How I'm Fixing the SEO for this Blog

I recently discovered Google Search Console, and learned to my absolute disappointment, that over two thirds of the pages on this blog aren't indexed by Google.  Well, it would explain why they don't get any traffic.  Worse still, it looks like nothing since 2020 was indexed.

So, here's what I've been doing to try and get my pages indexed, and attract more traffic to my blog.

1. Google Search Console - I've submitted my sitemap, several times, and added individual pages that have the best quality content (in my view).
2. Removed extraneous links on the page - the calendar of blog posts was diluting link juice and so that's gone.  Is it really relevant for you to know that I've been blogging here since 2011?  Probably not.
3. Tidied up the category labels - there was a whole cloud of these (literally) and I've reduced them to a manageable list, which I continue to prune.
4. Added group links on similar pages to show that they have a common theme - the Star Trek pages and calculator games pages first.  These now have 'Other pages you may be interested in' with links.  The Star Trek reviews and the Star Wars reviews all have links to the other episodes in the same season, showing Google that they're connected, they're not just 10 random pages.
5. Created static pages per category - Chess and Maths first - although these aren't getting much interest yet.
6. Submitted my site to Bing's webmaster tools and tracked traffic there - much easier, and much more straightforward.
7.  I've created external links to my site  - backlinks such as this one on Goodreads for Calculator Fun and Games.

Am I seeing an improvement in traffic? 

Not yet.

Am I giving up?

No!

Waterproof Electricity

Researchers at Oxford University proudly announced the development and successful testing of a new material which will conduct electricity even when underwater. The so-called 'waterproof electricity' is the result of a new type of plastic which will conduct an electric current but prevents any "leakage" of electric charges into the water.  Their findings, published in Materials Journal, mark a significant point in global materials development.

Historically, water has always been the biggest enemy of electrical devices. The only way to protect devices which are to be used underwater has been to physically coat them in a waterproof and airtight layer, leading to cumbersome and clunky devices, and as they to be operated underwater, this additional layer has made them particularly difficult to use.

Professor David Armstrong, the team leader at Oxford, explained, "As per recent information, we have been able to conduct electricity through our new material without any loss of current to the surrounding water. Clearly this opens up all kinds of applications, from underwater research to making domestic mobile devices waterproof." 

Dr Emily Turner, a senior researcher on the team, added, "The potential for this material is immense. We are looking at applications in underwater robotics, marine exploration, and even in everyday consumer electronics. The ability to have devices that are both electrically conductive and waterproof could revolutionize many industries."

Professor Mauro Pasta, another key researcher, emphasized the collaborative effort: "This project has been a true interdisciplinary endeavor, combining expertise from materials science, chemistry, and electrical engineering. The synergy between these fields has been crucial in achieving this breakthrough."

The research for the new polymer is based on PTFE (Teflon) which is water resistant, while having additional atom chains which enable it to conduct electricity.  Known as Fluoro-Ortho-Oxy Limonene, it's a highly oxygenated organic molecule formed from the oxidation of limonene. It features a unique structure that includes both closed-shell and open-shell peroxy radicals, which contribute to its exceptional properties.  Part of its structure is shown below. Its full chemical structure and further details will be released in an online article at noon today.

If you'd like to read more of my Chemistry articles, I can recommend my explanation of how I got into online A/B testing as a Chemistry graduate.

If this sounds like something out of Star Trek, there's probably a good reason for it.  

Airband Radio Aerials: Maths in Action

I've been interested in aircraft and airshows for over 40 years - anything military or civil, and I've blogged in the past about how to use a spreadsheet to track down where to watch the Red Arrows fly past on their transit flights.  You didn't think that post was about geometry without some real-life applications?  What is this - "Another day I haven't used algebra"?

Anyway - I've been particularly interested in the Red Arrows and their air-to-air chatter, and the communications between pilots and air traffic control.  Yes, I take my airband radio along to airshows and to airports, and listen to the pilots request and receive clearance to take off or land.  Getting to airports is more of a challenge than it used to be - my children aren't as interested as I am in the whole thing, and standing at the end of a runway in poor weather isn't as much fun as it sounds!

So, I've started developing my home-based receiver.  In other words, I spent my birthday money on an airband antenna and an extension cable to connect it from outside (cold and sometimes rainy) to my desk (warm and inside) so that I can listen to pilots flying nearby.

Now: nearby is a relative term.

From Stoke on Trent, I've been able to pick up pilot transmissions from about 35 miles away, on the southern edge of Manchester Airport.  That's with a very basic antenna, set on my garden gatepost and about two metres off the ground - not bad for a first attempt.

My dad, on the other hand, has been tracking radio transmissions for decades.  His main areas of interest are long wave (around 200 kHz), medium wave (500-1600 kHz), and TV (UHF, 300 Mhz to 3GHz).  

Airband falls into the Very High Frequency range, around 100-200 MHz.

Here comes the maths:
All radio transmissions travel at the speed of light, c = 2.998 * 10^8 ms-1.
c = f w

Where f (sometimes the Greek letter nu, ν) is the frequency, and w (usually the Greek letter lambda, ƛ) is the wavelength.

So, if we know the frequency range that we want to listen to, we can calculate the wavelength of that transmission.  And this is important, because the length of the antenna (or aerial) that we need will depend on the wavelength.  Ideally, the aerial should be the length of one full wavelength, for maximum reception effectiveness.  Alternatively, a half-wavelength or a quarter-wavelength can be used.

So:  we know the speed of light, c = 2.998 * 10^8 ms-1
And we know the frequency of the transmissions we want to receive, which is around 118 MHz.

c/ν = ƛ

ƛ =   2.5 metres

Which is feasible for an external, wall-mounted aerial.  Can you see where this is going?

Exactly.  And here it is:  

It's just over two metres from end to end, with a feed at the midpoint.  This is the Mark One; the Mark Two will be the same aerial but even higher up, and closer to vertical (with a bracket that will enable it to dodge the eaves of the roof!

Calculator Games: Ulam Sequences: Up, Up and Away!

 Up, Up and Away With Ulam Seqeunces

This article in the ongoing series of ‘mathematical puzzles you could investigate with a calculator’ (that’s why I just call it ‘Calculator Fun and Games’) is the Ulam Sequence.  Ulam sequences, named after mathematician Stanisław Ulam, are fascinating numerical sequences that begin with two specified integers. Each subsequent number in the sequence is defined as the smallest integer that is the sum of two distinct earlier numbers, where such a sum is unique within the sequence. This uniqueness constraint shapes the sequence's progression in an intricate way.

Ulam sequences are studied for their intriguing mathematical properties and their unpredictable, non-linear behavior, which challenges patterns typically found in additive sequences. They have applications in number theory and combinatorics, offering rich grounds for exploration and research.

Let's have a look at them in more detail, and start with the simplest.

How to Generate an Ulam Sequence

Take with two numbers (specifically, positive integers).  A good place to start is with a =1 and b=2.  To find the next number in the sequence, find the smallest integer that can be written as the sum of two distinct earlier numbers in just one way.  Continue with the next number, and the next.

For example, let’s start the Ulam Sequence with the numbers 1 and 2.  These are the first two terms in the sequence.

The next term is 3 (since 1+2=3).
After that comes 4 (since 1+3=4).

The next terms is not 5.  We can write 5 as 1+4 and as 2+3 using the terms that we’ve generated already.

The next term is 6 (since 2+4=6), and we can only write this in one way using our terms.

We can write 7 as 4+3 and as 6+1, so we skip 7.

The next term is 8 (since 2+6=8).

So, the beginning of the sequence is: 1,2,3,4,6,8,… (and it continues
1,2,3,4,6,8,11,13,16,18,26,28,36,38,47,48,53,57,62,69,72,77,82,87,97,99,102)

Note that 5, 7, 9, 10, 12, 14, 15, 17, 19, 20, 21 and 22 can be obtained in multiple ways using the terms before them.

However:  23 is not in the sequence because it cannot be obtained using the previous terms at all!    24 can be written as both 16+8 and 18+6, while 25 is not obtainable.

The sequence grows in an irregular, almost random pattern.  Let’s see what happens when we start with 1 and 3 instead of 1 and 2.

4 = 1 + 3 only
5 = 1 + 4 only
6 = 5 + 1 only
7 can be written as 4+3 and 6+1
8 = 5+3
9 is 4+5 and 6+3
10 = 6+4
11 is 6+5 and 8+3

The first 20 terms for the 1,3 Ulam Sequence are:

1,3,4,5,6,8,10, 12,17, 21, 23, 28, 32,34,39,43,48,52,59 and 63.

The Ulam Sequence is an interesting example of how simple rules can lead to complex and intriguing mathematical structures, which makes it ideal for calculator (or spreadsheet) exercises.  For example, here’s a comparison of the Ulam sequences for 1,2 compared with 1,3 (as I’ve calculated above) and then 1,4 and 1,5.  Interestingly, the 1,5 sequence does not race ahead of the 1,2 sequence as I had originally expected.

Term	Ulam (1,2)	Ulam (1,3)	Ulam (1,4)	Ulam (1,5)
1	1	1	1	1
2	2	3	4	5
3	3	4	5	6
4	4	5	6	7
5	6	6	7	8
6	8	8	8	9
7	11	10	10	10
8	13	12	16	12
9	16	17	18	20
10	18	21	19	22
11	26	23	21	23
12	28	28	31	24
13	36	32	32	26
14	38	34	33	38
15	47	39	42	38
16	48	43	46	40
17	53	48	56	41
18	57	52	57	52
19	62	59	66	57
20	69	63	70	69

There’s a balance between the ability to leap to larger numbers (1,5) initially – from 1 to 5 – and the need to fill in more numbers between 5 and 10 (because there are very smaller numbers that can be made in multiple ways).

A quick comparison of the Ulam Sequences for (2,b) is even more interesting.  We have to start with 2,3 since 2,1 is the same as 1,2 above, and 2,2 will only produce the even numbers (which is cute but dull).  In fact, any even number paired with 2 will produce uninteresting results!

Let’s compare 2,3 and 2,5:  These grow at a slower rate compared to the 1,b sequences.  Interestingly, they contain far fewer even numbers than the 1,b sequences; in fact 2,5 only contains the even numbers 2 and 12 in the first 20 terms (with no indication that there are any more even numbers further along the sequence).

Term	2,3	2,5
1	2	2
2	3	5
3	5	7
4	7	9
5	8	11
6	9	12
7	13	13
8	14	15
9	18	19
10	19	23
11	24	27
12	25	29
13	29	35
14	30	37
15	35	41
16	36	43
17	40	45
18	41	49
19	46	51
20	51	55

So there’s plenty of scope for investigation with a spreadsheet for the larger numbers.  For example, I haven’t found anybody else list the Ulam sequence for 10,11… so here it is:  the Ulam Sequence for (10,11)

10, 11, 21, 31, 32, 41, 43, 51, 54, 61, 62, 65…

After huge initial leaps of +10 or +11 between consecutive terms, the growth rate of the sequence starts to slow down.  There is only one term in the 20s, then two in the 30s, 40s and 50s, then three in the 60s.

Further reading:
Wolfram has lists and links for many of the 1,b and 2,b Ulam sequences.

Other articles on this blog on similar themes:
Snakes and Ladders (Collatz Conjecture)
Crafty Calculator Calculations (numerical anagrams with five digits)
More Multiplications (numerical anagrams, four digits)
Over and Out (reduce large numbers to zero as rapidly as possible)
Calculator Games: Front to Back
Calculator Games: The Kaprekar Constant

Game review: Pocket Mars

The game Pocket Mars is, in our experience, more of a puzzle than a game. We found it difficult to win (by getting all our colonists to Mars). The challenge wasn't beating the other players, it was achieving the win conditions at all.  Any thoughts of competition were dwarfed by the challenge of getting to the finish line by any means possible.

In fact, it was so difficult that we only played it twice before returning it to the charity shop from whence it came. 

In our opinion, a game needs to have a high level of player interaction.  My race to win should come at the detriment of your chances of winning.  Snakes and Ladders has no interaction, but at least it's easy and you can feel like you're competing.  However, a good game for us isn't Snakes and Ladders, where we have no interaction at all, and it's just you and the dice, then me and the dice, and we could add dozens of players without affecting the game at all.  No.  Player interaction is key - I want to win because I beat you, not just because I rolled bigger numbers or drew better cards compared to you.

And Pocket Mars feels like it has no player interaction at all - you take a turn, I take a turn.  We launch our men to Mars, we take turns to draw cards, and we see who can do the best.  It is, at best, a complicated puzzle where we each battle the rules and restrictions of the game in order to get the best outcome we can, and then compare against the other players.  The game talks about sabotaging the opponents, but this didn't happen when we played, and the game lacked any 'fun'.  It was too difficult to make any real progress in the game, and this meant that the return on effort was too high and we promptly gave up.

Nope, this game was not for us, which was a shame because it's been produced to a very high level of quality.  The box, cards and pieces all looked and felt great, but I feel that they could have been reorganised into a game that would be a lot more fun.  If you're looking for something simpler that has some interaction and is more enjoyable, I can recommend the Rollorama Football Dice Game which I review recently.

Calculator Games: Front to Back

This puzzle is not from the Calculator Fun and Games book, but another one that would be suitable (if you had time).  Early indications and preliminary reading indicate that this one could take some time to complete, but let's wade in anyway.

The game would be described in this way:

Take a number (three digits to start with)
Reverse the digits to form a new number
Add the two numbers together (this makes a change from subtracting...!)
If the new number is not a palindrome, continue reversing digits and adding.

Let's start with 456

456+654 = 1110

1110+0111 = 1221 so it's a palindrome.

Let's try 782

782+287 = 1069

1069 + 6901 = 10670

10670 + 7601 = 18271

18271 + 17281 = 35552

35552 + 25553 = 61105

50116 + 61105 = 111221

111221 + 122111 = 233332  which is a palindrome.

And let's try a smaller three-digit number, 165

165 + 651 = 816
816 + 618 = 1434
1434 + 4341 = 5775 which is a palindrome.

This is called the Lychrel process, and it's not named after a famous mathematician.  It makes a change!  It was named by a man called Wade van Landingham in 2002, who created the name as a rough anagram of his girlfriend's name, Cheryl.  Unlike most mathematical concepts, this one is not hundreds of years old - this also makes a pleasant change!

There is one number which has not yet been found to form a palindrom after many, many iterations of the Lychrel process, and that's 196.  Innocuous, isn't it?

196 + 691 = 887

887 + 788 = 1675

1675 + 5761 = 7436

7436 + 6347 = 13783

13783 + 38731 = 52514

52514 + 41525 = 94039

94039 + 93049 = 187088

187088 + 880781 = 1067869

1067869 + 9687601 = 10755470

10755470 + 7455701 = 18211171

18211171 + 17111281 = 35322452

35322452 + 25422353 = 60744805

60744805 + 50844706 = 

And at this point, my calculator says "Enough!"  I can't get all the digits any more, and this number still isn't reaching a palindrome.

My spreadsheet goes a little further:

60744805 + 50844706 = 111589511

111589511 + 115985111 = 227574622

227574622 + 226475722 = 454050344

454050344 + 443050454 = 897100798

897100798 + 897001798 = 1794102596

And then starts throwing "#VALUE!" messages at me, without reaching a palindrome.

The general definition for a Lychrel number is one that does not reach a palindrome in fewer than 500 iterations. This is easier to measure compared to 'never reaches a palindrome', and that means that the Lychrel numbers (more than 500 iterations) include 295, 394, 493, 592 and 689.

Some numbers immediately become palindromes after one iteration - these are trivial, commonplace and not very interesting!  For example, 110 + 11 = 121, and any other number where the units value is zero, and the hundreds and tens are both less than five.  The longer ones are definitely more interesting, because there's no obvious pattern (and it reminds me of the Collatz conjecture, which I'll be revisiting soon).  Larger numbers which need more than 500 iterations include 10538, 10553 and 10585.

So: can you find numbers which reach a palindrome before they make your calculator (or your spreadsheet) explode?

Further reading:

Lychrel Number Tester - dcode.fr
Lychrel Numbers - Geeks for Geeks

Other Calculator Fun and Games articles:

Snakes and Ladders (Collatz Conjecture)
Crafty Calculator Calculations (numerical anagrams with five digits)
More Multiplications (numerical anagrams, four digits)
Over and Out (reduce large numbers to zero as rapidly as possible)
Calculator Games: Front to Back
Calculator Games: Up, up and away with Ulam sequences
Calcualtor Games: The Kaprekar Constant

Calculator Fun and Games: Premium Prime Numbers

 PRIME NUMBERS

 Calculator Fun and Games is a maths puzzle book that’s worth its salt: it has a section on prime numbers, as any good maths book does.

Prime numbers are only divisible by themselves and 1, they have no other factors.  1 is not a prime number (by definition), but 2 is, making 2 the only even prime number.  The study of prime numbers dates back thousands of years. Some of the earliest records come from the ancient Greeks, particularly the mathematicians of Pythagoras' school (around 500 BC to 300 BC), who explored numbers for their mystical properties.  I recently looked at Pythagoras's Triples - there's always an overlap between different parts of maths.

One of the most significant contributions came from Euclid (circa 300 BC), who proved that there are infinitely many prime numbers. His work in Elements laid the foundation for number theory, including the Fundamental Theorem of Arithmetic, which states that every integer can be uniquely factored into prime numbers.  That's worth its own investigation, and is a puzzle in itself.

Calculator Fun and Games lists the prime numbers up to 1000, just for good measure.  But is there any relationship between them?  Are they connected?  Are there ways of finding (or generating) them?  And can all this be done with a humble calculator?  Maybe one day.

However, the book points out an interesting fact:  if you

Take a prime number (greater than 3)
Square it
Add 14
Divide by 12

Then the remainder is always 3.

For example:
5² = 25
25 + 14 = 39
39/12 = 3, with a remainder of 3

Let’s take a larger example: 577
577² = 332929
332929 + 14 = 332943
332943 / 12 = 27745 remainder 3

Alternatively, x² + 11 always divides exactly by 12.  I’m not sure why Ben Hamilton decided on adding 14… maybe he likes the remainder 3?

Does this apply in reverse?  If I take a number, and subtract 11 and take the square root, do we always get a prime? 

No.  Even if the square root is an integer, it doesn’t mean that the starting number was a prime.  For example, 92-11 = 81, and the square root of 81 is 9.  Only a subset of a = (SQRT(b-11)) will give a as a prime.

TWIN PRIMES

Twin primes are primes with a difference of two; for example, 11 and 13, or 17 and 19.  These are rarer than prime numbers, but it still seems that there will be an infinite number of twin primes.  An example of a larger twin prime is 971 and 973; while the largest known twin primes have 388,342 digits:  they are:  2996863034895×2^1290000±1.

Not a number you’d fit on your calculator.  The largest prime number which will fit on an eight-digit calculator (with a good old-fashioned LCD display) is 99,999,989.

Looking at all those nines, I’d like to play Over and Out with it, and see how long it would take to get it down to zero! 😊

Other recent Calculator Fun and Games articles:

Snakes and Ladders (Collatz Conjecture)
Crafty Calculator Calculations (numerical anagrams, five digits)
More Multiplications (numerical anagrams, four digits)
Over and Out (reduce large numbers to zero in as few steps as possible)