“Let’s have a look.”

He put his work down next to mine. And this is what they looked like:

His:

Proof

B and D are the respective images of A and C after translation F. The image of the line segment AC after translation F is therefore the line segment BD. I is on the line AC, therefore its image J is on the line BD.

I is the midpoint of AC, therefore IA = IC.

Translation preserves lengths, so JB = IA and JD = IC.

Therefore JB = JD.

If a point is on the line BD and is equidistant from B and D, it must be the midpoint of the line segment BD.

Conclusion: J is the midpoint of BD.

Mine:

Proof:

ABC is a triangle with one invisible side. I is the middle of the line BC. The invisible side AC touches the line IJ and the line DB.

Conclusion: J is more or less in the middle of the line BD.

Frankly, they didn’t have a lot in common. My work didn’t look much like anything at all. I’ve noticed that it’s often by comparison that mistakes become mistakes. I looked anxiously at Haisam. He showed my work to his father, who pulled a strange face as he put away the chess pieces.

“Did you do this on your own?” Haisam asked.

“Yes!” I said proudly.

“Oh, I see! That explains it.”

It was then that I realized he’d guessed the whole business with Marie. There was really no point in trying to hide anything from my honourable friend.

“You’d better ask her to help you revise geometry.”

“Why? Is it wrong?”

“Totally. But it’s quite funny. It’s even funnier than the real thing … one invisible side… That’s quite something!”

He slid the paper in front of me.

“Read me your conclusion. I think I must be hallucinating.”

“Well, yes, OK: ‘Conclusion: J is more or less in the middle of the line BD.’ Personally, I think it has a certain style.”

Haisam looked dismayed.

“You need to understand that ‘more or less’ doesn’t exist in maths… Here, copy down the solution.”

And that was when I said completely the wrong thing.

“Are you sure it’s right?”

I thought the world was going to stop turning on its axis and the snowman in the playground was going to melt away all in one go. Haisam’s father dropped all the chess pieces on the floor and they both stared at me as though I was some really peculiar creature. The Turkish father of my Egyptian friend made a triangle shape with his fingers and said, in an icy voice, “So you think, do you, that the people who built the pyramids would have trouble drawing a simple triangle?”