Musings

58

2023-11-22T00:00:00+01:00

111010₂ = 2011₃ = 322₄ = 213₅ = 134₆ = 112₇ = 72₈ = 64₉ = 58₁₀ = 53₁₁ = 4a₁₂ = 46₁₃ = 42₁₄ = 3d₁₅ = 3a₁₆ = 37₁₇ = 34₁₈ = 31₁₉ = 2i₂₀ = 2g₂₁ = 2e₂₂ = 2c₂₃ = 2a₂₄ = 28₂₅ = 26₂₆ = 24₂₇ = 22₂₈ = 20₂₉ = 1s₃₀ = 1r₃₁ = 1q₃₂ = 1p₃₃ = 1o₃₄ = 1n₃₅ = 1m₃₆ = 1l₃₇ = 1k₃₈ = 1j₃₉ = 1i₄₀ = 1h₄₁ = 1g₄₂ = 1f₄₃ = 1e₄₄ = 1d₄₅ = 1c₄₆ = 1b₄₇ = 1a₄₈ = 19₄₉ = 18₅₀ = 17₅₁ = 16₅₂ = 15₅₃ = 14₅₄ = 13₅₅ = 12₅₆ = 11₅₇ = 10₅₈

[image generated by AI]

57

2022-11-22T00:00:00+01:00

56

2021-11-22T00:00:00+01:00

Sum of six consecutive primes (3 + 5 + 7 + 11 + 13 + 17)

Sum of the first six triangular numbers (1 + 3 + 6 + 10 + 15 + 21)

Sum of the divisors of the first 8 positive integers

Maximum determinant in an 8 by 8 matrix of zeroes and ones

Only known number n such that φ(n-1)σ(n-1) = φ(n)σ(n) = φ(n+1)σ(n+1), where φ is Euler’s totient function and σ is the sum of the divisor function

Number of bits in a key used in the Data Encryption Standard

Number of signatories of United States Declaration of Independence

Number of curls in Shirley Temple’s hair

Much more personal significance, besides my current age, the atomic number of barium – no wonder I don’t lose weight! #GreekEtymologyJoke

[image generated by a 56-line script, using 56 open source fonts]

Greek flag

2021-03-25T00:00:00+01:00

For the celebrations of the 200 years of the start of the Greek revolution against the Ottoman empire, a lot of Greek flags were posted on social media.

Although the flag is not so simple like the French or German, for example, it can easily be constructed by using the basic drawing primitives of SVG.

Trying to draw it by hand-crafted SVG directives, I thought it would be fun to minimize. I had never delved into SVG before, so I had to read the specification.

The design of the Greek flag is defined in a law.

The result, as minimal as I could make it, was:

<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 27 18">
<rect fill="#005bae" width="27" height="18"/>
<path stroke="#fff" stroke-width="2" d="M0,5h10 M5,0v10 M10,3h17 M10,7h17 M0,11h27 M0,15h27"/>
</svg>

The SVG in a file can be found on this document.

Reading the commands, one can easily see that they create a blue rectangle (which will serve as background and therefore show all blue parts), and then draw thick white lines: first the cross and then the horizontal stripes.

Obvious, once one sees the final result.

Update

After publishing the above SVG, I was challenged by a friend to also animate it.

I believe SVG is not sufficient for this, but of course there is no need to use JavaScript. Some CSS animation primitives were enough.

I computed the parameters of 24 different positions (“frames”) for an animation loop. I also drew a flag pole, since it looked very strange without it.

The final result, which is a single file with no external dependencies, can be found on this link.

The Anti-N-Queens Problem

2021-02-28T00:00:00+01:00

The N-Queens problem is a well-known problem and it consists of placing N non-attacking queens on an NxN chessboard. It is solvable for any n > 3.

In February 2021, I explored a related similar problem, tentatively named Anti-N-Queens. A simple formulatin of the problem is the following:

What is the maximum number of “safe” squares when N queens are placed on a board?

A square on the board is considered safe when it is not threatened by any queen on the board. There is no restriction that the queens are not attacking each other, nor (obviously) that any rule of chess must be followed.

The complete results for two variations of the problem can be found on this document. The same paper has been published online and has a DOI of 10.5281/zenodo.4672187

Update

My results for the maximum number of safe squares when placing N queens on a standard 8x8 chessboard were accepted and published in the Online Encyclopedia of Integer Sequences (OEIS) as sequence A342151.

55

2020-11-22T00:00:00+01:00

Largest Fibonacci number to also be a triangular number.

2-Kaprekar number in base 10, but also in bases 6, 16, and 46.

Square pyramidal number heptagonal number, and centered nonagonal number.

54

2019-11-22T00:00:00+01:00

53

2018-11-22T00:00:00+01:00

In hexadecimal: 35 (same digits, reversed)

Prime number, self number (in base 10), Eisenstein prime, Sophie Germain prime, and it divides the sum of the first 53 primes (A045345).

Not a bad age to be in…

27

2018-09-07T00:00:00+02:00

27: age in which Brian Jones, Jimi Hendrix, Janis Joplin, Jim Morrison, Kurt Cobain, and Amy Winehouse died.

27: years that our marriage is alive.