What connects being late to class, computer games, and being a recluse in your mum’s house? No, it’s not the latest Grand Theft Auto game. These are, in fact, the approaches to successfully solving some of the longest standing ‘unsolved problems’ – those which have puzzled the world’s brightest for centuries, until a new approach sparked a breakthrough. Each has its own interesting story behind it…

**1. ****George Dantzig is late to class…**

An oldie but a goody – in 1933, a young phD student called George Dantzig was late to one of his statistics lectures. He saw two problems written on the board, which he assumed was the week’s homework. In the next few days, he did notice they were a bit harder than usual…he handed them in a few days late, apologizing to his professor. It turned out that these two problems weren’t homework at all, but two of the longstanding problems in statistics. You might also remember a distorted version of this story as being an introductory scene in the film *Good Will Hunting*.

Can we learn anything from this approach? Well, students for decades since have been trying this method of being late to math class, but without such great results. The story does, however, demonstrate the power of ‘not knowing what you can’t do’ – and the value of opening up problems to people with fresh perspectives, who may be unfamiliar with the perceived barriers that would put off others.

**2. ****Why does warm water freeze faster than cold water?**** **

It’s a seemingly simple question, and a counter-intuitive phenomenon – why does warm water freeze faster than cold water? Last year, the Royal Society of Chemistry decided that enough was enough, and that they would put up a prize of £1,000 for anyone who could provide a winning explanation of the ‘Mpemba effect’.

There was a huge response, as a staggering 22,000 people sent in their theories. Using the public to help filter these submissions down to a final 11, the judging panel eventually selected an explanation from Nikola Bregovic, a chemist from the University of Zagreb.

Running a prize was certainly a great approach in this case – focusing the attention of scientists in the field (Bregovic heard about it from a friend who said “this seems like something you could work out”), drawing the attention of innovators in adjacent fields, and capturing the attention of the general public, sparking an increased popular interest in chemistry.

**3. ****Fermat’s Last Theorem**

Holding the coveted title of ‘sexiest story in mathematics’, this one has it all: the age old problem, the young solver with a boyhood dream, the flaw in the theory, only for him to rise again like a phoenix from the ashes with an outsider idea….Another seemingly simple statement that had puzzled mathematicians for the past 350 years, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation *a ^{n}* +

*b*=

^{n}*c*

^{n}*for any whole number,*

^{ }*n*, greater than two. Originally conjectured by Pierre de Fermat in 1637, he cheekily claimed that he had a proof for the conjecture, but that it was too long to fit in his notes in the margin.

For centuries afterwards, a proof to the conjecture was pursued as a holy grail for mathematicians, much of this culture stirred by the numerous prizes that were put up at different times for its resolution. Despite the attention, Fermat’s Last Theorem established itself as the “mathematical problem for which the greatest number of incorrect proofs have been published”.

The story of its eventual proof by Andrew Wiles’ in 1995 is a great one, and I’d point you towards the BBC documentary on it, or the book, instead of trying to do it justice in one paragraph. In short, after years of dedicated work, Wiles unlocked the puzzle by proving and connecting the so called Taniyama-Shimura conjecture, which belongs to a completely different field of mathematics …another example of solutions coming from adjacent areas, which we often see in our InnoCentive Challenges.

**4. ****How gamers solved a problem scientists were stuck on for a decade**

In order to develop a cure for the M-PVM retroviral protease (an enzyme that plays a key role in the development of a virus similar to HIV), scientists needed to model it. The problem with modeling these proteins is that they ‘fold’ in frames measured in billionths of a second – the problem was described as “like trying to solve a million-sided Rubik’s Cube while it also spins at 10,000 r.p.m.”, and had eluded researchers and their existing software for the past 15 years.

Theorizing that their automated approach was lacking human intuition and ability to recognize patterns, they turned to Foldit – a site which turns folding proteins into a game, and has attracted hundreds of thousands of players.

Within 3 weeks, gamers had successfully modeled the protein, which researchers could take on and use to design anti-retroviral drugs, including AIDS drugs. Foldit had enabled researchers to leverage the power of crowdsourcing to deal with a complex problem, and then use gamification to motivate the crowd of gamers to work on that particular problem.

**5. ****“I know how to control the universe. Why should I run for a million?”**

First proposed in 1904, the Poincaré conjecture sought to prove that any shape without a hole can be formed into a sphere. After eluding mathematicians for close to a century, in the year 2000 the Clay Mathematics Institute recognized the puzzle as one of the 7 Millennium Prize Problems, and offered a $1,000,000 bounty for its successful proof. Its eventual proof by Grigori Perelman built closely on the work of another mathematician, Richard Hamilton, but added proof of how this theory could avoid ‘singularities’ (like exceptions), which had held back Hamilton’s attempt.

Painted by the media as a maverick recluse (there’s nothing wrong with living with your mother and not leaving the house!), Perelman would go on to reject the million dollar award. One explanation he gave was that he felt the Clay Mathematics Institute unjust for not sharing the prize with Hamilton, underlining the value of the idea sharing between the two. However, he gave a second explanation, which I find far more entertaining: “Emptiness is everywhere and it can be calculated, which gives us a great opportunity. I know how to control the universe. So tell me, why should I run for a million?”.

**Solutions come from thinking about things differently**

Looking at these 5 problems together, it’s not surprising that their resolution came from interesting approaches – using innovative crowdsourcing games, using prizes to capture the attention of new thinkers, and borrowing ideas from adjacent fields. There was a reason these problems had become longstanding puzzles – if they could have been solved the usual ways, they would have been solved already. If you do what you’ve always done, you’ll get what you always got!

* Authored By Harry Wilson, Program Manager*