moons' Laplace resonance (source):

Figure 2: The three-body Laplace resonance exhibited
by three of Jupiter's Galilean moons. Conjunctions are highlighted
by brief color changes.There are two Io-Europa conjunctions
(green) and three Io-Ganymede conjunctions (grey) for each
Europa-Ganymede conjunction (magenta). This diagram is not to scale.

The Laplace resonance is a particular type of orbital resonance and the eponymous Wikipedia article has a lots of examples of this and other types of resonances.

Intrinsic to the 1 : 2 : 4 proportions is the number 7. In fact (1, 2, 4) represents one of the 15 partitions of 7. This immediately to mind heptatonic musical scales:

A heptatonic scale is a musical scale that has seven pitches, or tones, per octave. Examples include:

• the major scale or minor scale in C major: C D E F G A B C

• the relative minor, A minor, natural minor: A B C D E F G A 

• the melodic minor scale, A B C D E F♯G♯A ascending, 

• the melodic minor scale, A G F E D C B A descending 

• the harmonic minor scale, A B C D E F G♯A 

• the Byzantine or Hungarian, scale, C D E♭ F♯ G A♭ B C.

So any partition of 7 can be viewed in terms of such a scale. Applying this to 1 : 2 : 4, there are six possible divisions. Figure 3 shows two of these:

Figure 3

I've written about musical scales more generally in my previous post titled Equal Temperament Tuning.