Problem 6

A pack of cards consists of 52 standard poker-size cards with rounded corners (the corner radius is 6mm). For the purposes of this puzzle, assume all cards to be flat and of negligible thickness. Herman puts all the cards on an A3 sheet of paper, observing two rules:

each card must cover the centre of the sheet;
the centre of the sheet must be at a distance of at least 10mm from any point on the edge of the Queen of Spades and at least 20mm from any point on the edge of any other card.

Herman marks then cuts out the area of the sheet which was covered by all the 52 cards. The rest of the sheet, i.e., parts of the sheet covered by 51 or fewer cards, is discarded.

What is the least possible perimeter of the cut-out shape? Enter the answer rounded to the nearest millimetre. You have to enter a whole number without decimal point.