Editor's note: In Thursday's programme Melvyn Bragg and his guests discussed Fermat's Last Theorem. As always the programme is available to listen to online or to download and keep - AI

    Hello

    No sooner had we finished what I thought was a brilliantly clear explanation by the three academics of Fermat's theorem than Marcus du Sautoy said, "the Travelling Salesman Problem is the next one you should tackle".

    It sounded like something out of Thomas Hardy, but I still had enough energy to say "what's that?" Marcus explained that the problem was how to find a mathematical formula which would enable you to visit a number of cities in the shortest possible time. This isn't too difficult to do when few cities are involved, but when an enormous number of cities are involved it becomes extremely difficult indeed to find the shortest way. Working it out would also solve another mathematical mystery, the N versus NP problem. The benefits that this could bring to commerce and computing would be enormous, not to mention allowing football supporters (Marcus is, like myself, a disappointed Arsenal fan this morning - we were all over the place, the House of Lords was stuffed with them) to calculate precisely where we'd end up in the Premier League.

    No, I mustn't trivialise it. The Travelling Salesman Problem will literally - Marcus again - "change the world". I've spent long enough now with what seem like abstruse or trivial mathematical problems not to believe that the consequences can be colossal for the future workings of the world. So watch out for the Travelling Salesman in a year or two on In Our Time.

    Yesterday I tried to wind up for this, for me, difficult feat of Fermat. I went on Hampstead Heath which I like to do as many mornings as I can. It was as if it were painted in mist. Which made it, of course, most mysterious (sorry about that, but I do feel rather euphoric after Fermat). Dog walkers came across the hills with their splay of eight or nine dogs. People seemed to drift along the short horizons to unknown destinations. The further I walked the more I came to the wooded parts of the Heath which are now carpeted in yellow gold, with leaves happily rotting into the earth. When I walked on the streets of London later, the same leaves will have been blown into machines or blown off the top of ponds, or nature being destroyed! Or at least, not to be too dramatic, removed from view.

    I saw a sign in Soho the other day which said: 'Waxing, £30, body and half leg'. Which leg? Which half? It really did say that!

    Last week I did a very long newsletter which taxed Ingrid a bit too much, so this time I will say little except that the Angel Awards earlier this week, founded by Andrew Lloyd Webber, brought to the Cambridge Theatre in London sixteen groups which had restored utterly derelict churches, or factories, or industrial buildings, or barns, or canals - an amazing array of places in this country brought back to function and to splendour by armies and armies of volunteers.

    The judges had a tough job. But in the end the four winners were: a derelict railway station in Tynemouth which is being refurbished and rebuilt to extraordinary standards, as high as you would expect from the engineering skills in the North East; a village church - congregation: 6 - which had restored a beautiful medieval tower and re-thatched the church roof and set in motion a thatching family; Droitwich Canals, at which one man had laboured voluntarily for more than fifty years and hundreds, even thousands, had helped him over that time; and the people who had withstood the flood in Cockermouth, and set their minds not only to restoring the town but to making it even better than it was before. This is the One Nation. This is the Big Society. People just getting on with it and not waiting to be told.

    I can feel that I have to stop now.

    Best wishes

    Melvyn Bragg

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