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Put three black squares into a 5x5 grid, such that the remaining white squares can be connected in only one way in a closed loop, visiting each white square exactly once (moving only up, down, right, or left.) One example is above; only one closed loop can connect all the white squares. How many different arrangements of three squares can determine a unique path? (I know of at least three more, but I expect there aren't very many beyond that.)

Source: Original. To compare solutions, rotate/reflect your grid to obtain the highest alphabetical report of the three squares. For example, the above grid should not be reported as "hmq", but should be rotated to report it as "gmn".

Solution

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