Month: July 2009

gently musing about marginalia and related issues

Whenever Wimbledon comes along, I am pleasantly reminded of a question I was asked at school.

The question was simple. If you have 128 people playing in a knockout tournament, how many matches will it take to complete the tournament? Assume no draws or replays.

When we were asked the question, everyone knew the traditional way to get the answer. 128 people. 64 pairs. 64 matches in the first round. Then 32. Then 16. Then 8. Then 4, then 2, then 1. Giving us 64+32+16+8+4+2+1 or 127.

Later on that day, a few of us got into conversation with the teacher, and an alternative route was broached. 128 people. How many winners? One. So how many losers? 127. So how many matches will that take? 1 loser generated per match. 127 matches.

How I loved the simplicity of that approach. Just work out the number of losers and you will have the number of matches. I could have danced all night.

I felt the same way when I first learnt about decimals. Why some decimals “terminate”. Why others “circulate”. How beautifully they do this.

The way I was taught it was something like this. Think of a fraction. Convert it into a decimal. Think of another fraction. Convert it into a decimal. Do this a dozen times with different fractions, and observe the results.

It soon became clear that most fractions didn’t terminate cleanly, they “recurred”. Fractions like 1/3, 2/7, 3/11. Some fractions, on the other hand, ended cleanly, fractions like 1/2 and 2/5 and 3/10. It didn’t take too long to see that the only fractions that terminated were those where the denominator contained factors of 2 and/or 5.

2 and 5. In a base 10 world. Oh yeah. That bored me. So what did I find interesting about decimals? Again, it took a question.

And the question went something like this. Other than those with 2 or 5 as denominator, all fractions that have a prime number as denominator recur when expressed as decimals. The length of the recurring number or numbers is called the period of recurrence. So for example 1/3, or 0.333333…. has a period of 1, 1/11 or .090909…. has a period of 2. What is the maximum period of recurrence of any fraction where the denominator is a prime other than 2 or 5?

Which led to a lovely meandering journey. To convert a fraction into a decimal you have to divide the numerator by the denominator. You keep carrying the residue over until it is zero (in which case the fraction terminates). Or you keep carrying over and over and over.

When would a fraction circulate? When it hits the same residue again and starts the same sequence of residues as a result.

So how long before the same residue must be encountered? That depends on how many different residues can be had before that. What is the maximum number of different residues? One less than the denominator prime.

Discovering that 1/97 indeed has a period of recurrence of 96 filled me with glee. 96 different residues before the same residue is encountered. How beautiful.

There are so many others, little stories and techniques and tricks and tips that have helped me keep a passionate amateur interest in mathematics, particularly in the theory of primes.

The trouble with being an amateur like me is that you start getting idealistic about the subject. And you believe in things like elegance and simplicity. Which means that I’m one of those guys who still believes that there must be an elegant solution to Fermat’s Last Theorem. A solution that could not be jotted down in a book about Diophantine equations, but a short and elegant solution nevertheless.

My love for mathematics exists because others put their love of maths into me first. They spent time with me and explained things to me and taught me and filled me with wonder and amazement.

It’s the same thing with so much in life. Whatever you love, whoever you love. Love has to be shared in order to grow.

So when I think about copyright and patent and stuff like that, when I think about music and art and stuff like that, I think about love. The love that a creative artist puts into his or her creation. And how love has to be shared in order to grow.

Thinking about teachers and learners

For some people, the internet, the web and social networks are all about A-lists and cabals and cliques and echo chambers. I don’t know about that, I’m not some people. I find the web very useful.

One of the things that distinguishes this continuing-to-emerge space from all that went before is in the context of learning. For anyone who wants to learn, the web is a wondrous place. I want to learn. So for me the web is a wondrous place.

Learners need teachers. On the web, this often means people who sacrifice incredible personal time and energy writing out what they’re thinking about, what they themselves are learning, so that they can teach others. And learn more themselves as a result; teachers need learners as well.

The best teachers are usually themselves lifelong learners; the reason they teach well is that they are learning as they teach, and they take care to do that.

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Bob Frankston and Dan Bricklin, the co-creators of Visicalc, are two such people. Lifelong learners, passionate about everything they’re interested in, selflessly sharing what they’re learning with anyone who wants to learn with them.

I discovered their blogs early on, and for a decade or so have been able to enjoy their teachings and learnings from a distance. Bob’s writings can be found here, while Dan writes here. If you haven’t already done so, start reading them today. I cannot recommend them strongly enough.

Over the years I’ve had opportunity to meet both Bob as well as Dan; they’ve always been willing to spend time with me. I now count them amongst my friends and feel privileged in being able to learn from them.

Recently Dan published a book, Bricklin on Technology. A solid two-handed read, 500 pages long. What he’s done is taken the essays he’s written, grouped them into logical sections, commented and enriched them here and there, and all in all produced a wonderful collection of essays and an eminently readable book.

Dan thinks very hard about everything he does, and it shows throughout the book. The principles that drive him are in evidence everywhere, principles that need propagating and embedding far and wide.

One, as technologists we are in the business of building tools. Tools that help people do things simply and easily. Tools that can be used in a variety of ways, in a variety of contexts, for a variety of reasons.

Two, the people that use the tools are also very diverse; they’re diverse in their ability to use the tools, the skills they bring with them, the environments and contexts they operate in and with. They’re diverse in the motivations that drive them to use the tools, diverse in the aims and objectives they have in using the tools. The tools we build have to support this diversity.

Three, our ability to build tools well is increasing. We’re learning more about building tools that others will use to build more tools; we’re learning more about building tools as open platforms upon which others can build over-the-top tools; we’re learning about building feedback loops that take the emotion out of many unnecessary discussions, ways of measuring what is happening easily and cheaply.

Four, all this is being done in a social context, in community rather than as individuals. So as designers we need to remind ourselves there are social and moral aspects to what we do.

There’s no point my quoting directly from his book; there’s too much I would want to quote, and my posts are long enough as it is. So I’m just going to quote one line from his summary:

It is usually the users of technology, not the inventors, who determine how tools are applied.

In some ways that’s what the book is about. Thoughtful, considered discussions on the user perspective. How to make sure we understand the motivations and context. How to build tools that work well yet are intrinsically “free”, versatile enough to let the user choose what to do and how. How to sidestep political/emotional debates by rigorous examination of the facts. And how to keep remembering that value is generated by the usage of the tools and not by the tools themselves.

An exhilarating read, well worth the effort. While I’d read many of the essays first time around, I was particularly taken with three sections: the discussions on what users will pay for; the views on the recording industry; the interview with Ward Cunningham.

I particularly particularly particularly enjoyed re-reading the essay on Book Sharing, a classic example of what happens when two clever and gifted people discuss important things. Thank you Dan, and thank you Bob for pushing him to write it and publish it.

So use your Saturday wisely. Order the book. Now.

[A coda. I’m looking forward to Bob taking a leaf out of Dan’s book and publishing a similar book. You listening, Bob?]