Mental Feats

Michael Curtis (who commented on a previous posting) has written several articles on mental feats involving memory and mathematics.

Coincidently, I too read about the Trachtenberg system (a method for performing mental arithmetic) many years ago.

Reading his site reminded me of a time when I'd relieve boredom during occasions such as high school assemblies by squaring 2-digit numbers in my head using the techniques I had read about.

Today, I'd still use Trachtenberg's method for numbers ending in 5, based on the equation (10 a + 5)^2 = 100 a (a+1) + 25. However I have since found faster methods for other numbers, which I haven't seen described on the web. They only require additions and subtractions, but one has more to memorize.

Let n be the 2-digit number to be squared. Then if n lies in the range:

1. 0-25: Memorize these answers.
2. 25-50: Work out how far n is from 50 and how far n is from 25. Then the answer is 100(n - 25) + (50 - n)^2.
3. 50-75: Compute 100((n-50) + 25) + (n-50)^2. (This is also Trachtenberg's method for squaring fifty-somethings.)
4. 75-100: Compute 100(100 - 2(100-n)) + (100-n)^2

While I'm at it, I'll record a method for finding square roots (of squares of 2-digit numbers):

1. Remove the last two digits of the square. Then the first digit of the answer is the largest digit whose square is less than this number.
2. The last digit of the square tells us what the last digit of the answer could be. If it is 0 or 5, then so is the last digit of the answer and we are done, otherwise:
3. Let the first digit of the answer is a. Compare the square with the square of 10a+5. If larger, then the last digit of the answer is between 6 and 9, and if smaller, it is between 1 and 4. Luckily, in base 10, the squares of 1 to 4 have distinct last digits. Also the squares of a and (10-a) end in the same digit, so it is now easy to determine the last digit of the answer.

The corresponding algorithm for cube roots is much simpler, because the cube of each digit has a distinct last digit (and similarly with other odd powers).

Mnemonic Major Systems

I discovered two things when I tried using the net to brush up on mnemonic major systems (aka phonetic number systems). (If you have no idea what they are read the article before continuing!) Firstly, the only encoding from digits to sounds I've seen online so far is the one published by Harry Lorayne. Secondly, it turned out I didn't need any brushing up at all. The mapping I learnt many years ago was so easy to remember that, not only can I still recall it, but the memory is so strong I cannot use Lorayne's system without confusion.

I came across the major system I use in a book by Jean Hugard. I argue that it is more natural and easier to learn. There were only a few rules to remember:

1,2,3 correspond to the consonants l,n,m respectively. This is easy to remember since the letters require 1,2, and 3 strokes to write. If you happen to know the British sign language alphabet, observe that you place 1, 2 or 3 fingers on the other hand's palm when signing l, n and m respectively.

Some mappings are based on the way you pronounce digits in English. 4 is r (think “fourrr”), 5 is f or v (think “five”). 0 is s or z (think “zero”).

Then there are the digits that look like letters. 6 is b or p, 7 is t, th or d. 9 is k or g. With sufficiently bad handwriting (or fonts), 6 and b are indistinguishable, as are 9 and g (or q, which in English is always pronounced starting with a k or g sound, and always as a k sound in several European languages). 7 and T are also similar.

The only rule I never liked much was the one for 8 (which is not a problem since being the odd one out makes it easy to remember!): “Eight” sounds like “aitch”, which hopefully helps you remember that the sh, ch (and j) sounds correspond to 8.

Digit	Consonant(s)	Reason
1	l	strokes
2	n	strokes
3	m	strokes
4	r	sound
5	f, v	sound
6	b, p	shape
7	d, t, th	shape
8	ch, j, sh	special
9	k, g, q	shape
0	s, z	sound

I also mostly prefer the mapping from playing cards to words as presented by Hugard. Although I dislike the aces been treated specially and agree with Lorayne assigning the names of the suits to the jacks, I believe thinking KH as a groom and QH as a bride for example is easier to learn.

However, there is at least one practical benefit to Lorayne's system. By Benford's law, it is more likely that a number one wishes to memorize begins with a 1, and it is easier to think of a word starting with d, t or th than to think of one starting with l.

Party Tricks

This page is developing a life of its own. It's getting updated frequently with useless random information. I'm embracing the inevitable and intend to make it look like what it has become: a blog.

Geeky party tricks:

1. The Doomsday Algorithm[Wikipedia article]: calculate the day of the week for any date instantly.
2. Do Cube Roots of 9-digit Numbers in Your Head

Drums, Engines, Sudoku, Poker Chips, Dodecahedrons

Some links I don't want to forget:

1. Drumkit implemented in Flash
2. Animated Engines.
3. Sudoku at Number-Logic.com: I've lost a lot of time here recently. I only recently discovered these frustrating puzzles, but according to the Wikipedia entry on Sudoku, they've been around since 1979.
4. Poker chip tricks have also been eating up my time of late. Note there are different ways to do poker chips tricks.
5. 12-sided calendar: print and construct your very own dodecahedral calendar.