2013 USAJMO Problems/Problem 2
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Each cell of an 
board is filled with some nonnegative integer. Two numbers in the filling are said to be adjacent if their cells share a common side. (Note that two numbers in cells that share only a corner are not adjacent). The filling is called a garden if it satisfies the following two conditions:
(i) The difference between any two adjacent numbers is either 0 or 
.
(ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to 
.
Determine the number of distinct gardens in terms of 
and 
.
