This is an excellent question, to me! I was shocked that the person who did it, did it, and it didn't really sit well with me. The explanation given didn't really feel true, and I found it confusing, but I dismissed it, again, because at this point I'm back to my reading-and-accepting mode.
But. I love that you bring up the discovery of the truth, and yes, I would have much preferred that Christopher be the one to do it, on his own. I think he's perfectly capable of making such a deduction and I would have liked to have read that scene. And to be inside his thinking as he began to realize it, and to be there too when he decides to confront the person. This was, perhaps, a missed opportunity by Haddon to further share the workings of Christopher's mind.
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Jennifer Graf Groneberg