I just finished my second victorious game of Civ5. I played as Hiawatha on an Earth-type standard-sized map. The starting place looked nice with a river, forests, plains and some hills but it quickly turned out to be a bit crammed: there was tundra to my north, mountains and then desert to my south and city-states to my east, west and south-west. I still managed to establish four cities before the borders of the city-states and my other neighbors came too close. My other neighbors were Gandhi to my east, Augustus Caesar to my west and Wu Zetian (Civ5) to my south, all behind the aforementioned obstacles.
With some open borders agreements I eventually found out that I had started the game somewhere in Russia -- mind you, I was playing on an Earth-type map. Augustus held Europe, Gandhi China and the Pacific region, Wu Zetian India, and so on. There were just two civs in the Americas, I never much interacted with them. Also, I never built a coastal city so I didn't get any proper ships, exploring the world with embarked military units and even a couple of workers. Also, no one ever settled into Australia, it stayed as the final frontier for the Barbarians.
Anyway, I figured a culture victory would be possible from where I stood so I focused on buildings that provide culture and didn't build that many units. At some point Alexander (who controlled most of the Africa), however, started getting hostile towards pretty much everyone, especially the city-states, so I built a few units and fortified them to my border. Luckily, nobody attacked ever attacked me in the game, except for the Barbarians. I actually felt like my economy relied on Barbarians: I was making a steady loss of one or two pieces of gold for the first half of the game but always managed to destroy Barbarian camps for some money. Halfway through the game I figured I should disband the useless units and even sell some unnecessary buildings and managed to fix my economy a bit. The golden ages were also very important for my economy, as during those I was easily making some 30-50 gold per turn.
In the 1970s or so I did some math and figured I'd never have the time to complete the Utopia Project, even if I managed to completely unlock the required five social policy branches. I didn't let this bother me that much as I felt was in a safe lead in the scores. The only thing that could prevent me from winning the game was if Alexander or Augustus Caesar would declare a war on me (which never happened). However, in 2010 I noticed something alarming: Gandhi actually had a few more points than I! I had about 1040 and he was at about 1060. Our closest competitor was Augustus Caesar who had less than a thousand points. So, what to do? The only possible way I saw was declaring a war on Gandhi. First I contacted Augustus and Alexander and bribed them to start a war against Gandhi, though.
Alas, Augustus and Alexander didn't have much common border with Gandhi so they couldn't do much about the situation. I quickly upgraded my musketmen, riflemen and other units to infantry units and other modern units and declared the war. My first target was their city that was close to the borders of my empire, to the east. Even though it was in a tight, mountainous spot it didn't take more than a few turns to capture with the help of one or two artillery units nicely placed within my own territory, three squares away from the target city.
After this Gandhi quickly mobilized his troops as well and provided me with an unpleasant surprise: he already had tanks! Luckily I had the technology to build anti-tank guns but one tank I had to destroy with a couple of infantry units and even a lancer (Civ5). Another tank killed another of my lancers but after finishing the first two anti-tank guns I quickly hunted it down.
My second target was their small city to my south, in a desert behind some mountains. I managed to attack this from two directions (as I had open borders with Wu Zetian) and took it quickly. India's capital was to the east from that city and to the south from the first city I took. I accidentally had an infantry unit killed before I was ready to take it because they had an artillery fortified there and could shoot further than I thought.
So anyway, there was also a city to the south-east from the city I first captured. It was behind some mountains so I employed a tactic where I would move and infantry unit by one hex so that it would see the target city, shot at it with my artillery units over the mountains and moved the infantry unit back. I managed to soften the city so much that I could eventually take it with my final lancer -- the infantry unit was killed when Gandhi attacked it with his infantry... Finally, Gandhi put up a good fight for Delhi and almost captured it back as well, but couldn't match my infantry, anti-tank guns and artillery. I unlocked the final social policy required for the Utopia Project in 2048 and would have taken another of Gandhi's cities in 2050 or 2051 but the game ended and I did indeed win by a fair margin. :)
I should mention that I puppeted all of the cities I captured and that Gandhi asked for peace two times: the first time he thought he was making a favor for me and demanded for all my luxury resources. Naturally I declined his offer, and at some point he came back, this time asking for peace with no extra strings attached. I declined that offer as well because I felt that Gandhi should at least give me a few cities before I could forgive him for acquiring more points than me... I think his next offer could have been more acceptable. ;)