Grid Tours
Carl G.
without visiting a cell more than once. You begin by selecting a starting
cell. Each cell has a number in it. To move to the next cell, you may move
up, down, across, or diagonally, a number of cells equal to the number in
the current cell. The number in the final cell is not used. For example,
consider the following 3 by 3 grid (best viewed with a fixed width font):
1 2 3
1: 2 2 1
2: 1 1 1
3: 1 1 2
(The rows and columns are labeled to assist in writing out the solutions.)
Let's assume you start in the cell at row 2, column 1, or (2:1). The number
in this cell is 1. This means you can move exactly one cell away; either
up, down, across, or diagonally. You can move to cell
(1:1),(1:2),(2:2),(3:1), or (3,2). If you decided to move to cell (1:1),
the next cell you selected would have to be exactly two cells away. Once
you have visited a cell, you can't visit that cell again.
One solution that visits every cell once is:
(2:1)(1:1)(1:3)(2:2)(3:3)(3:1)(3:2)(2:3)(1:2)
The final cell, (1:2), has a 2 in it, but the number is not used. There may
be more than one solution to a given puzzle (especially larger grids). You
may find it easier to print the puzzles on a piece of paper so that you can
cross-off visited cells and work out patterns.
Carl G.
Four cell puzzle:
1 2 3 4
1: 2 2 3 1
2: 1 2 1 1
3: 1 2 1 2
4: 2 3 2 2
Six cell puzzle:
1 2 3 4 5 6
1: 2 5 3 2 1 4
2: 4 2 1 1 1 4
3: 1 3 1 1 1 1
4: 1 2 1 2 1 5
5: 2 3 2 2 4 4
6: 1 3 4 1 5 5
Eight cell puzzle:
1 2 3 4 5 6 7 8
1: 5 3 2 1 7 2 2 7
2: 1 6 1 1 1 3 2 5
3: 2 2 4 2 2 4 6 2
4: 1 1 4 2 3 2 1 5
5: 1 2 2 1 3 2 4 4
6: 4 2 2 3 2 1 4 2
7: 1 5 2 2 3 1 2 1
8: 3 2 4 2 3 2 3 4
Ten cell puzzle:
1 2 3 4 5 6 7 8 9 10
1: 5 3 2 7 3 1 8 4 1 9
2: 5 1 1 5 5 1 1 5 7 5
3: 2 1 2 1 5 2 3 5 1 2
4: 2 4 5 2 1 3 2 6 5 1
5: 3 2 4 2 2 1 1 1 5 8
6: 4 2 2 1 1 4 4 2 3 5
7: 4 7 4 3 1 5 3 1 5 5
8: 4 2 2 1 3 3 1 5 3 5
9: 5 6 5 8 2 1 1 4 1 8
10: 1 3 3 3 1 4 1 8 8 7
il...@isgtec.com
SPOILER
Here are computer generated solutions, listing thw row and column for
each step:
4x4 grid:
1: (1, 1)
2: (1, 3)
3: (4, 3)
4: (2, 1)
5: (3, 1)
6: (4, 1)
7: (2, 3)
8: (1, 2)
9: (3, 2)
10: (3, 4)
11: (1, 4)
12: (2, 4)
13: (3, 3)
14: (2, 2)
15: (4, 4)
16: (4, 2)
6x6 grid:
1: (1, 1)
2: (1, 3)
3: (1, 6)
4: (1, 2)
5: (6, 2)
6: (3, 2)
7: (3, 5)
8: (2, 4)
9: (1, 4)
10: (3, 4)
11: (2, 5)
12: (3, 6)
13: (4, 5)
14: (4, 6)
15: (4, 1)
16: (4, 2)
17: (2, 2)
18: (4, 4)
19: (6, 4)
20: (6, 3)
21: (2, 3)
22: (3, 3)
23: (4, 3)
24: (5, 4)
25: (5, 6)
26: (5, 2)
27: (5, 5)
28: (1, 5)
29: (2, 6)
30: (6, 6)
31: (6, 1)
32: (5, 1)
33: (5, 3)
34: (3, 1)
35: (2, 1)
36: (6, 5)
8x8 grid:
1: (1, 1)
2: (6, 1)
3: (2, 1)
4: (1, 2)
5: (1, 5)
6: (8, 5)
7: (5, 8)
8: (1, 8)
9: (8, 1)
10: (5, 1)
11: (4, 1)
12: (3, 1)
13: (1, 3)
14: (3, 3)
15: (7, 7)
16: (5, 7)
17: (1, 7)
18: (3, 5)
19: (5, 3)
20: (7, 1)
21: (7, 2)
22: (2, 2)
23: (2, 8)
24: (7, 3)
25: (5, 5)
26: (8, 8)
27: (4, 8)
28: (4, 3)
29: (8, 7)
30: (8, 4)
31: (8, 2)
32: (6, 2)
33: (4, 2)
34: (3, 2)
35: (1, 4)
36: (2, 3)
37: (2, 4)
38: (2, 5)
39: (3, 6)
40: (7, 6)
41: (6, 7)
42: (2, 7)
43: (4, 5)
44: (7, 5)
45: (7, 8)
46: (6, 8)
47: (8, 6)
48: (6, 6)
49: (6, 5)
50: (6, 3)
51: (8, 3)
52: (4, 7)
53: (4, 6)
54: (2, 6)
55: (5, 6)
56: (3, 8)
57: (1, 6)
58: (3, 4)
59: (5, 2)
60: (7, 4)
61: (5, 4)
62: (4, 4)
63: (6, 4)
64: (3, 7)
10x10 grid:
1: (1, 1)
2: (1, 6)
3: (1, 7)
4: (9, 7)
5: (10, 8)
6: (2, 8)
7: (7, 3)
8: (3, 7)
9: (6, 10)
10: (1, 10)
11: (10, 1)
12: (9, 1)
13: (4, 1)
14: (4, 3)
15: (9, 3)
16: (4, 8)
17: (10, 2)
18: (7, 2)
19: (7, 9)
20: (2, 4)
21: (2, 9)
22: (9, 2)
23: (3, 8)
24: (8, 8)
25: (8, 3)
26: (10, 3)
27: (10, 6)
28: (10, 10)
29: (3, 10)
30: (5, 10)
31: (5, 2)
32: (3, 4)
33: (3, 5)
34: (8, 10)
35: (8, 5)
36: (8, 2)
37: (10, 4)
38: (10, 7)
39: (9, 8)
40: (9, 4)
41: (1, 4)
42: (8, 4)
43: (9, 5)
44: (7, 7)
45: (4, 10)
46: (5, 9)
47: (10, 9)
48: (2, 1)
49: (7, 6)
50: (7, 1)
51: (3, 1)
52: (1, 3)
53: (1, 5)
54: (1, 8)
55: (5, 4)
56: (7, 4)
57: (4, 7)
58: (2, 7)
59: (2, 6)
60: (3, 6)
61: (5, 8)
62: (6, 7)
63: (2, 3)
64: (1, 2)
65: (4, 2)
66: (4, 6)
67: (1, 9)
68: (2, 10)
69: (2, 5)
70: (7, 10)
71: (7, 5)
72: (6, 6)
73: (6, 2)
74: (4, 4)
75: (2, 2)
76: (3, 2)
77: (3, 3)
78: (5, 1)
79: (8, 1)
80: (4, 5)
81: (5, 5)
82: (5, 3)
83: (5, 7)
84: (6, 8)
85: (8, 6)
86: (8, 9)
87: (5, 6)
88: (6, 5)
89: (6, 4)
90: (6, 3)
91: (6, 1)
92: (10, 5)
93: (9, 6)
94: (8, 7)
95: (7, 8)
96: (6, 9)
97: (3, 9)
98: (4, 9)
99: (9, 9)
100: (9, 10)
__/\__
\ /
__/\\ //\__ Ilan Mayer
\ /
/__ __\ Toronto, Canada
/__ __\
||
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Share what you know. Learn what you don't.
Carl G.
il...@isgtec.com wrote in message <7q4m84$i1r$1...@nnrp1.deja.com>...
...
(Solutions to "Grid Tours")
...
Thanks for giving these puzzles a try and solving them. While the smaller
grids can be worked out without a computer, I am not surprised that a
computer was used to solve the larger grids. One of the main techniques to
solving these puzzles by hand is to "work backwards", i.e., figure out which
cells came before the current cell. One can then create a tree diagram of
the possible routes. By starting with the cells that have only one possible
connection (often corresponding to cells with larger numbers), one can
eliminate many of the connections. This will produce more cells with only
one possible connection and the process can be repeated. Finding the
correct route in the tree diagram is similar to solving a maze, except every
"room" must be visited at least once. I am surprised that more solutions
weren't posted. Perhaps including the larger grids scared people off.
Carl G.