A mind-reading trick to amaze at your Xmas party

Each card in this 4x4 array is marked with a number on its face-down side. You, the magician, ask a volunteer to take any card at random away from the array (keeping its number hidden) and discard the remaining cards from its horizontal row and vertical column.

This action is then repeated twice, again discarding the remainder of the row and column each time. When this has been done, only one card will remain. The volunteer also picks this up and adds it to the other three cards, leaving the discards in a heap to one side.

Now, lo and behold, you ask the volunteer to silently count up the four numbers and concentrate very hard for a moment. You then read his mind and correctly identify the total as 34!

The question is: Which sixteen numbers were marked on the cards?

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