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 Counting Ten Segment Paths (Posted on 2016-02-19)
Multiple paths can be drawn on the grid below following along ten of the twelve segments but never using any segment twice.
```+--+--+
|  |  |
+--+--+
|  |  |
+--+--+
```
How many such paths are there?

 No Solution Yet Submitted by Brian Smith No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
 computer solution | Comment 2 of 6 |
Below are the 200 paths that satisfy the requirements. These certainly include reflections, rotations and reversals; 200 is not divisible by 2*4*2=16, so some of these categories of duplication may overlap. Each digit pair should be considered as a pair of x-y coordinates; there are eleven such points on each route as ten segments are traversed:

01,00,10,11,01,02,12,11,21,20,10,
01,00,10,11,01,02,12,11,21,22,12,
01,00,10,11,01,02,12,22,21,11,12,
01,00,10,11,01,02,12,22,21,20,10,
01,00,10,11,12,02,01,11,21,20,10,
01,00,10,11,12,02,01,11,21,22,12,
01,00,10,11,12,22,21,11,01,02,12,
01,00,10,11,21,22,12,02,01,11,12,
01,00,10,11,21,22,12,11,01,02,12,
01,00,10,20,21,11,01,02,12,11,10,
01,00,10,20,21,11,01,02,12,22,21,
01,00,10,20,21,11,12,02,01,11,10,
01,00,10,20,21,22,12,02,01,11,10,
01,00,10,20,21,22,12,02,01,11,12,
01,00,10,20,21,22,12,02,01,11,21,
01,00,10,20,21,22,12,11,01,02,12,
01,02,12,11,01,00,10,11,21,20,10,
01,02,12,11,01,00,10,11,21,22,12,
01,02,12,11,01,00,10,20,21,11,10,
01,02,12,11,01,00,10,20,21,22,12,
01,02,12,11,10,00,01,11,21,20,10,
01,02,12,11,10,00,01,11,21,22,12,
01,02,12,11,10,20,21,11,01,00,10,
01,02,12,11,21,20,10,00,01,11,10,
01,02,12,11,21,20,10,11,01,00,10,
01,02,12,22,21,11,01,00,10,11,12,
01,02,12,22,21,11,01,00,10,20,21,
01,02,12,22,21,11,10,00,01,11,12,
01,02,12,22,21,20,10,00,01,11,10,
01,02,12,22,21,20,10,00,01,11,12,
01,02,12,22,21,20,10,00,01,11,21,
01,02,12,22,21,20,10,11,01,00,10,
01,11,10,00,01,02,12,11,21,20,10,
01,11,10,00,01,02,12,11,21,22,12,
01,11,10,00,01,02,12,22,21,11,12,
01,11,10,00,01,02,12,22,21,20,10,
01,11,10,20,21,11,12,02,01,00,10,
01,11,10,20,21,22,12,02,01,00,10,
01,11,12,02,01,00,10,11,21,20,10,
01,11,12,02,01,00,10,11,21,22,12,
01,11,12,02,01,00,10,20,21,11,10,
01,11,12,02,01,00,10,20,21,22,12,
01,11,12,22,21,11,10,00,01,02,12,
01,11,12,22,21,20,10,00,01,02,12,
01,11,21,20,10,00,01,02,12,11,10,
01,11,21,20,10,00,01,02,12,22,21,
01,11,21,20,10,11,12,02,01,00,10,
01,11,21,22,12,02,01,00,10,11,12,
01,11,21,22,12,02,01,00,10,20,21,
01,11,21,22,12,11,10,00,01,02,12,
10,00,01,02,12,11,10,20,21,11,01,
10,00,01,02,12,11,10,20,21,22,12,
10,00,01,02,12,11,21,20,10,11,01,
10,00,01,02,12,22,21,11,10,20,21,
10,00,01,02,12,22,21,20,10,11,01,
10,00,01,02,12,22,21,20,10,11,12,
10,00,01,02,12,22,21,20,10,11,21,
10,00,01,11,10,20,21,11,12,02,01,
10,00,01,11,10,20,21,11,12,22,21,
10,00,01,11,10,20,21,22,12,02,01,
10,00,01,11,10,20,21,22,12,11,21,
10,00,01,11,12,22,21,11,10,20,21,
10,00,01,11,12,22,21,20,10,11,21,
10,00,01,11,21,20,10,11,12,02,01,
10,00,01,11,21,20,10,11,12,22,21,
10,00,01,11,21,22,12,11,10,20,21,
10,11,01,00,10,20,21,11,12,02,01,
10,11,01,00,10,20,21,11,12,22,21,
10,11,01,00,10,20,21,22,12,02,01,
10,11,01,00,10,20,21,22,12,11,21,
10,11,01,02,12,11,21,20,10,00,01,
10,11,01,02,12,22,21,20,10,00,01,
10,11,12,02,01,00,10,20,21,11,01,
10,11,12,02,01,00,10,20,21,22,12,
10,11,12,02,01,11,21,20,10,00,01,
10,11,12,22,21,11,01,00,10,20,21,
10,11,12,22,21,20,10,00,01,02,12,
10,11,12,22,21,20,10,00,01,11,21,
10,11,21,20,10,00,01,02,12,11,01,
10,11,21,20,10,00,01,02,12,22,21,
10,11,21,20,10,00,01,11,12,02,01,
10,11,21,20,10,00,01,11,12,22,21,
10,11,21,22,12,02,01,00,10,20,21,
10,11,21,22,12,11,01,00,10,20,21,
10,20,21,11,01,00,10,11,12,02,01,
10,20,21,11,01,00,10,11,12,22,21,
10,20,21,11,01,02,12,11,10,00,01,
10,20,21,11,10,00,01,02,12,11,01,
10,20,21,11,10,00,01,02,12,22,21,
10,20,21,11,10,00,01,11,12,02,01,
10,20,21,11,10,00,01,11,12,22,21,
10,20,21,11,12,02,01,00,10,11,01,
10,20,21,11,12,02,01,11,10,00,01,
10,20,21,22,12,02,01,00,10,11,01,
10,20,21,22,12,02,01,00,10,11,12,
10,20,21,22,12,02,01,00,10,11,21,
10,20,21,22,12,02,01,11,10,00,01,
10,20,21,22,12,11,01,00,10,11,21,
10,20,21,22,12,11,10,00,01,02,12,
10,20,21,22,12,11,10,00,01,11,21,
12,02,01,00,10,11,12,22,21,11,01,
12,02,01,00,10,11,12,22,21,20,10,
12,02,01,00,10,11,21,22,12,11,01,
12,02,01,00,10,20,21,11,12,22,21,
12,02,01,00,10,20,21,22,12,11,01,
12,02,01,00,10,20,21,22,12,11,10,
12,02,01,00,10,20,21,22,12,11,21,
12,02,01,11,10,20,21,11,12,22,21,
12,02,01,11,10,20,21,22,12,11,21,
12,02,01,11,12,22,21,11,10,00,01,
12,02,01,11,12,22,21,11,10,20,21,
12,02,01,11,12,22,21,20,10,00,01,
12,02,01,11,12,22,21,20,10,11,21,
12,02,01,11,21,20,10,11,12,22,21,
12,02,01,11,21,22,12,11,10,00,01,
12,02,01,11,21,22,12,11,10,20,21,
12,11,01,00,10,11,21,22,12,02,01,
12,11,01,00,10,20,21,22,12,02,01,
12,11,01,02,12,22,21,11,10,00,01,
12,11,01,02,12,22,21,11,10,20,21,
12,11,01,02,12,22,21,20,10,00,01,
12,11,01,02,12,22,21,20,10,11,21,
12,11,10,00,01,02,12,22,21,11,01,
12,11,10,00,01,02,12,22,21,20,10,
12,11,10,00,01,11,21,22,12,02,01,
12,11,10,20,21,11,01,02,12,22,21,
12,11,10,20,21,22,12,02,01,00,10,
12,11,10,20,21,22,12,02,01,11,21,
12,11,21,20,10,00,01,02,12,22,21,
12,11,21,20,10,11,01,02,12,22,21,
12,11,21,22,12,02,01,00,10,11,01,
12,11,21,22,12,02,01,00,10,20,21,
12,11,21,22,12,02,01,11,10,00,01,
12,11,21,22,12,02,01,11,10,20,21,
12,22,21,11,01,00,10,11,12,02,01,
12,22,21,11,01,02,12,11,10,00,01,
12,22,21,11,01,02,12,11,10,20,21,
12,22,21,11,10,00,01,02,12,11,01,
12,22,21,11,10,00,01,11,12,02,01,
12,22,21,11,12,02,01,00,10,11,01,
12,22,21,11,12,02,01,00,10,20,21,
12,22,21,11,12,02,01,11,10,00,01,
12,22,21,11,12,02,01,11,10,20,21,
12,22,21,20,10,00,01,02,12,11,01,
12,22,21,20,10,00,01,02,12,11,10,
12,22,21,20,10,00,01,02,12,11,21,
12,22,21,20,10,00,01,11,12,02,01,
12,22,21,20,10,11,01,02,12,11,21,
12,22,21,20,10,11,12,02,01,00,10,
12,22,21,20,10,11,12,02,01,11,21,
21,11,01,00,10,11,12,22,21,20,10,
21,11,01,00,10,20,21,22,12,02,01,
21,11,01,00,10,20,21,22,12,11,10,
21,11,01,02,12,11,10,20,21,22,12,
21,11,01,02,12,22,21,20,10,00,01,
21,11,01,02,12,22,21,20,10,11,12,
21,11,10,00,01,02,12,22,21,20,10,
21,11,10,00,01,11,12,22,21,20,10,
21,11,10,20,21,22,12,02,01,00,10,
21,11,10,20,21,22,12,02,01,11,12,
21,11,10,20,21,22,12,11,01,00,10,
21,11,10,20,21,22,12,11,01,02,12,
21,11,12,02,01,00,10,20,21,22,12,
21,11,12,02,01,11,10,20,21,22,12,
21,11,12,22,21,20,10,00,01,02,12,
21,11,12,22,21,20,10,00,01,11,10,
21,11,12,22,21,20,10,11,01,00,10,
21,11,12,22,21,20,10,11,01,02,12,
21,20,10,00,01,02,12,11,21,22,12,
21,20,10,00,01,02,12,22,21,11,01,
21,20,10,00,01,02,12,22,21,11,10,
21,20,10,00,01,02,12,22,21,11,12,
21,20,10,00,01,11,12,22,21,11,10,
21,20,10,00,01,11,21,22,12,02,01,
21,20,10,00,01,11,21,22,12,11,10,
21,20,10,11,01,02,12,11,21,22,12,
21,20,10,11,01,02,12,22,21,11,12,
21,20,10,11,12,02,01,11,21,22,12,
21,20,10,11,12,22,21,11,01,00,10,
21,20,10,11,12,22,21,11,01,02,12,
21,20,10,11,21,22,12,02,01,00,10,
21,20,10,11,21,22,12,02,01,11,12,
21,20,10,11,21,22,12,11,01,00,10,
21,20,10,11,21,22,12,11,01,02,12,
21,22,12,02,01,00,10,11,21,20,10,
21,22,12,02,01,00,10,20,21,11,01,
21,22,12,02,01,00,10,20,21,11,10,
21,22,12,02,01,00,10,20,21,11,12,
21,22,12,02,01,11,10,20,21,11,12,
21,22,12,02,01,11,21,20,10,00,01,
21,22,12,02,01,11,21,20,10,11,12,
21,22,12,11,01,00,10,11,21,20,10,
21,22,12,11,01,00,10,20,21,11,10,
21,22,12,11,10,00,01,11,21,20,10,
21,22,12,11,10,20,21,11,01,00,10,
21,22,12,11,10,20,21,11,01,02,12,
21,22,12,11,21,20,10,00,01,02,12,
21,22,12,11,21,20,10,00,01,11,10,
21,22,12,11,21,20,10,11,01,00,10,
21,22,12,11,21,20,10,11,01,02,12,

DefDbl A-Z
Dim crlf\$, h As String, x, y, ct

Form1.Visible = True

Text1.Text = ""
crlf = Chr\$(13) + Chr\$(10)

For x0 = 0 To 2
For y0 = 0 To 2
x = x0: y = y0
placeIt
Next
Next

Text1.Text = Text1.Text & crlf & ct & " done"

End Sub

Sub placeIt()
s\$ = LTrim(Str(x)) + LTrim(Str(y))
h = h + s + ","
moved = 0
For dx = -1 To 1
For dy = -1 To 1
If dx * dy = 0 And dx + dy <> 0 Then
newx = x + dx: newy = y + dy
If newx >= 0 And newx <= 2 And newy >= 0 And newy <= 2 Then
x = newx: y = newy

news\$ = LTrim(Str(x)) + LTrim(Str(y))
a\$ = s + "," + news: b\$ = news + "," + s
If InStr(h, a) = 0 Then
If InStr(h, b) = 0 Then
placeIt
moved = 1
End If
End If

x = x - dx: y = y - dy
End If
End If
Next
Next

If moved = 0 Then
If Len(h) >= 33 Then
ct = ct + 1
Text1.Text = Text1.Text & h & crlf
End If
End If

h = Left(h, Len(h) - 3)
End Sub

 Posted by Charlie on 2016-02-19 13:29:41

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