Lieutenant Malcolm was a good chess player. He looked like any other young US Marines officer: skinny, shaven headed, although with a quite beaky nose. Anyway, you could always pick him out. He'd be the one with the chess board working out moves. I got to know him a little bit, as his bunk was opposite mine. I'd watch as he gave chess tips to those of his men who hadn't completely given in to poker or hearts.
About five hours into the battle, Lieutenant Malcolm was killed. He was the weapons officer in Alpha Company, 1st Battalion, 8th Marine Regiment, the unit I'd joined as an "embed". Just before dawn, Alpha Company blew a large hole in an outer wall, and entered the police station right in the heart of Fallujah. As the sun rose the marines found themselves surrounded and under attack from all sides.
Lieutenant Malcolm's squad went up on the highest rooftop they could find - but not higher than two minarets on either side with snipers. There was a wall about 15 inches tall for cover. Everyone tried to get close to it while bullets skipped across the paving stones. When he heard his men were in trouble - the men to whom he'd been giving chess tips just the day before - Lieutenant Malcolm went to get them.
As he ran onto the roof, one of the sniper's bullets hit his helmet, bouncing off. He kept going, and didn't leave until he had shepherded all his men down. He was killed by the second bullet. It got him in the back, just below the flak jacket, as he jumped down the stairwell.
Yesterday, the number of deaths for US-led forces stood at 51, with some 450 injured. These figures represent the coalition's worst losses in any battle in Iraq since the invasion.
I asked another young officer, Lieutenant Bahrns, about the massive firepower that the marines would bring to bear on Fallujah, he told me: "If there are civilians in there, they are non-combatants, then by no means do we want to hurt a woman or a child. We're here to protect them, we're here to keep them safe and we're here to turn over Fallujah back to them."
Lieutenant Bahrns was leading a squad responsible for clearing insurgents from the southern tip of Fallujah. It was by now more than a week into the battle, the longest continuous period of urban, house-to-house fighting since the Vietnam War. Alpha company was holed up in a house right on the edge of the desert. You could really see that the insurgents had nowhere else to go. Every night, though, the insurgents would attack.
Half an hour after sunset the first rocket-propelled grenades made yellow streaks across the sky, and exploded just behind us. The marine snipers would try to pick off the insurgents circling around the building. The next morning, we saw their bodies, splayed out at odd angles, already starting to bloat, the flies thick on their faces. Lieutenant Bahrns told me how he'd lost his machine gunner. The gunner had been first into a house, and been shot and killed by those inside. There was a long battle. For three hours they couldn't even get the dead marine's body out. When the marines finally stormed the house, they found three other bodies inside, each holding weapons: one was a boy, "maybe ten years old". You could tell that Bahrns was sickened about his, almost in anguish. "They were shooting at my marines, what could we do?" he asked.
Throughout the week, we saw only two glimpses of civilians. One was a group with white flags running away. Another was a shell-shocked man who was brought into the marines base on a stretcher after being found wandering the streets. The marines saw many dead bodies - some being gnawed at by dogs in the streets - but they were all fighters, even if in this case the "fighter" was a child.
Paul Wood, the BBC's Middle East Correspondent, was embedded with the 1st Battalion of the Eighth Marine Regiment during the battle of Fallujah. His film can be seen on BBC2 Newsnight at 10.30pm tonight.