An Invisible Maze is a square room with a tiled floor, in which the tiles form a grid. You may walk only to adjacent tiles (no diagonal moves). There is a number on the wall for each row and column of tiles. An Invisible Maze can have any numbers on the walls provided that it has at least one True Path. A True Path will take you from the northwest corner to the southeast corner, and the number of tiles you touch in each row and column is equal to the corresponding number on the wall.
There is an NxN tiled Invisible maze that has at least two different True Paths. Minimize N and then, using that N, minimize the sum of all the numbers on the wall.
Important: Two paths are considered the same even if they touch the exact same tiles in a different order.
When the output of the below program is sorted and examined, no valid results are found for sz = 4, and the lowest total found for sz = 5 is indeed 26, for which there are three basic solutions, each with a possible flip about the main diagonal.
CLEAR , , 25000
DIM SHARED sz
sz = 4
DIM SHARED gr(sz, sz), wall(2 * sz), row, col, path(sz * sz)
OPEN "mzpath" + LTRIM$(STR$(sz)) + ".txt" FOR OUTPUT AS #2
gr(1, 1) = 1
row = 1: col = 1: path(1) = 1
addOn 2
SUB addOn (lv)
FOR drow = -1 TO 1
FOR dcol = -1 TO 1
IF ABS(drow) + ABS(dcol) = 1 THEN
newrow = row + drow: newcol = col + dcol
IF newrow > 0 AND newrow <= sz AND newcol > 0 AND newcol <= sz THEN
IF gr(newrow, newcol) = 0 THEN
oldcol = col: oldrow = row
row = newrow: col = newcol
gr(row, col) = 1
path(lv) = sz * (row - 1) + col
IF row = sz AND col = sz THEN
walltot = 0
FOR r = sz TO 1 STEP -1
t = 0
FOR c = 1 TO sz
IF gr(r, c) THEN t = t + 1
NEXT
wall(sz - r + 1) = t
walltot = walltot + t
NEXT
FOR c = 1 TO sz
t = 0
FOR r = 1 TO sz
IF gr(r, c) THEN t = t + 1
NEXT
wall(sz + c) = t
walltot = walltot + t
NEXT
PRINT #2, USING "### "; walltot;
FOR i = 1 TO 2 * sz
PRINT #2, USING "### "; wall(i);
NEXT
FOR r = 1 TO sz: FOR c = 1 TO sz
IF gr(r, c) THEN PRINT #2, "x";: ELSE PRINT #2, " ";
NEXT: NEXT
FOR i = 1 TO lv
PRINT #2, USING "###"; path(i);
NEXT
PRINT #2,
ELSE
addOn lv + 1
END IF
gr(row, col) = 0
row = oldrow: col = oldcol
END IF
END IF
END IF
NEXT
NEXT
END SUB
The relevant output lines are (with annotations added):
wall numbers occupied positions path
total left(bottom to top) top(left to right) (reading order-- by position number
top-to-bottom and as at left
left to right)
26 1 1 3 5 3 2 2 3 2 4 x xx xxxxx xx x x x 1 6 7 12 13 8 3 4 9 10 15 20 25
26 1 1 3 5 3 2 2 3 2 4 xxx xxxxx xxx x x 1 6 7 2 3 8 13 14 9 10 15 20 25
26 1 3 5 3 1 3 2 3 2 3 x x xx xxxxx xx x x 1 6 11 12 17 18 13 8 9 14 15 20 25
26 1 3 5 3 1 3 2 3 2 3 x xxx xxxxx xxx x 1 6 11 12 7 8 13 18 19 14 15 20 25
26 2 2 3 2 4 1 1 3 5 3 xxxx xx xxx xx xx 1 2 3 4 9 10 15 14 13 18 19 24 25
26 2 2 3 2 4 1 1 3 5 3 xxxx xx xxx xx xx 1 2 3 4 9 8 13 14 15 20 19 24 25
26 3 2 3 2 3 1 3 5 3 1 xxx xx xxx xx xxx 1 2 3 8 9 14 13 12 17 18 23 24 25
26 3 2 3 2 3 1 3 5 3 1 xxx xx xxx xx xxx 1 2 3 8 7 12 13 14 19 18 23 24 25
26 3 5 3 1 1 4 2 3 2 2 x x x xx xxxxx xx x 1 6 11 16 17 22 23 18 13 14 19 20 25
26 3 5 3 1 1 4 2 3 2 2 x x xxx xxxxx xxx 1 6 11 16 17 12 13 18 23 24 19 20 25
26 4 2 3 2 2 3 5 3 1 1 xx xx xxx xx xxxx 1 2 7 8 13 12 11 16 17 22 23 24 25
26 4 2 3 2 2 3 5 3 1 1 xx xx xxx xx xxxx 1 2 7 6 11 12 13 18 17 22 23 24 25
The diagrams below represent the three basic solutions, which can be flipped about a diagonal. The paths are shown below in hexadecimal to alleviate crowding.
2 2 3 2 4 2 2 3 2 4
3 1 7 8 3 1 4 5
5 2 3 6 9 A 5 2 3 6 9 A
3 4 5 B 3 7 8 B
1 C 1 C
1 D 1 D
3 2 3 2 3 3 2 3 2 3
1 1 1 1
3 2 8 9 3 2 5 6
5 3 4 7 A B 5 3 4 7 A B
3 5 6 C 3 8 9 C
1 D 1 D
1 1 3 5 3 1 1 3 5 3
4 1 2 3 4 4 1 2 3 4
2 5 6 2 6 5
3 9 8 7 3 7 8 9
2 A B 2 B A
2 C D 2 C D
Edited on August 18, 2013, 6:45 pm
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Posted by Charlie
on 2013-08-18 14:31:49 |
computer verification of minimum
