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Puzzle for Today #272: Arranging coins

From BBC R4 “Today” programme, 19th July 2018 [link]

I have a tray of length 5 and width 2 so 10 round coins of width 1 will fit in it snugly without overlaps. No room for another. Similarly, a tray of length 50 will accommodate only 100 coins. Things get more interesting with a longer tray; a tray of length 500 and width 2 can accommodate at least 1001 coins.

What is the smallest integer value of N, such that a tray of length N (and width 2) can accommodate 2N + 1 coins?


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