Prove that e is irrational.

Find the connection between the constructed length and the original rectangle.

Find the radius of the circle inscribed inside the isosceles triangle.

Show how the values 1, 2, 4, 8, 16, 32, 64, 128, and 256 can be placed in a 3x3 square grid so that the product of each row, column, and diagonal gives the same value.

Prove that the roots of the polynomial, xⁿ + c_n-1x^n-1 + ... + c₂x² + c₁x + c₀ = 0, are irrational or integer.

Prove that the minimum number of moves to completely reverse the positions of the coloured counters can never be square.

Prove that π is irrational.

Which fits better... a round plug in a square hole or a square plug in a round hole?

Prove that cos(x) is algebraic if x is a rational multiple of Pi.

Find the side length of the square inscribed inside the right angled triangle.

What fraction of the large red circle do the infinite set of smaller circles represent?

Key concepts [More]

The infinite! No other question has ever moved so profoundly the spirit of man,” said David Hilbert, one of the most influential mathematicians of the 19th century. A subject extensively studied by philosophers, mathematicians, and more recently, physicists and cosmologists, infinity still stands as an enigma of the intellectual world. Thinkers clash over questions such as: Does infinity exist? Can it be found in the physical world? What types of infinity are there? Through an interdisciplinary discussion with some of the world’s leading thinkers, this program will delve into the many facets of infinity and address some of the deepest questions and controversies that mention of the infinite continues to inspire.

Probably Approximately Correct: Nature's Algorithms for Learning and Prospering in a Complex World [More]

From Nature magazine

From Nature magazine

From Simons Science News ( find original story here ).

Few of us have not paused at one time or another to marvel at the beauty of a soap bubble. The iridescent, evanescent orbs, which can persist for minutes before vanishing in an instant, have captivated bubble-blowing children and pensive bathtub recliners alike.

When summer hits, I dread jogging outside. But a study finds that elephants can be in true danger in the heat.

From Simons Science News ( find original story here ).

In 2008, for the first time, U.S. women earned more doctorates in biology than men did. But advanced degrees in other core disciplines of science, technology, engineering and mathematics (STEM) remain stubbornly gender-imbalanced. In chemistry, for instance, women now garner 49 percent of bachelor's degrees but only 39 percent of Ph.D.s. What dissuades so many from further study?

Among the winners in Pres. Barack Obama’s 2014 federal budget: the U.S. Geological Survey, Department of Education and National Science Foundation (NSF). The losers include agribusiness and the Environmental Protection Agency. Obama’s plan, announced April 11, is less a vision for a progressive science agenda than a reflection of political reality--specifically, the fact that he still must work with a conservative Congress. The budget gives a boost to science education and research, long-standing administration priorities. In a nod to fiscal conservatives it trims administrative costs and slashes subsidies for agribusiness. On the matter of fossil fuels, however, it largely maintains the status quo, preserving funding for drilling and fracking. Here’s a quick list of winners, losers and how the new White House budget would affect them.

A heavy metal concert might be a tough place to think about physics, especially in the mosh pit, where some audience members dance violently. But the mosh pit itself is actually an interesting place to find physics in action. And not just force equals mass times acceleration.

From Simons Science News ( find original story here ).

I take issue with the celebration of Paul Erd s’ 100 th birthday.

From Simons Science News ( find original story here ).

What do polar ice caps, guinea worm disease and wildfires have in common? All are being modeled with cutting-edge mathematics. Mathematical societies and institutes around the world are participating in " Mathematics of Planet Earth ," or MPE, this year. They aim to study the math that underpins geologic and biological processes on our planet as well as encourage more math researchers to tackle these problems. Events are planned for the year 2013, but the organizers hope that the initiative will have lasting effects.

It has been four decades since Belgian mathematician Pierre Deligne completed the work for which he became celebrated, but that fertile contribution to number theory has now earned him the Abel Prize, one of the most prestigious awards in mathematics.

This story was originally published by Inside Science News Service .

[More]

Camping enthusiasts and aspiring modern sculptors take heed: researchers have achieved a breakthrough in understanding and controlling overcurvature, which is found in such disparate settings as pop-up tents, DNA plasmids and curved origami. Overcurvature occurs when a ring is too curved to lie flat in a plane the way a normal circle does. For example, if you detached a segment of a Slinky and connected its ends to make a closed loop, you would have a hard time getting the whole thing to lie flat on the floor. The intrinsic curvature of the Slinky would cause the ring to buckle and assume a three-dimensional saddle shape.