Fibonacci nim

Fibonacci nim is played by two players, who alternate removing coins or other counters from a pile.

According to the normal play convention, the player who takes the last coin wins.

[1] The game was first described by Michael J. Whinihan in 1963, crediting its invention to Oregon State University mathematician Robert E. Gaskell.

[4] The game strategy also involves a number called the "quota", which may be denoted as q.

[2] Based on these definitions, the player who is about to move can win whenever q is greater than or equal to the smallest Fibonacci number in the Zeckendorf representation, and will lose (with best play from the opponent) otherwise.

In a winning position, it is always a winning move to remove all the coins (if this is allowed) or otherwise to remove a number of coins equal to the smallest Fibonacci number in the Zeckendorf representation.

When this is possible, the opposing player will necessarily be faced with a losing position, because the new quota will be smaller than the smallest Fibonacci number in the Zeckendorf representation of the remaining number of coins.

Fibonacci nim is played with a pile of coins. The number of coins in this pile, 21, is a Fibonacci number, so a game starting with this pile and played optimally will be won by the second player.
Visual representation of the Zeckendorf representations of each number (a row of the image) as a sum of Fibonacci numbers (the widths of rectangles intersecting that row). An optimal strategy in Fibonacci nim removes the smallest rectangle in the row for the current pile of coins, leaving a pile described by the remaining rectangles from that row.