A particle travels on a grid of lattice points. Every trip starts at (0,0) and stops at (1,0). It can travel in any orthogonal direction provided:
* it remains in the first quadrant (i >= 0, j >= 0);
* it never visits the same lattice point more than once during the same trip.

During each trip it earns income of i*j dollars for each lattice point visited, but must pay a tax of one dollar for each step taken. For example, if it travels {up, right, down}, its earnings would $1, taxes $3, for a loss of $2. In this trip it visited 4 lattice points if you include (0,0), but there were only 3 steps.

(1) Determine the maximum profit the particle can make for a trip of exactly 7 steps?
(2) Same question for a trip of 13 steps.
(3) What is the minimum number of steps a trip would need to be for a profit of 100 or more?
(4) Same question for a profit of 1000 or more.

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Posted by Mona on 2025-05-07 23:10:17