This year has seen a flurry of results as mathematicians hunt down the elusive proof of the twin prime conjecture. Will they get their wish for Christmas this year?

Are there objective chances in the world?

The fact that a sizeable proportion of the financial workforce is made up of physicists is one of the industry's best-kept secrets. We talk to Laura Tadrowski, who has made the leap from physics to finance.

The Fibonacci sequence – 0, 1, 1, 2, 3, 5, 8, 13, ... – is one of the most famous pieces of mathematics. We see how these numbers appear in multiplying rabbits and bees, in the turns of sea shells and sunflower seeds, and how it all stemmed from a simple example in one of the most important books in Western mathematics.

You don't need to count to see that five apples are more than three oranges: you can tell just by looking. That's because you were born with a sense for number. But is that sense related to the mathematical abilities you develop later on?

If chemistry makes you think of white lab coats and green liquids then think again. This year's Nobel prize goes to three researchers who "took chemical experiment into cyberspace".

The 2013 Nobel prize in physics goes to Peter Higgs and François Englert for proposing the mechanism that gives things mass.

Why does time only ever move in one direction? We talk to philosophers of physics Jeremy Butterfield and David Wallace, as well as the eminent Roger Penrose about the puzzle time poses to physicists and what it has to do with the Big Bang and the second law of thermodynamics.

Is the Universe finite or infinite? Is there infinity inside a black hole? Is space infinitely divisible or is there a shortest length? We talk to philosophers and physicists to find out.

Images are now being taken on the world's most powerful digital camera. For over 500 nights over the next five years the Dark Energy Camera will photograph the light from more than 100,000 galaxies up to 8 billion light-years away in each image.

A commonly held belief about medieval Europe is that academic pursuits had fallen into a dark age. The majority of scholars were churchmen, and their enquiry often related to some principle of church practice. But is there a value to respecting the tenacity of historic mathematicians?