Corrected entry: The old trick of posting "evidence" to yourself by registered mail, thereby "proving" the date upon which it came into your possession is a legal urban myth, and as a trained and experienced lawyer Goodwin would know that. A number of attempted stings in the 1940s where con-men posted empty envelopes to themselves and later carefully steamed them open to insert the "evidence" meant the technique has no legal standing at all. Goodwin is not attempting a bluff - he is genuinely delighted when he is handed the envelope.
Corrected entry: Dick Goodwin's boss gives him one week in New York to find proof. On the day he meets Charles Van Doren, Van Doren has just appeared on the Today Show where he said that he's been on Twenty-One for 9 weeks. However, by the time Goodwin interviews Herb Stempel; Stempel mentions that Van Doren has been on Twenty-One for 11 weeks. This indicates that Goodwin has been in New York two weeks and would've had to have returned to Washington, D.C. and give up his investigation.
Corrected entry: The opening sequence of Twenty One is shown twice. The first time, Jack Barry stands at the podium awaiting his cue to begin the show, but the second time, the band strikes up, and after the announcer makes introductions Barry runs to the podium from backstage.