Teaser 2903: What’s my card?

From The Sunday Times, 13th May 2018 [link]

Four expert logicians play a game with a pack of six cards, numbered from 1 to 6. The cards are shuffled and each draws a card, holding it in front of him so that he cannot see his own number but can see the numbers held by the others. The winner is the player who can first deduce what number he is holding.

Each player in turn is asked to announce the sum of two numbers that he can see on the other cards. One game started with Alf announcing 6. After a long pause, no-one claimed victory, so Bert then announced 5. There was a significant pause, but before Charlie spoke, someone claimed victory.

Which cards did Alf and Bert hold, and which cards remained on the table?

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