Tuesday, December 28, 2010

Triple Digit Mental Multiplication

Note: This post originally appeared on my discontinued website maartensity.com. The published date and time has been adjusted to mirror the original.

After the piece on double digit mental arithmetic I wrote not long ago, I wondered if the method could be extended to 3 digit multiplication. However, as I found out while trying, it's all too easy to create temporary numbers beyond all practicality, clogging short term memory and pushing the brain onto a rollercoaster of calculation and recalculation to recover numbers from earlier phases!

Nevertheless, it turns out there is indeed a way to build on the 2 digit method that yields results, and requires only a bit more from short term memory than the 2 digit method. (Remember, the goal is to do the calculation without the aid of a pen and paper!).

Without further ado, let's use for our example

x 378

The first stage is a repeat of the method we employed in the previous article. We take the two rightmost digits and multiply them (have a look at the previous article to see how this is done).

x 78



This forms the foundation. Our mind has been primed by these numbers, making the subsequent calculations slightly easier to do.

Let's take another look at the original three digit numbers:


In the second stage we multiply the (thus far unused) number in the top left corner with the (already used) bottom two right digits: 5 x 78. That gives 390. Likewise, we then multiply the (unused) number in the bottom left corner with the top right two digits: 3 x 69. That gives 207, and added to 390 gives 597.


The final step in stage two is to multiply the number in the top left with the bottom left: 3 x 5 gives 15.

Now 597 actually represents 59700, and 15 represents 150000, so to calculate the addition we can invoke them visually as follows:


Note: 15+5=20 followed by 97 gives 2097.

The final calculation we do is to add the earlier result (5382) to the latest (2097). Remember that they represent different multiples of 10, so to add them we follow the same kind of visual method as above:



And that is the answer!

Seems easy? Actually, it does take more than twice longer than the 2 digit method, but that is largely down to the number of steps involved, and not due to the intrinsic difficulty of any individual step.


This article and its contents is made available freely under the Creative Commons License v3. If you find it useful and want to share, please mention me or link back. Thanks :-)

Monday, April 12, 2010

Double Digit Mental Multiplication

Note: This post originally appeared on my discontinued website maartensity.com. The published date and time has been adjusted to mirror the original.

I have often wondered why mental arithmetic is so hard beyond the simple tables from one to twelve we learned at school. Take for instance something like: 84 x 32. Seems simple enough, but how long will it take you to work it out in your head? Will you even be able to? Can you do it under pressure? Note that I am referring to real world scenarios, for instance doing the calculation while talking to someone, without the aid of any tools such as pen and paper.

The basic stumbling block appears to be the brain's inability to hold on to this information in consciousness, in real-time, for long enough to get going. It is significantly easier even just to write down the initial figures on paper, leaving your brain free to do the calculations. Doing it all in your head, you are soon spending more time trying to remember the original digits than doing the actual calculation!

It follows that any method of complex mental arithmetic should minimise the information your brain needs to keep in consciousness, instead relying on long term memory and existing skills, i.e. your old arithmetic tables.

At school, this is how we learned to do double digit arithmetic:

x  32
  168    -> starting from the right: 

              4 x 2 = 8
              8 x 2 = 16
              => 168
+2520    -> starting from the right: 

              0 x 10 = 0
              4 x 3 = 12,(write 2 keep 1); 
              8 x 3 = 24, add 1 left over = 25
              => 2520
 2688    -> starting from the right: 

              add 8+0 = 8
              add 6+2 = 8
              add 1+5 = 6
              2 = 2
              => 2688

This is straightforward on paper, but as pure mental arithmetic, it requires working with far too many numbers at once: 84, 32, 8, 16, 168 (throw away 8 and 16 at this point), 12, 1, 24, 25 (throw away 12, 1, and 24 at this point), 2520 (throw away everything except 168 and 2520)

By the time you calculate 2520 your consciousness is cluttered with 9 figures, 5 of which you can throw away, but still have the potential to confuse through the brain's policy of indiscriminate retention. 9 is considered just about the maximum number of discrete elements we can keep in consciousness at any one time (bar advanced memorising techniques of course, which is outside scope).

That is why it becomes necessary to memorise some digits while in the middle of calculation, and this is a sheer waste of time.

But once you've arrived at both 168 and 2520, it's relatively straightforward to add them and provide the answer: 2688.

In conclusion, given enough time to memorise digits as they are being calculated, all double digit arithmetic is possible without any aids using this method. However, in my experience, it takes far longer than is practical in any real life situation.

This is how I came to see that using a slightly different method, there's far less chance of forgetting things. It takes a bit of practice to get used to, but most double digit arithmetic can then be done in less than 20 seconds.

The trick lies in handling the "middle" digits immediately after the easy right-hand digits, thereby stringing all the results together and making the flow of the calculation more natural.
Let me illustrate with the same example above:

 x  32
     8    -> starting from the right: 4 x 2 = 8
   28     -> diagonals: (4 x 3) + (8 x 2) = 28
  24      -> now the left: 8 x 3 = 24

  2688    -> let the three figures fall into one, 
             adding where more than one digit appears => 2688

The first phase is always more difficult, but this time there are less big figures to remember, making it easier: 84, 32, 8, 12, 16, 28, 24 (throw away 84 and 32).

This time you only needed to keep in mind 7 figures - and what's more, the largest number is only two digits - compared to 168 and 2520 in the standard method. Now keep the three new figures (8, 28, and 24) in their positions, and let them "fall" to the bottom into one figure. In other words, from the right, 8 fall to the bottom, 8 fall to the bottom, 2 collects 4 = 6 fall to the bottom, 2 fall to the bottom => 2688.

Now, as a final enhancement, consider changing the first set of double digits to a single figure of four digits in a predictable order that will always yield the same results. Eg. 84 x 32 becomes 3428, which makes the all-important "middle digits" calculation easier - just work it from left to right: (3 x 4) + (2 x 8) = 28. Then digit 2 and 3, conveniently next to each other: 4 x 2 = 8 => 288; then the outer digits 1 and 4: 3 x 8 = 24 => 2688.

Suddenly it's easy!


This article and its contents is made available freely under the Creative Commons License v3. If you find it useful and want to share, please mention me or link back. Thanks :-)

Tuesday, March 23, 2010

The Status of Truth in Edgar Allan Poe's Poem "To Helen"

Note: This post originally appeared on my discontinued website maartensity.com. The published date and time has been adjusted to mirror the original.

When Edgar Allan Poe revised his childhood poem “To Helen”, he must have known it had enough potential to be added to his catalog of more mature and well-known works. Two oft-quoted lines, from the second stanza, are in fact from the revised edition:

On desperate seas long wont to roam,Thy hyacinth hair, thy classic face,Thy Naiad airs have brought me homeTo the glory that was Greece,And the grandeur that was Rome.

They are an obvious improvement over the earlier version. Compare the naïve descriptive style of the following, with the much more striking statement above:

To the beauty of fair Greece,And the grandeur of old Rome.

In the revised version, Greece and Rome become symbols, metaphors for glory and grandeur. The comparisons compel the reader to draw on known associations of Greece and Rome, and revise their hitherto held notion of those two concepts. A nostalgic sense of what grandeur and glory is supposed to mean is evoked because of the distance in time and space between the reader and those civilisations. After all, the reader has no first hand experience, and most readers would not have read any significant book of history to authorise (or cast doubt) on that statement.

The combination of limited knowledge and nostalgia allows the reader to project an imaginative presence of Rome and Greece onto the more abstract canvas of the concepts of glory and grandeur.
The statement suggests a definitive measure of those two concepts, and the remoteness of those measurements (Rome and Greece) makes it all the more plausible. The reader's imagination is given free reign. In this sense the poetic statement relies much on what the reader doesn't – can't – know, giving the poet a privileged status and authority as having a more direct and intuitive understanding.

While 20th century philosophy has sought to prove, at times, the inability of language to accurately reflect truth, these lines – in particular when contrasted to the less effective, albeit pleasant enough, original version – is a keen reminder that language does not always seek to clarify or explicitly delineate truth. Instead, the reader may be enticed by suggestion, and given a lead to explore without a firm factual conclusion being drawn.

The poet's endeavour in this case is a form of mythmaking, or myth enhancement, but its value should be sought in context: it lends the Helen of the poem--its proper subject--higher status and credibility.

Sunday, March 21, 2010

Review: Just After Sunset by Stephen King

Note: This post originally appeared on my discontinued website maartensity.com. The published date and time has been adjusted to match the original.

Stephen King remains one of my favourite writers. When he's on form he's unbeatable. He writes page-turners like very few other people, yet keeps his characters live and fleshy, so that you always feel that they're in the room with you. Or you're out on the road with them, as the case may be, smelling the tar and the whiff of gasoline as you pass a lonely gas station next to the highway.

King has delivered some highly memorable short stories over the years. Leaving aside his novellas for the moment, eg. those in Different Seasons (which included Rita Hayworth and Shawshank Redemption and Stand By Me, both adapted to film) and Four Past Midnight; his short works have included some very creepy stories indeed such as The Raft and The Monkey, and an implausible but very frightening story in The Jaunt. These are from Skeleton Crew. There is also the earlier compilation Night Shift, which contained such wonderful stories as Sometimes They Come Back, I Know What You Need, and Children of the Corn.

His more recent novels still contain elements of the supernatural in abundance, but they serve to enhance the more familiar, mundane realities of family life and relationships. The horror is no longer the focus of those stories. Lisey's Story is a good example.

The short stories in Just After Sunset are of this variety too. I felt the influence of Ray Bradbury (for instance in Willa) and a kind of homage to Lovecraft--of sorts--in N. (yes, that's the title of the story), but in general the stories are about people and their relationships, and the lineage of Raymond Carver is also discernible.

My verdict on Just After Sunset is that despite a few duds, there are genuine gems among them. What's more, their subject matter means re-readings could be rewarding.

On to the stories themselves.

Willa is one of my favourites. It is suitably dreamy, and the twist does not cheapen it--instead it produces the meaning of the story. There is something both sad and joyous in Willa and David's gradual realisation of their situation, and their authenticity amidst a backdrop of Americana is like something Edward Hopper might have painted: a glimpse of a genuine exchange of feelings between characters whom we will never get to know any better.

The Gingerbread Girl is an all-action story which has King doing what he does best--creating a story in which you simply have to know what happens next. By the time you're halfway through the story, you'll find yourself starting to turn the next page before you've even started the current one. It doesn't have much else to recommend it, but it won't matter.

Harvey's Dream was genuinely scary. It is masterfully written. King says he wrote it in one sitting, and I suppose it is one of those conceived in its entirety. Subtle yet devastating.

Rest Stop is about a chance witness of domestic abuse at a public restroom, who suddenly uses the unreality of the situation to toy with a different persona. A fascinating story.

Stationary Bike probes the unresolved conflicts that lurk in the mind, and the uncertainties that underscore apparent success in a new endeavour. A man tries to lose weight and reduce his cholesterol after a warning by his doctor, but the more successful he becomes in doing so, the more he finds there is a backlash from the part of him that benefited from being obese. It is an imaginative story, and I don't want to give away too many details.

The Things They Left Behind, although apparently supernatural, probably has more to do with the residue of emotions that do not disappear just because that person has gone. The "things" of the title is the objectified locus of those residual traumas that need to be returned to those who are most capable of looking after them, perhaps dissolving them. Not my favourite story, but not bad.

Graduation Afternoon had me smiling. It has a wonderfully over-the-top ending (of the tried and tested variety) that in effect renders some of the more serious utterances by the characters comical. Comedy-horror?

N. reminded me of Lovecraft, but it is fair to say it wasn't my favourite story. I found it both predictable and derivative, in the sense that King tried to emulate Lovecraft rather than take him as a starting point. As a result the outcome was more or less what I would have expected, and the journey wasn't particularly interesting. Pass.

The Cat From Hell. This story is gold. I loved it from start to finish. It's classic King and harks back to the good old days.

The New York Times at Special Bargain Rates is heartfelt with poignant moments. Once again, it is more about the relationship between husband and wife than about anything supernatural. It almost felt like the story could have developed more, but as it is it still works.

Mute is not quite as effective as it could have been. This might be because part of the ending is predictable, and the other half relies on it gaining psychological depth that I wasn't sure was there all along. As a yarn it was well-written though, as always.

Ayana reminded me a bit of The Green Mile - there is even a moment wherein the main character complains of urinary tract infection. However the denouement of the story is different, and more mysterious. A satisfying story and with plenty left to the imagination. Just the way we like it.

A Very Tight Place is more like The Gingerbread Girl than it is like either Ayana or Willa, in other words it is more about action and tension than about subtlety and emotion. It is quite effective though, another fine page turner of a story, and there was something more English than American - Roald Dahl perhaps - in the ending.

King's stories are never merely cerebral exercises. He always gives them texture, and the characters are seldom less than vivid. You can practically hear the twang in their voices, and the smell of cigarettes and beer on their breath. This most recent compilation of King short stories continues his long-standing flirtation with the short fiction form, and is full of variation. It shows him in fine form. There are one or two stories the occasional King reader might prefer to skip - N. would be first on my list - but on the whole it is a rewarding experience. Several stories also felt like they had movie potential. It's amazing how cinematic his story-telling feels.

Saturday, March 20, 2010

'You Take This Path' published in Streetcake Magazine

Note: This post originally appeared on my discontinued website maartensity.com. The published date and time has been adjusted to match the original.

When I came across Streetcake Magazine a few months ago, I was pleased to see a small, independent magazine that publishes experimental fiction. The magazine's banner reads "The magazine for innovative, visual and experimental writing". The writing tends to be poetry, short prose pieces, or extracts from longer works.

The magazine is run by Nikki Dudley and Trini Decombe, who is from Chile (the latest issue is dedicated to Chile following the devastation caused by the earthquake). The magazine often features their original work (I particularly liked What do I wanna paint? by Decombe in issue 9), and my impression is of passionate writers who run the magazine out of much more than editorial interest.

They were kind enough to publish my visual poem You take this path, which you can view in Issue 9.

you take
this path that you
have taken
every day you take
you take to the sea
i have become
the image that you
want this image
that you see
but today
i am alone
today, this day
        i de
 li    be   rate
to     di ss olve
 sos olve my
        sos love
      l i k e  a
clam         S.O.S
  l i k e
  ommmmm re
    r e turned
          same p a t h
to find you
    but you had r e