15

Part of 2021 BMT

Problems(2)

BMT 2021 Guts Round p15

Source:

\frac{\cos \left(\frac{\pi}{12}\right)\cos \left(\frac{\pi}{24}\right)\cos \left(\frac{\pi}{48}\right)\cos \left(\frac{\pi}{96}\right)...}{\cos \left(\frac{\pi}{4}\right)\cos \left(\frac{\pi}{8}\right)\cos \left(\frac{\pi}{16}\right)\cos \left(\frac{\pi}{32}\right)...}

BMT 2021 General p15

Source:

2\times 2

grid, which he proceeds to place chips on. One by one, he places a chip on one of the unit squares of the grid at random. However, if at any point there is more than one chip on the same square, Benji moves two chips on that square to the two adjacent squares, which he calls a chip-fire. He keeps adding chips until there is an infinite loop of chip-fires. What is the expected number of chips that will be added to the board?