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[Fossaman]

For a term like 45a⁵ you would just tear apart both parts. So you get {3, 3, 5, a, a, a, a, a} as factors for the completely factored form. Mix and match any factors you like to get something like 9a⁴.

[lordnincompoop]

For a term like 45a⁵ you would just tear apart both parts. So you get {3, 3, 5, a, a, a, a, a} as factors for the completely factored form. Mix and match any factors you like to get something like 9a⁴.

Thanks!

[Eagle_eye]

does anyone know an equation that can be used to calculate pi?

[Shinziril]

Here's the Leibniz formula for pi.  This is only an approximation using a non-infinite number of terms, of course, but you can keep going for as long as you like to get as much accuracy as you like.  Unfortunately, it converges very slowly, but it will work.

[Christes]

If you want something faster (but harder to understand), you can look into Machin's formula.

Of course, you have to know how to turn arctan into a series in order to use it to compute pi...

[Arzgarb]

And for something extremely fast and without fancy trigonometric functions, there's Ramanujan's formula. 8 decimal places with each iteration!

[Vector]

Goddammit, Ramanujan, why do you have to show all of us up... again?

[Eagle_eye]

so there's no non-infinite equation that gets it exactly? 

[Vector]

so there's no non-infinite equation that gets it exactly? 

No.  That's one of its more interesting qualities.

Look up transcendental numbers.

[Bouchart]

Ok there's this math problem that I've had in the back of my mind for some time.

You have a 10 by 10 grid.  In each square, you can place an A or a B.  They each have the following values:

A always has the value of 1.
B has a value of 1, plus 1 for each A in the surrounding 8 squares.

So a 3X3 grid like this:

ABA
AAA
BAA

has a value of:
7 for the A's
plus 10 for the B's for a total of 17.

Maximize the value of the 10x10 grid. 

[Jim Groovester]

I think a checkerboard pattern would maximize that but those are my initial thoughts looking at the problem.

[ed boy]

Let's try a few patterns and see what we get.

Code: [Select]

AAAAAAAAAA   /10/   AAAAAAAAAA   /10/   AAAAAAAAAA   /10/   ABABABABAB   /19/   ABABABABAB   /24/   
ABAABAABAA   /34/   ABABABABAB   /50/   BBBBBBBBBB   /66/   BBBBBBBBBB   /38/   BABABABABA   /29/   
AAAAAAAAAA   /10/   AAAAAAAAAA   /10/   AAAAAAAAAA   /10/   ABABABABAB   /19/   ABABABABAB   /29/   
ABAABAABAA   /34/   ABABABABAB   /50/   BBBBBBBBBB   /66/   BBBBBBBBBB   /38/   BABABABABA   /29/   
AAAAAAAAAA   /10/   AAAAAAAAAA   /10/   AAAAAAAAAA   /10/   ABABABABAB   /19/   ABABABABAB   /29/   
ABAABAABAA   /34/   ABABABABAB   /50/   BBBBBBBBBB   /66/   BBBBBBBBBB   /38/   BABABABABA   /29/   
AAAAAAAAAA   /10/   AAAAAAAAAA   /10/   AAAAAAAAAA   /10/   ABABABABAB   /19/   ABABABABAB   /29/   
ABAABAABAA   /34/   ABABABABAB   /50/   BBBBBBBBBB   /66/   BBBBBBBBBB   /38/   BABABABABA   /29/   
AAAAAAAAAA   /10/   AAAAAAAAAA   /10/   AAAAAAAAAA   /10/   ABABABABAB   /19/   ABABABABAB   /29/   
ABAABAABAA   /19/   ABABABABAB   /25/   BBBBBBBBBB   /38/   BBBBBBBBBB   /24/   BABABABABA   /24/   
Score: 205          Score: 275          Score: 352          Score: 271          Score: 280
I'm pretty sure the middle one is the best. If you swap one of the As for a B, you gain two points for the surrounding As, but lose 6 points from the surrounding Bs (lose 3 on the top row). If you swap one of the Bs for an A, you gain two points from the surrounding Bs, but lose six from the surrounding As (different values on the ends, but it's still not worth it.

Of course, there could be a completely different arrangement that has a higher score, But I believe this is the highest scoring such pattern.

[Bouchart]

Yeah my initial thought was alternating rows of A's and B's but I couldn't prove it out in any way.

I also don't know if it would hold for a grid with an odd number of squares.

[ed boy]

I'm almost certain it would hold. No matter where a grid square is, it is better off in the alternating rows (or columns) layout than any other.

[eerr]

Alternating diagonals.

Man I am just not on the ball for this stuff.

Kind of reminds me of a capacitor.

Also, with the details given, there is no difference in value between A and B.
A=B