maths problem for DS

[chezlaw]

[ QUOTE ]
Could you post some problems for skalansky to break down into simple logic?

Skalansky flexes his math logic alot so i think it would be interesting to see if he approaches things as uniquely as he makes it sounds like he does.

[/ QUOTE ]
here's a very simple maths problem, so simple a kid can solve it. (so simple there's a danger DS may solve it with recourse to nothing but simple logic). Hopefully it will illustrate the point.

What's the sum of the first n natural numbers? i.e

1+2+3+....+n =

[soon2bepro]

This is what I came up with: (in white)

<font color="white">
1 | *1
2 | *1.5
3 | *2
4 | *2.5
5 | *3
6 | *3.5
7 | *4
8 | *4.5
9 | *5
10 | *5.5
11 | *6

and goes on forever...

That means that n plus all the numbers before it = n [(n/2) + 0.5]

So for example the sum of

1+2+3+4+5+6+7+8+9+10 = 10 * [(10/2) + 0.5)

= 10 * 5.5
= 55 </font>

[chezlaw]

[ QUOTE ]
This is what I came up with: (in white)

<font color="white">
1 | *1
2 | *1.5
3 | *2
4 | *2.5
5 | *3
6 | *3.5
7 | *4
8 | *4.5
9 | *5
10 | *5.5
11 | *6

and goes on forever...

That means that n plus all the numbers before it = n [(n/2) + 0.5]

So for example the sum of

1+2+3+4+5+6+7+8+9+10 = 10 * [(10/2) + 0.5)

= 10 * 5.5
= 55 </font>

[/ QUOTE ]
but was that by simple logic? Yes its simple but thats because its a simple problem.

Break it down into the steps you made and the reasoning behind them.

chez

[soon2bepro]

[ QUOTE ]
Break it down into the steps you made and the reasoning behind them.

[/ QUOTE ]

Ok, first I made a list of n and the result up to n11, calculating each result independently.

It looked like this:

1 | 1
2 | 3
3 | 6
4 | 10
5 | 15
6 | 21
7 | 28
8 | 36
9 | 45
10 | 55
11 | 66

Then I tried to spot some pattern in them. Some formulae that would work for all of them, depending on n of course. I started finding patterns until about 5 minutes after starting I got the whole deal and broke it down into the math.

At first I noticed that the odd numbers's result was a multiple of n. for 1, it was 1. for the second odd number, 3, it was 2. For 5, it was 3, and so on. Then I noticed that 1.5, 2.5 etc, worked for the even ones. It was all pretty simple from there.

[David Sklansky]

First off I think you misunderstood my original comment about math and logic. I didn't mean that advanced problems can be easily solved without recourse to advanced math. I meant that they could in theory be solved that way. After all advanced math is just a logical progression from less advanced math. (Which is why it is ridiculous to postulate someone who is extremely adept at everything through calculus could have more trouble with higher math than someone who wasn't.)

Meanwhile unless you have some trick up your sleeve your problem is a perfect example of my point. Add them up two at a time from the outside inwards. n/2 pairs of n+1 totals.

[nightwood]

It's idle to debate whether a step is "simple" or not; for some it will be for some not, depending on various factors.

Once a problem is solved and the proof is tested it's quite easy to say "Aw, yes, all but simple steps." - now, getting to this solution is a completely different thing.

Let's just take a slightly more difficult problem - Euclids Proof of the infiniteness of prime numbers. It's very easy to grasp .. but I'm quite interested if anyone's willing to argue that it's easy to come about it.

[chezlaw]

[ QUOTE ]
Meanwhile unless you have some trick up your sleeve your problem is a perfect example of my point. Add them up two at a time from the outside inwards. n/2 pairs of n+1 totals.

[/ QUOTE ]
That requires an idea. Once you've had the idea the rest is simple logic. Many people would never have that idea but can grasp it if its pointed out because integers are simple and its a simple idea.

To the point, the idea and understanding are not reducible to simple logic and its the inability to grasp the concepts and ideas that prevents people from doing advanced maths.

All of us are limitation in this respect and its why although anyone of moderate intelligence can do maths methods (differential equations etc) most struggle with advanced abstract algebra or number theory - these require very high intelligence to graps the concepts and ideas.

chez

[bigpooch]

Technically, if n is odd, you won't have "n/2 pairs".

A very simple solution, one that most children understand,
is based on the commutativity of addition and is really the
same idea:

1+2+3+...+n = S
n+(n-1)+...+1 = S

Adding the two equations above,

(n+1)+...+(n+1) = 2S

or n(n+1) = 2S
or S = n(n+1)/2.


IDEAS
=====

An idea can be very simple (such as above) or complex (such
as Kummer's idea of numbers of the form a+bi in attempt to
solve FLT). Even when an idea fails to solve a problem, it
can create a body of theory. What differentiates great
problem solvers from ordinary ones is not only perseverance,
but finding the "key ideas". Often, solving problems
require many ideas; that is why mathematicians often need
to have exposure to analysis and abstract algebra as well as
their area(s) of research.

Also, some ideas for proofs are very beautiful, but often
require some background. For example, I am still enamored
with Furstenberg's proof of the infinitude of primes (based
on point-set topology), but obviously it is not as easily
accessible as the common proof attributed to Euclid.

[Enrique]

Furstenberg's proof is beautiful. Here's the link

[nightwood]

[ QUOTE ]
Furstenberg's proof is beautiful. Here's the link

[/ QUOTE ]

Very much OT but - it definitely is.