perplexus.info :: General : Counting Ten Segment Paths

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.

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How many such paths are there?

No Solution Yet Submitted by Brian Smith

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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

Private Sub Form_Load()

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

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

x = x - dx: y = y - dy

End If

If moved = 0 Then

If Len(h) >= 33 Then

ct = ct + 1

Text1.Text = Text1.Text & h & crlf

End If

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

End Sub

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

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