Jeff the fly starts at the point  in the coordinate plane. At each step, Jeff takes a step to the right, left, up, or down. After  steps, how many different points could Jeff end up at?

 Jan 22, 2020

The answer is 286. It comes from the following sum:

1+3+6+10+15+21+28+35+45+55+66 = 286.

It is the number of ordered partitions of 10 as the sum of four numbers taken from the set 0, 1, 2, 3, ..., 10 . If we write these partitions as 4-tuples such as (3, 2, 0, 5), and take the coordinates to mean, say,

(3 units to the right, 2 units up, no movement to the left, 5 units down), then each partition would be uniquely associated with a lattice point in the plain (or is that plane?!). I have carefully counted the number of such ordered partitions and have a list of them, but am too tired at the moment to list them. 


So the answer is 286.

 Jan 23, 2020

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