Bar-Ilan's weekly mathriddle competition, 2024

• Question 10 - It's Crowded Here!

  55 gears are placed on the game board in the shape of a 'pyramid':

  10 gears in the bottom row, 9 gears in the row above, and so on.

  In this state, the gears cannot rotate freely (convince yourself why!)

  Remove gears to allow free movement.

  What is the maximum number of gears that can remain on the board so that they can all rotate?

  

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  Topics:

  Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry