## Ken's Puzzle of the WeekUnique 5x5 Paths

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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".

Solutions were received from Kirk Bresniker, Richard Mathar, Bernie Erickson, and Joseph DeVincentis.  There are 12 solutions.  Bernie Erickson points out the following data:

There are 2300 possible ways to choose 3 squares out of 25 (25 x 24 x 23 / 6).  Rotations and reflections reduce the number of possibilities to 319, kind of a strange number.  Inspecting each of them, I find 285 with no path, 22 with multiple paths, and 12 with only one path.

Multiple Paths:  ABC ABE ABI ABM ABO ABQ ABS ABU ABW ABY AEH AHI AHO AHS AHY ANO ANS CHQ CLQ CLW GHI GHS

Single Paths:   AHM AHW AMN ANW CHM CHW CLM CLS CMR CQR GHM GMN

[KD: Thanks to Kirk for the formatting below to provide a compact way to view the 12 grids.]

```0 1 1 1 1     0 1 1 1 1     0 1 1 1 1     0 1 1 1 1     1 1 0 1 1     1 1 0 1 1
1 1 0 1 1     1 1 0 1 1     1 1 1 1 1     1 1 1 1 1     1 1 0 1 1     1 1 0 1 1
1 1 0 1 1     1 1 1 1 1     1 1 0 0 1     1 1 1 0 1     1 1 0 1 1     1 1 1 1 1
1 1 1 1 1     1 1 1 1 1     1 1 1 1 1     1 1 1 1 1     1 1 1 1 1     1 1 1 1 1
1 1 1 1 1     1 1 0 1 1     1 1 1 1 1     1 1 0 1 1     1 1 1 1 1     1 1 0 1 1

1 1 0 1 1     1 1 0 1 1     1 1 0 1 1     1 1 0 1 1     1 1 1 1 1     1 1 1 1 1
1 1 1 1 1     1 1 1 1 1     1 1 1 1 1     1 1 1 1 1     1 0 0 1 1     1 0 1 1 1
1 0 0 1 1     1 0 1 1 1     1 1 0 1 1     1 1 1 1 1     1 1 0 1 1     1 1 0 0 1
1 1 1 1 1     1 1 1 0 1     1 1 0 1 1     1 0 0 1 1     1 1 1 1 1     1 1 1 1 1
1 1 1 1 1     1 1 1 1 1     1 1 1 1 1     1 1 1 1 1     1 1 1 1 1     1 1 1 1 1```

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