Saturday, 20 August 2016

Modulo Arithmetic and Days of the Week

Yesterday, I noticed something curious about the Wikipedia entries for days of the year. Here's an example from January 1:

The text that caught my eye is:
This date is slightly more likely to fall on a Tuesday, Friday or Sunday (58 in 400 years each) than on Wednesday or Thursday (57), and slightly less likely to occur on a Monday or Saturday (56).
Huh? At first I thought this was an obvious mistake; surely the days of the week are evenly distributed. After all, there are 365 or 366 days in a year and 7 days in a week, and these numbers are relatively prime.

Here's my (erroneous) line of reasoning:

  1. Neither 365 nor 366 are divisible by 7. [True]
  2. Therefore, January 1 can be any day of the week. [True]
  3. The "leap year" cycle repeats after 400 years. [True]
  4. But 400 is not divisible by 7. [True]
  5. Therefore, the "leap year and January 1 weekday" cycle repeats every 2800 years. [False]
My mistake was trying to divide 7 (the number of weekdays) into 400 (the number of years in a "leap year" cycle). What I should have done is try to divide 7 into the number of days in a "leap year" cycle.

So, how many days are there in 400 full years? Well, there are (100 - 4 + 1) = 97 leap days, so there are (365 * 400 + 97) = 146097 days.

Amazingly, 146097 is divisible by 7, so the "leap year and January 1 weekday" cycle repeats every 400 years, not every 2800 years. As we know, 400 is not divisible by 7, so this cycle cannot possibly have a uniform distribution of weekdays for January 1.

Indeed, even leap days (February 29) are not immune from this lack of uniformity because 97 is not divisible by 7. In any 400 year period, there are:
  • Thirteen each of Mondays, Wednesdays and Fridays,
  • Fourteen each of Saturdays and Sundays, and
  • Fifteen each of Tuesdays and Thursdays.

At this point in the discussion, someone usually witters on about calendar reform.

So why not...

Imagine reforming the calendar such that January 1 is always a Saturday but without forsaking the 7-day weekly cycle. One scheme would be to have a 52-week standard year (364 days) with periodic "leap weeks" (371 days). If we define a leap year to be any year that is divisible by 6 or 62 we end up with a 186-year leap cycle with 153 standard years and 33 leap years, making a total of (153 * 364 + 33 * 371) = 67935 days. That implies that the mean year length is (67935 / 186) = 365.2419355 which is closer to the solar year of roughly 365.242181 days than the current Gregorian calendar (365.2425).

So here's the new scheme:

  • A standard year is 364 days, divided into 52 weeks of seven days.
  • A month is defined as 4 weeks or 28 days. There are thirteen months in a year.
  • A leap year consists of 371 days with the extra "special" week of festivities tacked on to the end.
  • A year is a leap year if it is divisible by 6 or 62.
  • All years, months and weeks begin on Saturday.
Strange, but this seems vaguely familiar...

Saturday, 23 July 2016

Saturday, 16 July 2016

Blok Clok

When I was a boy, I was given a curious present: a perpetual desk calendar made from wooden cubes:
You can still buy them, so there must be some enduring appeal. The calendar interested me because of a strange quirk that isn't immediately obvious.

There are four blocks: one for the day of the week (with seven labels), two for the day of the month (1 to 31) and one for the month (with twelve labels). The "weekday" and "month" blocks are easy to label. For example, two months on each face fills the "month" block and, providing the container covers up the lower half of the block, all is peachy.

The two date blocks are a little more involved. Dates run from "01" to "31":


By inspection, we see that the only digits that appear twice on a single date are "1" and "2". However, if you sit down with a pencil and paper, it quickly becomes apparent that you'll also need two zeroes. That means that the following digits are required:


But, hang on! That's thirteen digits. Two cubes have only twelve faces between them. How do they fit? The clever "quirk" is that "6" and "9" occupy the same face; you simply flip the block around 180 degrees. This gives you the following faces for the two blocks:


This arrangement allows to you represent all integers between "00" to "32" inclusive; just enough! It's deceptively clever. And that's probably why I like my wooden perpetual calendar so much.

Fast forward several years, if not decades, and my current fascination with clocks led me to consider whether you could tell the time with a similar arrangement. I knew representing the hours would not be a problem: as we've seen, two blocks can represent all integers between "00" to "23" inclusive for a twenty-four-hour clock. But representing minutes is harder.

It is fairly obvious that you need at least three blocks to represent all integers between "00" to "59" inclusive. But are three sufficient? It turns out that they are; you don't even need to use the upside-down "9" trick. The solution uses an additional cube to display a colon between the hours and minutes digits. For example:


The "tens minutes" digit takes up a single block: "0" to "5" inclusive. So the problem simplifies to trying to represent the "units minutes" digit ("0" to "9") and a colon with two blocks. The following labelling is one solution:


Using three blocks to represent minutes between ":00" to ":59" inclusive can be repeated for the seconds. This gives us eight blocks:

  hours-a    0,1,2,3,4,5
  hours-b    0,1,2,6,7,8
minutes-a    :,0,1,2,3,4
minutes-b    0,1,2,3,4,5
minutes-c    :,5,6,7,8,9
seconds-a    :,0,1,2,3,4
seconds-b    0,1,2,3,4,5
seconds-c    :,5,6,7,8,9

Not only do the blocks need to rotate to any of their six faces (and in the case of "hours-b", the "6" sometimes has to be flipped to become a "9") but pairs of blocks also have to be swapped periodically:

hours-a <=> hours-b
minutes-a <=> minutes-c
seconds-a <=> seconds-c

Some extra fettling is needed to make the rotations aesthetically pleasing, but the result is a twenty-four-hour clock that twists and turns at a fair old lick.

Minimal Rotor Clock

The Rotor Clock I posted about last month is all very well, but there are a heck of a lot of moving parts: thirty-two rotors, each with ten faces. You could physically build it, but things would be a lot easier if there were fewer components. So I set about trying to reduce the number of rotors for a twelve-hour clock that displays the time as words.

The extreme (degenerate) case is a single rotor with 12 * 60 = 1440 faces. This would produce a working clock, but the rotor would need to be enormous (or the text very small). The next idea I had was a clock with two rotors: one for hours and one for minutes. You could repeat the twelve hours five times and make both rotors have 60 faces. This would still require very large rotors. Also, telling the time in a "natural" British English manner is quite tricky. Consider:


The solution I came up with (which may not be optimal, but is possibly close) is to use four rotors, each with at most twelve words or phrases plus a blank face:

    Rotor A      Rotor B      Rotor C         Rotor D
    =======      =======      =======         =======
    THIRTEEN     ONE          O'CLOCK         ONE
    FOURTEEN     TWO          MINUTE PAST     TWO
    EIGHTEEN     FIVE         HALF PAST       FIVE
    NINETEEN     SIX          QUARTER TO      SIX
    TWENTY       SEVEN        MINUTES TO      SEVEN
                 EIGHT        MINUTE TO       EIGHT
                 NINE                         NINE
                 TEN                          TEN
                 ELEVEN                       ELEVEN
                 TWELVE                       TWELVE

Note that rotor B is identical to rotor D.

This produces a clock that can tell the time as hoped for, but with the insertion of another blank face in rotor A and a bit of duplication in rotor C, I was able to reduce the number of large rotations quite considerably. This produces a series of relatively smooth transitions. The final rotors are organised thus:

    Rotor A      Rotor B      Rotor C         Rotor D
    =======      =======      =======         =======
    (blank)      (blank)      (blank)         (blank)
    THIRTEEN     ONE          O'CLOCK         ONE
    FOURTEEN     TWO          MINUTE PAST     TWO
    (blank)      THREE        MINUTES PAST    THREE
    EIGHTEEN     SIX          HALF PAST       SIX
    TWENTY       EIGHT        QUARTER TO      EIGHT
    (blank)      NINE         MINUTES TO      NINE
    (blank)      TEN          MINUTE TO       TEN
    (blank)      ELEVEN       (blank)         ELEVEN
    (blank)      TWELVE       (blank)         TWELVE

If you look at the JavaScript code, you'll see that control of the rotors is via two data tables. This scheme lends itself to efficient compression. Indeed, a slightly cut-down variation of the clock can be compressed into a single 1017-byte HTML page (providing you're using a highly-compliant browser such as Google Chrome or Mozilla Firefox).

Sunday, 26 June 2016

Word Time 1KB

One of my inspirations for the Rotor Clock was the Instructables wordclock that tells the time (and temperature) based on a 16-by-16 grid of letters:

It looked very much like a puzzle to me, so I immediately set about trying to work out the minimum size of grid that could tell the time to an accuracy of one minute. I managed to get the grid down to 13-by-13:


[I have a suspicion you can get it even smaller if you use tricks such as spelling words vertically; but, after a considerable amount of effort, I plumped for 13-by-13]

In my usual modus operandi, I then tried to optimise the heck out of the problem by hand and attempt to fit a solution into a one-kilobyte web page that works on Chrome, Firefox and Internet Explorer. For added frisson, I didn't allow myself to use non-ASCII characters nor the evil JavaScript "eval()" or its ilk. The result, World Time, takes up just 1023 bytes:

Here's a quick breakdown of the contents:

We can omit a large number HTML elements such as "DOCTYPE", "head" and "body" and dive straight into the style definitions, which are assumed by browsers to be CSS. We set all backgrounds to black, centre the table in the middle of the page and format each table cell so that a 13-by-13 grid takes up most of the viewport. The default text colour is set to very dark grey and a shadow effect is added to simulate "light bleed":

We then define a table that will hold the 169 grid characters and be accessed in the JavaScript via a identifier named "t". The string is split into 169 table cells (each one 10 characters of HTML) which are then divided into 13 rows:

Next we decode a 78-character string that has been btoa-encoded (so we adhere to our ASCII restriction). The 78 characters represent 39 pairs of offsets into the "t" grid. The first of each pair is the "start" offset (0 to 168); the second is the "end" offset plus one. These 39 pairs define tokens with even numbers identifying them thus:

           #00 = "O'CLOCK"
           #02 = "MINUTE"
           #04 = "MINUTES"
           #06 = "PAST"
           #08 = "TO"
           #10 = "HALF"
    #12 to #38 = "ONE" to "FOURTEEN" (upper portion)
           #40 = "QUARTER"
    #42 to #50 = "SIXTEEN" to "TWENTY"
           #52 = "MIDNIGHT"
    #54 to #74 = "ONE" to "ELEVEN" (lower portion)
           #76 = "NOON"

These 38 tokens constitute all the words used by the clock. See below for details of the call to "setInterval":

Next, we define a function "d()" that takes two arguments (the third argument declaration is just a short way of declaring a variable - see 140bytes for other such techniques), The first argument is the hour as an integer. The second argument is either "00" (the token for "O'CLOCK") or "" depending on requirements. The "d()" function therefore returns the hour as a string of two-digit tokens (e.g. "7400" for "ELEVEN O'CLOCK"). It has a special case for "MIDNIGHT" and "NOON" which never take the "O'CLOCK" suffix:

We define another function "g()" that takes an integer number of minutes (0 to 59) as its only argument. It returns a string of tokens representing the appropriate number of minutes as words (e.g. "(something) MINUTES" or "TWENTY (something) MINUTES", taking into consideration the exceptional cases for "ONE MINUTE", "QUARTER" and "HALF". Note that we're playing fast and loose with the type system here: the number 1202 will be converted to the string "1202" elsewhere in the program.

Next, we define an anonymous function (let's call it "storeColours") that takes one true argument ("b" is a local variable declaration again). This is called with the result of another anonymous function ("calculateColours") that takes three arguments: the current local hour, the current local minute and an empty dictionary. The "calculateColours" function constructs a full string of tokens that describe the current time. If the minute value is zero, we use the "d()" function to construct "MIDNIGHT", "NOON" or "(hour) O'CLOCK". If the minute value is less than or equal to thirty, it constructs "(something) PAST (hour)". Otherwise, if constructs "(something) TO (hour+1)". The string of tokens are split into two-digit numbers and looked up in the "s" table described above. For every letter in every word, "calculateColours" stores an HSL colour in the "c" dictionary keyed by the appropriate offset into the grid. The dictionary is then passed to "storeColours" which modifies the style text colour of each table cell in turn. Setting the colour to "" results in the text colour reverting to the colour defined in the CSS entry for "td" (i.e. very dark grey):

Finally, we invoke the "calculateColours" and "storeColours" function at most every 1000 milliseconds via the "setInterval" function mentioned above, and gracefully close the HTML tags.

A lot of the JavaScript compression came from Google Closure Compiler.

The result is a colourful clock that spells out the time in English words, uses a responsive HTML layout and takes the local timezone, changes in daylight savings and even leap-seconds into consideration.

UPDATE: 2016-06-29

I've managed to shave an additional ten bytes off the size and updated the page, but the structure and technique are essentially the same as above.

Sunday, 19 June 2016

Rotor Clock

There seems to be a bit of fad for clocks that spell out the time. There are even some online examples utilising a variety of languages. And jolly good they are too!

Spelling out the time in British English is fairly simple. Just take the local time and extract the minutes. If the value is less than thirty, express the time as

  (minutes) PAST (hour)

Otherwise, use

  (60-minutes) TO (hour+1)

There are some exceptional cases; if the minute value is zero, use

  (hour) O'CLOCK

Whereas minute values or 15, 30 and 45 are generally expressed as

  HALF PAST (hour)
  QUARTER TO (hour+1)

It turns out that, using the above scheme, the maximum number of characters needed to spell out the time is 32. For example:


If we are a little bit lenient with the spacing between the words, we can prune the number of distinct characters in each position to ten (including space). This leads to an interesting configuration for a mechanical device that can spell out the time using 32 rotors of only 10 facets each.
With a little bit of CSS3 and HTML5, it's possible to produce a web page that brings such a device to life. If you further optimise the HTML, CSS and Javascript, you can present it in a single file of only 2191 bytes.

Sunday, 22 May 2016

A Short History of Long Ships

With the recent maiden voyage of "MS Harmony of the Seas," there's been a lot of talk in the press about large and long ships. So here's my attempt at a summary:

The blue whale is the largest extant animal.

La Santa María, launched in 1460, was largest of three ships used by Christopher Columbus to cross the Atlantic for the first time in 1492. She ran aground later in 1492.

The Nao Victoria, launched in 1519, was the first ship to circumnavigate the globe in 1522. She disappeared in 1570.

HMS Victory, launched in 1765, was Nelson's flagship at the Battle of Trafalgar in 1805. She is still in service.

SS Great Western, launched in 1837, was the largest passenger ship until 1839. She was broken up in 1856.

SS Great Britain, launched in 1843, was the longest passenger ship until 1854. She was finally scuttled in the Falklands in 1937.

SS Great Eastern, launched in 1858, was the longest ship built of any type until 1899. She was scrapped in 1889.

RMS Titanic, launched in 1911, was the largest ship afloat until she sunk on her maiden voyage in 1912.

RMS Queen Elizabeth, launched in 1938, was the largest (by displacement) passenger liner for almost fifty years. She finally caught fire and sunk in Hong Kong harbour in 1975.

USS Iowa, launched in 1942 and now a museum ship, is the longest battleship ever built. The Japanese Yamato was more massive, but slightly shorter.

The aircraft carrier USS Enterprise, launched in 1960, is the longest naval vessel ever built. She was decommissioned in 2012, but, being nuclear powered, she will take a long time to be completely dismantled.

SS France, launched in 1960, took over from RMS Queen Elizabeth as being the longest passenger ship. She was sold for scrap in 2005.

Batillus, launched in 1976, was the largest supertanker by gross tonnage ever built. She was scrapped in 1976.

Seawise Giant, launched in 1979, was the longest ship ever built. She was scrapped in 2010.

Dmitri Donskoi, launched in 1980, is a Typhoon-class submarine, the largest in the world. She is still in active service.

RMS Queen Mary 2, launched in 2003, is the largest ocean liner ever built. She is still in service.

Vale Brasil, launched in 2010, is the largest bulk carrier ever built. She is still in service.

Pieter Schelte, launched in 2013, is the largest vessel ever constructed in terms of gross tonnage, breadth and displacement. She is still in service.

MV Barzan, launched in 2014, is the largest container ship ever built. She is still in service.

MS Harmony of the Seas, launched in 2015, is the largest passenger ship in the world by gross tonnage. She entered service in May 2016.

The maximum length of ships in service is usually limited by canal sizes and port facilities.

See here for a larger image.