Place the numbers 1 to 64 in the grid, so that each number is a knight's move away from its neighbour. The numbers 1 and 64 are also a knight's move away from each other. So, for example, the numbers 7 and 9 would each be a knight's move away from number 8.

The letters c and s to the right and below the grid indicate where the cubes and squares are situated.

The following clues will help you place the numbers.

C. Only one number in the twenties.
D. Nothing divisible by 21.
F. Only two numbers in the thirties. Total 210.
G. Three numbers in the thirties.

2. Nothing less than 20.
3. No number in the thirties.
4. Total 214.
6. Nothing divisible by 6 or 11. Only one number in the thirties.
7. Only one number divisible by 10.
8. Only one number divisible by 10.

1

2

3

4

5

6

7

8

A

40

54

c

s

s

B

c

C

6

s

D

E

c

s

F

1

14

c

s

s

G

37

59

s

H

19

34

s

s

c

s

c

c

s

s

s

s

c

s

s

The following three knight's tours satisfy the diagram (including the square and cube counts), but only the last one also satisfies the clues:

 40 25 54  7 42 27 46  9    40 25 54  7 42 27 52  9    40 25 54  7 42 27 46  9
 55 62 41 26  3  8 43 28    55 62 41 26 53  8 45 28    55 62 41 26 53  8 43 28
 24 39  4 53  6 45 10 47    24 39 16 63  6 43 10 51    24 39 16 63  6 45 10 47
 61 56 63  2 15 52 29 44    61 56  5  2 15 46 29 44    61 56  5  2 15 52 29 44
 38 23 60  5 64 11 48 13    38 23 60 17 64  3 50 11    38 23 60 17 64  3 48 11
 57 20  1 16 51 14 33 30    57 20  1  4 47 14 33 30    57 20  1  4 51 14 33 30
 22 37 18 59 32 35 12 49    22 37 18 59 32 35 12 49    22 37 18 59 32 35 12 49
 19 58 21 36 17 50 31 34    19 58 21 36 13 48 31 34    19 58 21 36 13 50 31 34
 

 A compiled VB 5.0 program took
 
 4664.25 seconds = 1 h 17 m 44.25 s
 
 to find the
 
 3 solutions.
 
Run time would vary of course from computer to computer.
 
 Only in the last solution does row F total 210.
 Also, only in the last solution does column 4 total 214.

In VB 5.0:

Dim b(8, 8), moveNo, sCt, ways, cubeRow(8), sqRow(8), cubeCol(8), sqCol(8)

Private Sub Command1_Click()
b(1, 1) = 40
b(1, 3) = 54
b(3, 5) = 6
b(6, 3) = 1
b(6, 6) = 14
b(7, 2) = 37
b(7, 4) = 59
b(8, 1) = 19
b(8, 8) = 34

sqCol(2) = 1
sqCol(3) = 1 ' excludes 1
sqCol(4) = 2
sqCol(5) = 1
sqCol(8) = 2
sqRow(1) = 2
sqRow(3) = 1
sqRow(5) = 1
sqRow(6) = 1 ' excludes 1
sqRow(7) = 1
sqRow(8) = 1

cubeCol(3) = 0 ' excludes 1
cubeCol(5) = 1
cubeCol(6) = 2
cubeRow(1) = 1
cubeRow(2) = 1
cubeRow(5) = 1
cubeRow(6) = 0 ' excludes 1

moveNo = 1
tm = Timer

place 6, 3

Print sCt
Print Timer - tm
Print ways
End Sub

Sub place(frow, fcol)
  DoEvents
  For dr = -2 To 2 Step 4
   For dc = -1 To 1 Step 2
    GoSub tryMove
   Next
  Next
  For dc = -2 To 2 Step 4
   For dr = -1 To 1 Step 2
    GoSub tryMove
   Next
  Next
  Exit Sub

tryMove:
  moveNo = moveNo + 1
  trow = frow + dr
  tcol = fcol + dc
  If trow > 0 And trow <= 8 And tcol > 0 And tcol <= 8 Then
    good = 1
    If moveNo = 4 Or moveNo = 9 Or moveNo = 16 Or moveNo = 25 Or moveNo = 36 Or moveNo = 49 Or moveNo = 64 Then
      If sqCol(tcol) < 1 Or sqRow(trow) < 1 Then
        good = 0
      End If
    End If
    If moveNo = 8 Or moveNo = 27 Or moveNo = 64 Then
      If cubeCol(tcol) < 1 Or cubeRow(trow) < 1 Then
        good = 0
      End If
    End If
    If good Then
      If b(trow, tcol) = moveNo Then save = 1 Else save = 0
      If b(trow, tcol) = 0 Or save Then
        b(trow, tcol) = moveNo
          Select Case moveNo
            Case 6, 14, 19, 34, 37, 40, 54, 59
             good = save
          End Select
          If good Then
            Select Case moveNo
              Case 4, 9, 16, 25, 36, 49, 64
                sqCol(tcol) = sqCol(tcol) - 1
                sqRow(trow) = sqRow(trow) - 1
              Case 8, 27, 64
                cubeCol(tcol) = cubeCol(tcol) - 1
                cubeRow(trow) = cubeRow(trow) - 1
            End Select

        If moveNo < 64 Then
            place trow, tcol
         Else
          ways = ways + 1
          Print "w "; ways
          Open "traveling knight.txt" For Append As #2
          For rp = 1 To 8
           For cp = 1 To 8
            f$ = Right$("   " + Format(b(rp, cp), "##"), 3)
            Print f$;
            Print #2, f$;
           Next
           Print
           Print #2,
          Next
          Print
          Print #2,
          Close 2
        End If

            Select Case moveNo
              Case 4, 9, 16, 25, 36, 49, 64
                sqCol(tcol) = sqCol(tcol) + 1
                sqRow(trow) = sqRow(trow) + 1
              Case 8, 27, 64
                cubeCol(tcol) = cubeCol(tcol) + 1
                cubeRow(trow) = cubeRow(trow) + 1
            End Select
          End If

        If save = 0 Then b(trow, tcol) = 0
      End If
    End If
  End If
  moveNo = moveNo - 1
 Return

End Sub

 

Edited on September 3, 2007, 1:15 am

Posted by Charlie on 2007-09-03 01:14:18