Troop withdrawals begin as the Americans train their ARVN allies to take over the war. Gery Benedetti patrols the hostile waters of the Mekong Delta. James Anderson leads a battalion into Cambodia after Nixon's controversial order. Don DeVore struggles to survive a fierce night attack, and Anne Purcell gets promising news about her husband.

Donald Trump is the 45th President of the United States and commander in chief of America's armed forces, the most powerful position in the world. But who is Donald Trump? What's he really like? And what does he really believe? To find the answer, we’re going back to his roots. This documentary tells the remarkable story of Donald Trump's family, an extraordinary immigration success story.
It reveals how Trump's penniless German grandfather arrived in America in the 1880s, and ended up running bars and brothels in the gold rush. And how his Scottish mother travelled from the Outer Hebrides to New York and worked as a maid before marrying Donald's tycoon father. We will meet his driven parents, his wayward brother, and his three wives.

The turmoil of 1968 helps Nixon win the presidency. Troop strength in Vietnam peaks and the draft accelerates. Karl Marlantes endures bitter jungle fighting on Hill 484, and draftee Arthur Wiknik gets a grim introduction to combat on 'Hamburger Hill'. But shortly after winning both hills, US troops abandon them.

The enemy gains ground when the massive Tet Offensive catches the Americans by surprise. At Khe Sanh and Pleiku, U.S. troops are under siege. Americans mount a counteroffensive but the shock of the assault ignites anti-war fervour at home.

American troops launch widespread 'search and destroy' operations; body count, not territory, becomes the measure of success in Vietnam. Charles Brown fights for survival on the bloody slopes of Hill 875 and at home, the American public begins to question U.S. military strategy.

Dr Hannah Fry travels down the fastest zip wire in the world to learn more about Newton's ideas on gravity. His discoveries revealed the movement of the planets was regular and predictable. James Clerk Maxwell unified the ideas of electricity and magnetism, and explained what light was. As if that wasn't enough, he also predicted the existence of radio waves. His tools of the trade were nothing more than pure mathematics. All strong evidence for maths being discovered.
But in the 19th century, maths is turned on its head when new types of geometry are invented. No longer is the kind of geometry we learned in school the final say on the subject. If maths is more like a game, albeit a complicated one, where we can change the rules, surely this points to maths being something we invent - a product of the human mind. To try and answer this question, Hannah travels to Halle in Germany on the trail of perhaps one of the greatest mathematicians of the 20th century, Georg Cantor. He showed that infinity, far from being infinitely big, actually comes in different sizes, some bigger than others. This increasingly weird world is feeling more and more like something we've invented. But if that's the case, why is maths so uncannily good at predicting the world around us? Invented or discovered, this question just got a lot harder to answer.