## Sunday, 31 October 2021

### Generalised Golden Ratios

Back in The Games and Puzzles Journal, #13 p.223 (May 1996), [the first issue of volume 2 which came out 25 years ago] the first puzzle was on "A Question of Proportion". It was about rectangles of various shapes including the standard paper sizes and the golden rectangle.

The 'Numberphile' videos on YouTube, some by Ben Sparks, have recently been featuring questions of this nature. I also borrowed from our local library the book One to Nine by Andrew Hodges which includes similar material, including the "Plastic" number.

Towards the end of the book Hodges implies that "Computable" numbers as defined in Alan Turing's famous paper are denumerable, that is they can be matched with the set of natural numbers.

This also raises the philosophical question of whether any other "real" numbers "really" exist, if they cannot be calculated  by a finite set of rules. Such numbers are those that appear in Cantor's diagonal proof of the nondenumerability of the continuum.

All the usual suspects like root 2, root 3, the golden section, e and pi and Euler's constant gamma are all computable numbers. This just means that they can be computed to any number of decimal places and that the calculation by whatever method always gives the same digit in the nth place for any n.

They really are real real numbers!

Going back to my puzzle in the G&PJoutrnal the last question asked was: What shape of rectangle gives a smaller rectangle of the same proportions when a rectangle of m squares by n squares is removed from one end?

The answer was given in G&PJournal #14 p.245 (December 1996). By applying the same methods as for the golden ratio I found a formula for such a generalised golden ratio, in terms of m and n.

It now occurs to me that we can denote the larger and smaller golden ratios by capital and lower case Phi as before, but with a suffix 1, and the generalised golden ratios by the same letters with a suffix m/n. This also indicates that these golden ratios all form a denumerable set since they can be placed in one-to-one correspondence with the ratios m/n, which are known to be denumerable.

The formulas are: Phi suffix m/n = [root (m^2 + 4.n^2) +/- m]/2.n

For example Large Phi suffix 1/2 = (root 17 + 1)/4 = 1.2807764...

Small phi suffix 1/2 = (root 17 - 1)/4 = 0.7807764...

Their difference is 1/2 and their product is 1 (or 0.99999...).

Some of these generalised ratios work out to be rational. The simplest case is when m=3, n=2 which gives the two Phi with suffix 3/2 as 2 and 1/2, but most of the ratios will be irrational.

ADDENDUM: It now occurs to me that the ratio m/n of the rectangle removed can in fact be any number not just  a rational fraction. And conversely the shape of the whole rectangle can be given by any number x, the rectangle removed being x - 1/x.

## Saturday, 9 October 2021

### Another Onitiu 18x18 Attempt

Here is another attempt at solving the Onitiu problem of constructing a knight tour with 90 degree rotational symmetry on the 18x18 board with the squares in a knight path (the red line). It uses two wazir moves in each quarter, equalling my previous result, making it an Emperor tour.

Onitiu Problem 18x18 Quaternary Emperor Tour

The pattern is considerably different from the previous example. So perhaps a solution is possible, if the exactly right configuration can be found. Or maybe two wazir moves per quarter is the best that can be done. Maybe someone could programme a computer to settle it.

## Wednesday, 22 September 2021

### Onitiu Problem 30x30 Solution

This was the most difficult to solve so far. It shows a knight tour with 90 degree rotational symmetry with the square numbers in a closed knight path (the red line). There are sixteen cells (marked by squares) where the four circuits intersect. The initial and final cells of each circuit are marked by a darker square (001 and 900 in the case of the circuit of square numbers).

Solution to Onitiu Problem 30x30

The thick black lines mark single-move links between the circuits. The dotted cells mark the links of two or three moves. These short links place considerable restrictions on the way the paths can be arranged.

Numerologists may notice that the numbers 666 and 216 (6 cubed) occupy diametrically opposite cells. This is because their difference is 450, half of 900.

## Friday, 17 September 2021

### Locked Out of Twitter Again

Twitter is again asking for a mobile phone number from me to sign in, but there doesn't seem to be any allowance for people who don't have mobile phones. I hope this is not becoming a general thing. It will mean that people without mobile phones are becoming second class citizens. I was also made to pass an "I'm Not A Robot" test, but apparently that was not sufficient!

## Thursday, 9 September 2021

### Another Onitiu Solution

After a couple of months I have at last solved another of the knight tours with 90 degree rotational symmetry and with the square numbers in a knight circuit (the red line). The type face for the numbers is necessarily vey small. They range from 001 to 676 which is the board size 26x26.

Onitiu 26x26 Birotary Solution

## Saturday, 28 August 2021

### Enumeration of Grid Points

I've been reading all my books that include some Number Theory, and doing some Study of Numbers, which I'm thinking of publishing under the title "Numerology: The Wisdom and Folly of Numbers".

Here is a little item that is probably not new, but I can't find it in any of my sources. Can anyone locate it in some publication? I've a vague idea I've seen something like it somewhere.

I call numbers of the form (2^r)x(3^s) "Basals". In other words they are any numbers that exclude prime factors greater than 3. .The sequence runs: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 98, 108, 128, ...

The corresponding powers of 2 and 3 in the sequence run: (0,0), (1,0), (0,1), (2,0), (1,1), (3,0), (0,2), (2,1), (4,0) and so on. Every pairing occurs and they are listed in a unique order.

This scheme thus provides an enumeration of the grid cells of an endless board, as partially illustrated in this tour diagram, from (0,0) = 1 to (9,0) = 512.

Enumeration of Coordinate Points

It is interesting that the path appears never to cross itself.

Further: This diagram shows the similar result obtained using (2^r)x(5^s) to determine the sequence. This goes up to 5^4 = 625 at (0,4). The moves in the upward direction tend to be knight moves.

Enumeration using primes 2 and 5.