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What is cutting edge maths? - Mathematics Stack Exchange My maths teacher always keeps telling me about this 'cutting edge maths' that is going on in the world, amazing maths research, etc A lot of the google searches I've done for 'Cutting Edge Mathematics' hasn't returned much useful information, so I've taken to mathematics stack exchange
When you randomly shuffle a deck of cards, what is the probability that . . . The situation is not the same as in the birthday paradox The birthday paradox works because the two identical birthdays may appear between any two of the persons However, in your experiment, you demand that you are one of the two persons involved in the same card deck A situation analogous to the birthday paradox would be given by the question "what is the chance that over the last 600
geometry - Why is the volume of a cone one third of the volume of a . . . The volume of a cone with height h and radius r is 1 3πr2h, which is exactly one third the volume of the smallest cylinder that it fits inside This can be proved easily by considering a cone as a solid of revolution, but I would like to know if it can be proved or at least visual demonstrated without using calculus
Can the MRB constant be interpreted via alternating sums of edge . . . The MRB constant is the limit point defined by lim x → ∞(∑2xn = 1(− 1)nn1 n) This sequence involves the nth roots of n, which can be interpreted geometrically as when across all hypercubes with hyper volume equal to n, the even partial alternating sum of these edge lengths converges to the MRB constant Since a unit hypercube's longest diagonal in n dimensions is equal to √n, and
arithmetic - Can a piece of A4 paper be folded so that its thick . . . If, however, the pressure created by making the folds (and cutting the paper into tiny rectangles and whatnot) compresses the paper so that it becomes less than ~0 1mm thick, then our exponent will no longer be valid Say that during the cutting process, the cellulose fibers unravel somewhat, leaving only two layers of fiber
Why is the volume of a sphere $\\frac{4}{3}\\pi r^3$? As the plane cutting through the solids moves, these blue squares will form 4 small pyramids in the corners of the cube with isosceles triangle sides and their apex at the edge of the cube Moving through the whole bicylinder generates a total of 8 pyramids
Is it possible to draw this picture without lifting the pen? This means not only can you trace it without lifting your pen, but that you can do so in such a way that each edge is crossed only once, and also that you end where you began! Rory's answer shows one way this can be done
Visually stunning math concepts which are easy to explain Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain but are
Calculus proof for the area of a circle - Mathematics Stack Exchange I was looking for proofs using Calculus for the area of a circle and come across this one $$\\int 2 \\pi r \\, dr = 2\\pi \\frac {r^2}{2} = \\pi r^2$$ and it struck me as being particularly easy The only