memory and answers

Visualisation of the (countable) field of alge...

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I attended a public lecture yesterday by Professor E.Victor Flynn on some fields within Algebraic Geometry…It was incredibly fascinating. Although maybe 20% of it went over my head, it did feel like i’d simply have to jump up to be able to reach it. So when I got home, I decided to google some of the ideas that had been expressed.

I always knew number theory was a diverse field, but I never realised HOW diverse it was. At one point, I had 6 or 7 tabs open, with each one linking to one of the others, sharing ideas and definitions – you could not read any one page completely without knowing content from the others. Most of the time, it was simply a case of definitions – when talking about an algebraic number field, you need to know about field extensions, fields and Rational numbers, and knowing the definitions of these terms allows you to understand the definition of an algebraic number field – and so this bought up some interesting questions for me.

Before I’d started looking into these ideas, I’d done my normal routine of checking 20 or so math based blogs for new content. one blog –  “Godel’s lost letter and P=NP” – spoke about the importance of memorisation. Now, if you’ve read my previous blogs, you may notice that I find memorisation of theorems and definitions to be a complete waste of time.

I think now I am beginning to see the error of my ways…

without knowing definitions, we cannot hope to know other definitions that depend on the earlier ones. If an algorithm F works because algorithm G works, we need to know how G works to show F works.

On analysing these thoughts (and ideas expressed by others), I re-evaluated what I thought and why. I think one of the comments on the other blog found the real issue – ‘rote’ memorisation.

however, my thoughts are still developing on this front, and maybe at another point I shall come back to the idea. I agree – and think I have thought this way for some time, but failed to notice it –   that knowing definitions and theorems etc are incredibly important. I think what I have issue with is how we learn them, and how we learn to apply them.

now, moving on to the questions from the last post:

I still don’t have much of an idea for the second question – in truth I’m a  bit bored of it, so I’m just going to leave it.

but as for the first, this one appeals to me!

so: what do we know?
the number plate is only 4 digits long and contains 2 unique digits, so it’s of the form aabb or abba or abab. As the eldest child is 9 years old, it must be divisible by 9, so

9|(2a+2b)

⇒ 9|(a+b) with  0 ≤ a,b ≤ 9

⇒a+b=9

now, what else do we know? 8 children, each with a different age. the eldest is 9, which means that the other 7 children are either 1,2,3,4,5,6,7 or 8. therefore, there is either a child of 4, or a child of 8, which means the number is divisible by 4.

this tells us the number plate was one of:

9900, 1188,7272,2772, 3636,6336 or 5544.

this gives my smith an age of 00, 88, 72, 36 or 44. logic would dictate that 00 is impossible, 88 and 72 highly improbable. we shall include them for now, but we won’t include 00.

now notice that none of these possible numbers are divisible by 5, which means the children’s ages are 1,2,3,4,6,7,8 and 9. so the number must be divisible by 504=(9*8*7).

simple calculation gives:

1188 mod 504=180

7272 mod 504=216

2772 mod 504=252

3636 mod 504=108

6336 mod 504=288

5544 mod 504=0 (504*11=5544)

so the number plate was 5544, the children’s ages were 1,2,3,4,6,7,8,9 and Mr Smith is a (presumably) very tired 44-year-old.

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Rambling puzzles…Number plates, shoelaces and you.

Square root of x formula. Symbol of mathematics.

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I recently read 2 rather interesting questions online. The first was this:

Mr Smith has 8 children, and each one is a different age. His entire family is in the car, when his eldest, aged 9, shouts out “Daddy! That number plate has only two numbers, and each number is repeated twice! And the number is divisible by each of our ages!” “You’re Correct!” Mr Smith shouts as he whips out his iPhone to check. “and look, the last two digits are my age!”

What is Mr Smith’s age?Actually the question wasn’t like that, it gave you choices and you had  to say which of those numbers were not the age of one of his kids, but this question is a bit harder and more interesting.
The second question:there are 30 shoelaces in a closed box with all of their ends sticking out (so 60 ends). You tie one end to another, and keep doing this until each end is connected to one other.
How many ways are there of doing this?
What is the expected numberof loops?This was on the “Mind Your Decisions” Blog, and I gave a comment which got beautifully shot down by a fellow commentator, And I’m still uncertain who was right.
But anywho, have a go at the puzzles and see if you can solve them.
The first question I don’t know if it’s possible to answer without extra info. If it isn’t, try work out what extra info you need.
As for the second, the only decent suggestion I have is you start with a smaller number and see what happens.
I also recently discovered that Cambridge University has a very different teaching approach to mathematics. They arrange for 2 students to meet up with a lecturer to go over mathematics that they’ve been doing. I’m quite jealous of this! And yet I wonder if I’d actually take the opportunity…doubtless there are people somewhere who will hear about Auckland University and be amazed I haven’t taken more opportunities that have been offered to me, yet from my point of view, they don’t seem like options.
One thing I’ve always been told to do is go and speak to my lecturers. (In fact, I’ve told people to do it myself). Yet I never have. I always think “what would I talk to them about?” I have one query for one of my lecturers, but it seems incredibly trivial and it has no point other than to give me something to talk about with him for about 43 seconds (it takes about 13 seconds to say hello, how are you etc normally). And then? Do I just leave or what? This is why I would like the chance to have a supervisor, but the truth is I do have the chance, I’m just not taking it.I was aiming to speak to one of my lecturers this last week, but have been unable to attend uni this last week and a half…so I think I’m going to aim to go and speak to some of them next week. With some good questions. Hopefully.
I’ve been reading a fair amount of mathematics recently,online and print, about mathematics and mathematicians. There are a variety of mathematicians, some are ‘gifted’ (or ‘genius’) and others persevere and work hard at it – and of course there are varying degrees of both in some mathematicians, and some great mathematicians have neither. But no matter what ‘category’the mathematician falls into, one key characteristic is passion, passion for mathematics. Another, a willingness to forge a new path that no ones been down before.
It is this second that strikes me most. Too often we do what we are expected to do, and not what we want to do. The joy is doing what we want. If we feel forced into doing it, there is not much joy there.
Some mathematicians will tell you to memorise important theorems and proofs – that is the way to be a good mathematician. Others will tell you to discover the proofs and theorems yourself – that is the way to be a good mathematician. And yet others will tell you find a good mathematician and be their ‘apprentice’ – that is the way to be a good mathematician.
But I guess it all comes down to us as individuals. What do you want to do? How do you want to do it? Forget about the marks or opinions,what do you want to do?