Sunday, February 17, 2013

How to Memorize π to 100 Decimal Places

One of my undergraduate courses introduces the foundations of cognitive science.  It first portrays three different approaches to cognition (classical, connectionist, embodied), emphasizing their differences and their mutual antagonism.  It then explores how to unify these three views. 

I end the course with a lecture on memory, because this topic is one that promotes theoretical unification.  Mnemonic techniques for converting numbers into words are clearly in the realm of classical cognitive scientists interested in symbol manipulation.  Techniques that assert associations between memories, so that a recalled item cues the recall of the next item, have definite connectionist attributes.  Techniques that promote the use of the real world, such as using well-known locations to store to-be-remembered material, evoke embodied cognitive science’s interest in cognitive scaffolding.   

I set the stage for my lecture by performing a feat of memory.  Recently I recalled π to 100 decimal places under the watchful eyes of my students. (As I wrote the digits across the board in front of the class, they used the internet to track my progress – without me asking!)   This performance is not particularly notable – world record holder Lu Chau has recited π to 67,890 digits – but it is grand enough for me, and it possibly impresses my students. 

Surprisingly, anyone interested in performing this memory feat who peruses the internet for how-to advice is often steered in inefficient and laborious directions.  Some websites recommend learning π by repetition:  “Repeat 4 digits out loud at least 20 times” says one.  Other sites recommend looking for numerical patterns within a block of digits, and then for other patterns that link one block to the next.  Others provide piems – poems that represent the desired digits – for memorization, and for later decoding.  A handful of sites refer to classic mnemonics for converting digits to words; but cautions the reader about difficulties;  the words have to be linked together into a (memorisable) story in order to recite π’s digits in the right order, which is quite challenging because the required words do not typically come in meaningful patterns! 

My own technique for this memory feat seems more efficient.  I employ two different mnemonic techniques in concert.  First, I use a standard technique for converting digits into words.  However, I deal with the sequencing of the words using a second technique, the ancient method of loci, in which I move from location to location in a ‘memory palace’, placing to-be-remembered images at each location.  Several of my ‘π words’ are chunked together in a single image.  My memory palace is a very familiar place – my house – and I can memorize 100 digits of π using only 16 different images.  To perform my feat, I simply walk through my memory palace, retrieving the stored images in the necessary order.  Each image recalls a phrase; I (knowing the Major method described below) translate the phrase into numbers; I then write out the numbers in front of the class. 

Of course, as this memory feat is part of a lecture, I am compelled to explain my methods to my students.  I begin by telling them the following story, which illustrates my use of the method of loci.  If you read the story, concentrating on creating each image (placing it in your own memory palace), then you will be able to perform this memory feat as well: 

I begin at the front sidewalk to my house. I look at the sidewalk, covered in snow, and place upon it an image of a running car motor resting on top of a red tulip; the tulip vibrates from the action of the engine. This image represents the concept Motor-Tulip. 

Next, I see the top of the stairs that lead to my front door. I imagine seeing NBA star Steve Nash limping down the stairs as he leaves my house. I place that image on the stairs, standing for the sports headline Nash! Lame! Leave! 

I open the front door, step into the vestibule. There I imagine an unknown person throwing a pickaxe (instead of a dart) to choose one of a number of nametags stuck to the wall. Each tag holds the name of a poem. This dynamic image conveys the meaning Pick-Poem-Name. 

Stepping beyond the vestibule, I look into the walk-in closet. I imagine seeing automatic conveyor of the type used by drycleaners; many fur coats hang from this device. My image activates, and the coats spin round and round the closet. This is clearly a Fur-Changer. 

Moving along, I look into the main floor bathroom. I add to this room an image of my mother in an incoherent rage, so angry that volumes of foam spew from her mouth. Yes, I see Mom-Foaming. 

The next room that I reach is the kitchen. On the counter, I see a large, overflowing bowl of dog kibble. I imagine making each of these kibbles even smaller by cutting them into pieces using a large, sharp knife. Setting this image into motion encodes Kibbles-Knife. 

Moving towards the end of the kitchen, I peer into the back entry and notice the upright freezer. I imagine opening the freezer, gazing inside; in my mind’s eye, I see an aluminum bucket being flipped up and down on a hot frying pan. To me, this image means Fried-Bucket. 

Beside the back entry is a pantry connected to the kitchen. On the small counter within it, I imagine seeing an old, torn, dirty roadmap; attached to the map is a lit fuse with its bright spark moving dangerously close to the end. This is a strange device, perhaps invented by one of my cats; it is a Shabby-Map-Bomb. 

I leave the kitchen, and enter the dining room. I imagine that the dining room table supports a large cooler of ice. On this ice are many of seal fins, eerily flipping up and down on their own. Tonight, dinner appears to be Cooled-Seal-Fins. 

I pass from the dining room into the living room. There, in my imagination, an unknown woman works, taking a black hockey puck and laboriously winding colorful ribbons around it. I do not know who this woman is, but I do know her role in the household: She is the Puck-Wrapper. 

I turn away from the living room, and begin to climb the stairs to the second floor. At the first stairs landing I imagine seeing a person with a very large earlobe from which dangles a strange earring. The earring consists of several different gnomes and a cup of coffee. To me the image represents Earlobe-Gnomes-Coffee. 

I climb the second flight of stairs, turn left and gaze into the laundry room. There I imagine a number of my former teachers all standing around using cloths to polish the nose of a large fish. Thus in the laundry room I place the concept Teachers-Shine-Fish-Nose. 

Beside the laundry room is a linen closet. I open it and imagine seeing a beehive from which swim a number of long, lanky pipefish. Apparently, the linen closet contains a Hive o’ Pipefish 

The next room that I enter is a second floor office. There I imagine seeing Knives Chau, a character from the movie Scott Pilgrim vs. the World. She is using a hammer to smash all my old LPs into bits. The concept in this room is Knives-Hammer-Vinyl. 

The office leads into the master bedroom. I imagine seeing on the bed a desert island surrounded by ocean. The island is small, but covered with a mass of moving ticks. The bed contains Marooned-Ticks. 

My final stop is the bathroom attached to the master bedroom. On the bathroom floor, I imagine seeing a single shoe that has the appearance of a cob of corn. This is my missing Shoe-Cob. 

I designed the images in the story above to be novel, bizarre, and dynamic in order to make them easy to remember. In fact, after selecting the images, I learned each of them as well as their position in my memory palace after only a couple of imaginary walk-throughs. 

I selected the words encoded by the images with a different mnemonic technique in mind. This technique was the Major method, for converting digits into consonant sounds. For instance, the consonant sound n is associated with the digit 2 because n has two downward strokes; the consonant sound m is associated with the digit 3 because m has three downward strokes. The table below provides the full Major method; I highly recommend reading about it in Lorayne, H. and J. Lucas (1974). The Memory Book. New York, Stein and Day. 

To use the Major method to remember a long number, take the number, convert its digits into consonant sounds, and then add vowels to make meaningful words and expressions. To recall the number, recall the expression – then work backwards to convert its consonant sounds back into digits. 

Consonant Sounds
sh, ch, g, j
p, b

 The images that I placed in the various loci in the story that I told above now make perfect sense: each represents words to translate into digits of π with the aid of the Major method, as shown in the table below:  

Words From Image
Nash! Lame! Leave
Hive O’ Pipefish
With a little practice using the Major method – I usually do so by memorizing the digits of license plates on other cars as I drive – converting words into digits becomes automatic.  This ability, combined with the method of loci and vivid mental imagery, permits you to perform memory feats like mine without too much effort.  However, if I ever want to reach Lu Chau’s level of performance, then I will have to move into a much, much larger memory palace!

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