Geoffrey Case was a much more difficult challenge. He had died a year after graduating. I had long ago noted his basic details from the reunion website. He had not married and had no (known) children.

Meanwhile the DNA samples trickled back. Two doctors, both in New York, declined to participate. Why would medical practitioners not participate in an important study? Did they have something to hide? Such as an illegitimate daughter in the same city that the request came from? It occurred to me that, if they suspected my motives, they could send a friend’s DNA. At least refusal was better than cheating.

Seven candidates, including Dr von Deyn, Jr, returned samples. None of them was Rosie’s father or half-sister. Professor Simon Lefebvre returned from his sabbatical and wanted to meet me in person.

‘I’m here to collect a package from Professor Lefebvre,’ I said to the receptionist at the city hospital where he was based, hoping to avoid an actual meeting and interrogation. I was unsuccessful. She buzzed the phone, announced my name, and Professor Lefebvre appeared. He was, I assumed, approximately fifty-four years old. I had met many fifty-four-year-olds in the past thirteen weeks. He was carrying a large envelope, presumably containing the questionnaire, which was destined for the recycling bin, and his DNA.

As he reached me, I tried to take the envelope, but he extended his other hand to shake mine. It was awkward, but the net result was that we shook hands and he retained the envelope.

‘Simon Lefebvre,’ he said. ‘So, what are you really after?’

This was totally unexpected. Why should he question my motives?

‘Your DNA,’ I said. ‘And the questionnaire. For a major research study. Critical.’ I was feeling stressed and my voice doubtless reflected it.

‘I’m sure it is.’ Simon laughed. ‘And you randomly select the head of medical research as a subject?’

‘We were looking for high achievers.’

‘What’s Charlie after this time?’

‘Charlie?’ I didn’t know anyone called Charlie.

‘All right,’ he said. ‘Dumb question. How much do you want me to put in?’

‘No putting in is required. There is no Charlie involved. I just require the DNA … and the questionnaire.’

Simon laughed, again. ‘You’ve got my attention. You can tell Charlie that. Shoot me through the project description. And the ethics approval. The whole catastrophe.’

‘Then I can have my sample?’ I said. ‘A high response rate is critical for the statistical analysis.’

‘Just send me the paperwork.’

Simon Lefebvre’s request was entirely reasonable. Unfortunately I did not have the required paperwork, because the project was fictitious. To develop a plausible project proposal would potentially require hundreds of hours of work.

I attempted an estimate of the probability that Simon Lefebvre was Rosie’s father. There were now four untested candidates: Lefebvre, Geoffrey Case (dead), and the two New Yorkers, Isaac Esler and Solomon Freyberg. On the basis of Rosie’s information, any one of them had a twenty-five per cent probability of being her father. But having proceeded so far without a positive result, I had to consider other possibilities. Two of our results relied on relatives rather than direct testing. It was possible that one or both of these daughters were, like Rosie, the result of extra-relationship sex, which, as Gene points out, is a more common phenomenon than popularly believed. And there was the possibility that one or more of my respondents to the fictitious research project might have deliberately sent a false sample.

I also had to consider that Rosie’s mother might not have told the truth. It took me a long time to think of this, as my default assumption is that people will be honest. But perhaps Rosie’s mother wanted Rosie to believe that her father was a doctor, as she was, rather than a less prestigious person. On balance, I estimated the chance that Simon Lefebvre was Rosie’s father was sixteen per cent. In developing documentation for the Asperger’s research project I would be doing an enormous amount of work with a low probability that it would provide the answer.

I chose to proceed. The decision was barely rational.

In the midst of this work, I received a phone call from a solicitor to advise me that Daphne had died. Despite the fact that she had been effectively dead for some time, I detected in myself an unexpected feeling of loneliness. Our friendship had been simple. Everything was so much more complicated now.

The reason for the call was that Daphne had left me what the solicitor referred to as a ‘small sum’ in her will. Ten thousand dollars. And she had also left a letter, written before she had gone to live in the nursing home. It was handwritten on decorative paper.

Dear Don,

Thank you for making the final years of my life so stimulating. After Edward was admitted to the nursing home, I did not believe that there was much left for me. I’m sure you know how much you have taught me, and how interesting our conversations have been, but you may not realise what a wonderful companion and support you have been to me.

I once told you that you would make someone a wonderful husband, and, in case you have forgotten, I am telling you again. I’m sure if you look hard enough, you will find the right person. Do not give up, Don.

I know you don’t need my money, and my children do, but I have left you a small sum. I would be pleased if you would use it for something irrational.

Much love,

Your friend,

Daphne Speldewind

It took me less than ten seconds to think of an irrational purchase: in fact I allowed myself only that amount of time to ensure that the decision was not affected by any logical thought process.