PT_: I assume you're Dutch?

Definitely

I haven't solved it, but in hope it may give a help to someone else:
I think it must be significant that all of these are primes.
A is: 1st prime, 1st prime, 2nd prime, 791st prime, 25th prime, 4th prime, 95th prime, 4th prime.
B is: 3rd prime, 1st prime, 115th prime, 3rd prime, 4th prime, 24th prime, 15th prime, 2nd prime.

Which has the effect of reducing it to two new sequences, which I can't make sense of!

Well, my brother has the solution for both sequences
a) are the first X digits of... sqrt(5) !
b) are the first X digits of... 10/19 !
It's so easy now that I know it I only need to figure out of how many digits the next number consists

Going by what OldMathTeacher said, each entry concatenates more digits until it becomes prime. A's next value is 89, while B's is 6842105263157894736842105263157.

Well done guys! I'm delighted to see these solutions, even if my share in finding them was almost insignificant. I love this kind of problem, although this one was definitely too hard for me.

Let's have more sequences!

a) V, G, C, R, D, J, F, Z, I, A, N, H, B, ?, ?, ?

b) 3, 22, 1, 9, 18, 10, 21, 27, 23, 55, 66, 90, 5, ?, ?, ?

^^ Very funny PT_

Easy way to find squares of #'s near 50:
I will use 54 as an example!

Step 1:
Find how far away it is from 50 (Subtract!)

Code:


Step 2:
Add that number to 25, and multiply by 100

Code:


Step 3:
Square the number from Step 1, and add it to the number you got in Step 2

Code:


And that's it!
54^2 is 2916, and it works with any number within (but not including) 10 of 50.

I am not sure if this works with all numbers (I.E. instead of 25, you do n/2), but feel free to correct me!

Define x to be the number you found in Step 1.

(50 + x)² = 2500 + 100x +x².

Yes, this is the easiest way to do it and, as the expansion here shows, it doesn't have to be within 10 of 50.
E.g. to find 67², so x = 17.
Add it to 25: 42, then multiply by 100: 4200.
Square the x: 289, and add to the previous result:
67² = 4489.

It works for negative x too.

I have to say this because nobody I know has as much appreciation for advanced mathematics than me.
If you are looking for a good book to read on mathematics that is more of a history lesson while teaching you a little bit about math, I recommend Fermat's Last Theorem by Amir D. Aczel. It goes on about the history of math that relates to FLT from Pythagoras to Andrew Wiles who proved FLT(no, not a spoiler). It touches on advanced geometry, the complex plane, modular forms, and elliptical curves. It is inspiring me to study advanced mathematics, and like many past mathematicians, as a side hobby.
FLT:(paraphrased)
a²+b²=c²
There are no whole numbers greater than two in which the base can be split into the sum of two whole number bases with the same power.
Example:
a³+b³=c³ does not exist in which a, b, and c are whole numbers.

PT_: Is there a graphing program for the ti83 plus that can graph equations like the ones on the cup, that have only x on one side?

Like x=2abs(sin(y))

You can indirectly achieve this using parametric graphing, where

Code:

Make sure that Tmin and Tmax in the window settings are set to the minimum and maximum Y values.

Alternately, you could use a program (such as this one that I found after a cursory search) to graph the equation.

I'd have thought you'd be better off with Simon Singh's book of the same title, based as it is on his substantial interviews with Wiles and some of his Princeton colleagues. All the reviews I can find which compare the two books prefer Singh's. I wish too that the BBC documentary which foreshadowed his book was available on DVD, as it brings out the human side of mathematical discovery better than anything else I have seen (better even than "The Man who Knew Infinity", since that is a partly fictionalised account rather than interacting directly with the mathematicians themselves).

Sounds interesting also. I haven't read that book.
Anyone know a really good multivariable Calculus book under $50?

Have fun!

The Carillonveld is a popular park at the university where many students spend their lunch break. This park is interesting because its shape resembles a regular hexagon (polygon DEFGHI).
During a lunch break, Amber, Brenda and Cees walk through the park, each of them choosing a different path (A, B and C). Paths A, B and C are parallel to the sides of the park (see the figure below). The hexagon is divided into seven sections, four of which are equilateral triangles (side lengths indicated in the figure below).
Determine:
a. What is the circumference of the entire park?
During another lunch break, Amber takes a longer walk through the park. She first walks along the path from A1 to A2; and then back to A1 via D, E, F, and G.
b. What is the size of the area surrounded by this path?