The Duke of Cambridge has said he is still “working on” a name for his four-day-old son.
The third Cambridge baby, who was born at 11.01am on Monday, has not yet been named, with an announcement expected at any moment.
The Duke, who made a surprise appearance with his brother Prince Harry at a sports centre, promised people would learn their final choice for a name “soon”.
But the Duke, now a father-of-three, suggested a decision has not even been made, as he undertook a public engagement with Prince Harry at the Greenhouse Centre, London.
Asked about the name, he said: “I’m working on that one. You’ll know soon, you’ll know soon.”
After watching judo and playing table tennis with young people at the new sports centre, the Duke was given a bag of presents for the baby.
Leiban Hussein, 19, a graduate of Greenhouse said: "There was a lot of banter between the brothers about the table tennis.
"William said he liked it although his back hand needs more work but Harry said he wasn’t very good. Prince William was teasing Harry and saying the last time hey played he beat him.
"William looked really pleased with the gift for the baby and thanked us all very much. But no, he didn’t say anything about it or it’s name!"
Michael Sherwood, Greenhouse chairman, publicly congratulated the two princes on their respective good news and thanked them for coming at such busy times in their lives.
The Duke rolled his eye theatrically and smiled when it was mentioned by Mr Sherwood that is was a busy time for him and he grinned at Harry when his wedding was mentioned.
The newborn Prince has remained nameless for longer than his older brother and sister, with the full names of both Prince George and Princess Charlotte announced by their proud parents two days after their birth.
The Duke fuelled speculation about the name during a public engagement on Wednesday.
Told by Alexander Downer, the Australian High Commissioner, that he favoured his own name as a front-runner, the Duke replied: "Funny you should say that It's a good name."
The Dean of Westminster, introducing the Duke to Sir Jerry Mateparae, Mr Downer's New Zealand counterpart, joked: "Jerry would like it to be Jerry."
"It's a strong name, I have to say," the Duke conceded smiling.
Asked how the Duchess and the baby were doing, the Duke said: "They're very well, thanks."
The father-of-three said they are "in good form", adding: "Sleeping is going reasonably well, he's behaving himself, which is good news."
While the Duke's words stoked speculation, Alexander is not likely to be a final contender as it is one of Prince George's middle names.
The Queen will be informed of the name before it is made public.
Bookmakers, who reported a last minute rush on bets, still favour Arthur followed by James, Albert, Philip, Louis, Henry or Edward.
The Prince joins siblings Prince George Alexander Louis, born on July 22 2013 and Princess Charlotte Elizabeth Diana, born on May 2 2015.