April 5th, 2009


Reclaiming þorn

I was very amused to come across þis essay, advocating þe return of þe old letter þorn, instead of "th".

In fact, "th" represents two different sounds in English, a voiced and an unvoiced consonant. (Þink of þe difference between "them" and "thick".) But we never really notice þe distinction; þey operate as one phoneme in English. Þe only modern language þat uses þe letter is Icelandic, but in þeir spelling "þ" is always "thick"; þe softer, voiced consonant is spelt ð.

Apparently þe reason þat "þ" was dropped from English is þat when printing began, þe font sets arriving from France and Germany had all þe oþer letters except þis one, so we slipped into þe situation of spelling þe sound "th". (For þe same reason, we dropped þe old letters wynn - ƿ - and yogh - ȝ - þough þe latter survives in þe name of Sir Menȝies Campbell.) Þis process has left English as a language using very few diacritical marks in its version of þe alphabet, which I have always felt is raþer dull.

Þere are a number of oþer languages which also have þe "th" spelling mainly for þe unvoiced version - Welsh and Albanian are þe two I am most familiar wiþ - þough in Castilian Spanish, "c" and "z" are often used for it, and in Turkmen, oddly enough, it is represented by þe Latin letter "s". In a number of languages (including again Castilian Spanish) "d" generally represents þe voiced sound; in Albanian it is spelt "dh" and in Welsh "dd" (and in Fijian, apparently, "c").

I am not committing to write all my future livejournal entries using þis neglected letter. But I am not going to ignore it eiþer!

Square numbers ending in repeated digits

If you consider the square numbers in base 10, very few of them end in repeated digits, apart from the trivial case of all squares of multiples of 10 ending in multiple 0's.

Square numbers which end in 1 or 9 always have an even number in the 10s column. (01, 09, 49, 81, 121, 169...) Square number which end in 6 always have an odd number in the 10s column. (16, 36, 196, 256...) Square numbers which end in 5 always in fact end with 25, so never with 55.

Square numbers which end in 4, however, always have an even number in the 10s column (04, 64) and so we get plenty ending in ..-44 (starting with 144 which is 12 squared) and even ...444 (starting with 1444 which is 38 squared). But there it stops. If the last three digits are 4's, the number in the thousands column is odd: 462 squared is 213444, 538 squared is 289444, etc.

What about other bases? Well, in binary, every even number's square ends with at least two zeros (because they are all multiples of four), and every odd number's square ends in ...001 (because they are always one more than a multiple of eight).

In hexadecimal we start off promisingly:

324 (18 squared) in base 10 is 144 in hex
529 (23 squared) in base 10 is 211 in hex
2116 (46 squared) in base 10 is 844 in hex
3481 (59 squared) in base 10 is E99 in hex
4761 (69 squared) in base 10 is 1299 in hex
6724 (82 squared) in base 10 is 1A44 in hex
11025 (105 squared) in base 10 is 2B11 in hex
12100 (110 squared) in base 10 is 2F44 in hex

91204 (302 squared) is 16444 in hex
326041 (571 squared) is 4F999 in hex
521284 (722 squared) is 7F444 in hex
762129 (873 squared) is BA111 in hex
1380625 (1175 squared) in base 10 is 151111 in hex
5522500 (2350 squared) is 544444 in hex

Now we have to go up considerably in scale, and there aren't any squares ending in more than five repeated 4's, as the sixth last digit is always odd (1's and 9's, however, continue)

2088941230489 (1445317 squared) in base 10 is 1E65E999999 in hex

48209289910681 (6943291 squared) in base 10 is 2BD899999999 in hex (I haven't found any smaller squares ending with seven repeated digits)

And I guess that we can keep going with steadily increasing numbers of 1's and 9's (this is probably trivial enough to prove).

Counting in an odd-numbered base, it is also pretty easy to build up repeated digits as the distinction between odd and even numbers no longer applies.

4 (2 squared) expressed in base 3 is 11
121 (11 squared) expressed in base 3 is 11111

Just thought you ought to know.


I've been watching The Prisoner over the last few months (having bought a complete set of videos at auction from the estate of the much-missed David Stewart) and reached the end this evening, treating myself to a double bill of Once Upon a Time and Fallout.

There are those who say that Fallout, the final episode is a true work of genius, and those who say that it is total nonsense. (I suppose there may be some who maintain both points of view.) Myself, I love it to bits. IMHO the only moment where it falters is where Number Six, The Kid and The Former Number Two are shooting down the bad guys to "All You Need Is Love" - I think that is in somewhat poor taste. But all the rest is brilliant: a triumph of showing rather than telling, leaving the viewer to put together his or her own interpretation. (Even if that interpretation may be "What a load of rubbish!")

My favourite moments: are: 1) when the Controller puts on his mask and robe and goes to join the other masked, robed guys; 2) when they all echo Number Six's first person pronoun to the point of drowning out what he is saying; and 3) at the end when Number Six and the Butler cross the road together. But I think it is almost all sheer genius.

What about you?

And what reference book about The Prisoner should I get?