This is my representation of the
Minus Cube
Consider a 2 unit cube being composed of 8 unit cubes but with one removed.
The blue circles are the centres of where the unit cubes may be placed.
The unlabelled circle is the centre of the unoccupied space.
Any cube that is horizontally or vertically adjacent to the void (ie, connected by a straight line) may be moved into it (ignore gravity).
Your task is to report on three challenges:
1. The least number of moves to return the left graphic to that shown on the right, the "home" position.
After setting the
HOME position:
2. The least number of moves to create a reflection of the shown left cube, ie, exchange the
2 with the
4 and the
6 with the
void.
3. The least number of moves so that the sums of opposing vertices on the four major diagonals (
red) is
9; allow the blank to be
8.
Having played with the interactive, have you found another challenge we could share in Comments?
Note: Please exercise the utmost of care with the interactive as programming does not trap careless entries, so
on browser freeze you may need to wait about 30 secs for a pop-up dialog advice.
Part 2:
DECLARE SUB makeMove (mno!)
DECLARE FUNCTION checksol! (mno!)
CLEAR , , 25000
OPEN "minus cube 2.txt" FOR OUTPUT AS #2
DIM SHARED cube(1, 1, 1), currx, curry, currz, hist(50), maxmove
maxmove = 19
cube(0, 0, 0) = 6 ' to 0
cube(1, 0, 0) = 7
cube(1, 1, 0) = 3
cube(0, 1, 0) = 2 ' to 4
cube(0, 0, 1) = 5
cube(1, 0, 1) = 0 ' to 6
cube(1, 1, 1) = 4 ' to 2
cube(0, 1, 1) = 1
currx = 1: curry = 0: currz = 1
makeMove 1
FUNCTION checksol (mno)
c = 0
IF currx = 0 AND curry = 0 AND currz = 0 THEN
IF cube(1, 0, 1) = 6 THEN
IF cube(1, 0, 0) = 7 THEN
IF cube(1, 1, 0) = 3 THEN
IF cube(0, 1, 0) = 4 THEN
IF cube(0, 0, 1) = 5 THEN
IF cube(1, 1, 1) = 2 THEN
IF cube(0, 1, 1) = 1 THEN
c = 1
FOR i = 1 TO mno
PRINT #2, hist(i);
NEXT
PRINT #2,
END IF
END IF
END IF
END IF
END IF
END IF
END IF
END IF
checksol = c
END FUNCTION
SUB makeMove (mno)
' x-axis
tox = 1 - currx
hist(mno) = cube(tox, curry, currz)
IF hist(mno) <> hist(mno - 1) THEN
SWAP cube(currx, curry, currz), cube(tox, curry, currz)
prex = currx
currx = tox
IF currx = 0 THEN ok = checksol(mno): ELSE ok = 0
IF ok = 0 AND mno < maxmove THEN
makeMove mno + 1
END IF
currx = prex
SWAP cube(currx, curry, currz), cube(tox, curry, currz)
END IF
' y-axis
toy = 1 - curry
hist(mno) = cube(currx, toy, currz)
IF hist(mno) <> hist(mno - 1) THEN
SWAP cube(currx, curry, currz), cube(currx, toy, currz)
prey = curry
curry = toy
IF curry = 0 THEN ok = checksol(mno): ELSE ok = 0
IF ok = 0 AND mno < maxmove THEN
makeMove mno + 1
END IF
curry = prey
SWAP cube(currx, curry, currz), cube(currx, toy, currz)
END IF
' z-axis
toz = 1 - currz
hist(mno) = cube(currx, curry, toz)
IF hist(mno) <> hist(mno - 1) THEN
SWAP cube(currx, curry, currz), cube(currx, curry, toz)
prez = currz
currz = toz
IF currz = 0 THEN ok = checksol(mno): ELSE ok = 0
IF ok = 0 AND mno < maxmove THEN
makeMove mno + 1
END IF
currz = prez
SWAP cube(currx, curry, currz), cube(currx, curry, toz)
END IF
END SUB
finding:
5 1 4 5 1 6 7 3 2 4 5 1 6 5 1 2 3 7
5 1 4 5 7 3 2 4 1 7 5 2 3 6 7 5 6 7
5 1 4 3 2 6 7 2 6 4 1 5 2 6 3 2 6 7
5 1 4 3 2 4 3 2 7 6 4 3 1 5 6 7 3 4
5 1 4 3 7 6 1 4 2 1 4 5 6 7 3 2 1 4
5 1 2 3 7 6 3 7 4 2 1 5 6 3 7 4 3 7
5 1 2 3 7 5 4 2 1 4 5 6 4 5 6 7 3 4
5 1 2 3 7 5 4 2 1 6 5 4 6 5 4 7 3 4
5 1 2 3 4 2 3 4 7 6 4 3 1 5 6 7 3 4
5 1 2 6 7 3 6 2 4 6 2 4 1 5 6 2 3 7
5 1 2 6 7 3 4 2 1 7 6 4 3 6 7 5 6 7
5 1 2 6 7 3 4 2 6 4 2 6 1 5 6 2 3 7
5 6 7 3 2 7 6 1 4 2 3 6 7 4 1 5 6 7
5 6 7 3 2 1 4 5 6 4 5 2 3 7 4 5 1 4
5 6 7 3 4 5 6 1 2 4 3 6 5 2 1 5 6 7
5 6 7 5 6 1 4 6 5 3 2 4 1 5 6 2 3 7
5 6 7 5 6 1 2 7 5 3 4 2 1 5 7 4 3 7
5 6 2 1 6 2 7 3 4 6 2 5 6 2 1 4 3 7
4 1 5 4 7 6 4 7 6 3 2 4 7 5 1 2 3 7
4 1 5 4 7 3 2 5 4 6 5 4 1 2 3 7 6 5
4 1 5 6 7 3 1 4 6 5 4 1 2 4 1 2 3 7
4 1 5 6 7 3 1 4 6 5 2 1 4 2 1 4 3 7
4 1 5 6 7 4 6 7 4 3 2 4 7 5 1 2 3 7
4 1 2 3 7 6 5 4 1 2 4 1 6 7 3 4 1 5
4 3 2 1 5 6 7 4 3 2 4 3 6 5 1 4 3 7
4 3 7 6 5 4 6 5 4 1 2 4 5 7 3 2 1 5
4 3 7 6 5 1 3 4 6 7 2 3 4 2 3 4 1 5
4 3 7 6 5 1 3 4 6 7 4 3 2 4 3 2 1 5
4 3 7 4 5 1 2 7 4 6 7 4 3 2 1 5 6 7
4 3 7 4 5 6 4 5 6 1 2 4 5 7 3 2 1 5
7 6 2 3 6 2 5 1 4 6 2 7 6 2 3 4 1 5
7 6 5 7 6 3 2 5 7 1 4 2 3 7 5 4 1 5
7 6 5 7 6 3 4 6 7 1 2 4 3 7 6 2 1 5
7 6 5 1 4 7 6 3 2 4 1 6 7 2 3 7 6 5
7 6 5 1 2 3 4 7 6 4 7 2 1 5 4 7 3 4
7 6 5 1 2 5 6 3 4 2 1 6 5 4 3 7 6 5
7 3 2 6 5 1 4 2 6 4 2 6 3 7 6 2 1 5
7 3 2 6 5 1 4 2 3 5 6 4 1 6 5 7 6 5
7 3 2 6 5 1 6 2 4 6 2 4 3 7 6 2 1 5
7 3 2 1 4 2 1 4 5 6 4 1 3 7 6 5 1 4
7 3 2 1 5 7 4 2 3 6 7 4 6 7 4 5 1 4
7 3 2 1 5 7 4 2 3 4 7 6 4 7 6 5 1 4
7 3 2 1 5 6 1 5 4 2 3 7 6 1 5 4 1 5
7 3 4 1 5 6 3 4 2 3 4 7 6 5 1 2 3 4
7 3 4 1 2 4 1 2 5 6 4 1 3 7 6 5 1 4
7 3 4 1 2 6 5 2 6 4 3 7 2 6 1 2 6 5
7 3 4 7 5 1 2 4 3 5 7 2 1 6 5 7 6 5
7 3 4 7 3 6 5 1 2 4 7 3 6 7 3 2 1 5
(48 ways of going from the Home position to the mirror of the Home position in 18 moves.)
Part 3:
Rather than allowing the blank to be 8, and adding to 9, the calculations continue to treat the blank as zero and seek to make opposing corners add to 1 mod 8. The moves, of course, will the the same no matter which way you look at it.
DECLARE SUB makeMove (mno!)
DECLARE FUNCTION checksol! (mno!)
CLEAR , , 25000
OPEN "minus cube 3.txt" FOR OUTPUT AS #2
DIM SHARED cube(1, 1, 1), currx, curry, currz, hist(50), maxmove
maxmove = 10
cube(0, 0, 0) = 6
cube(1, 0, 0) = 7
cube(1, 1, 0) = 3
cube(0, 1, 0) = 2
cube(0, 0, 1) = 5
cube(1, 0, 1) = 0
cube(1, 1, 1) = 4
cube(0, 1, 1) = 1
currx = 1: curry = 0: currz = 1
makeMove 1
FUNCTION checksol (mno)
c = 0
IF (cube(0, 0, 0) + cube(1, 1, 1)) MOD 8 = 1 THEN
IF (cube(0, 0, 1) + cube(1, 1, 0)) MOD 8 = 1 THEN
IF (cube(0, 1, 0) + cube(1, 0, 1)) MOD 8 = 1 THEN
IF (cube(1, 0, 0) + cube(0, 1, 1)) MOD 8 = 1 THEN
c = 1
FOR i = 1 TO mno
PRINT #2, hist(i);
NEXT
PRINT #2,
END IF
END IF
END IF
END IF
checksol = c
END FUNCTION
SUB makeMove (mno)
' x-axis
tox = 1 - currx
hist(mno) = cube(tox, curry, currz)
IF hist(mno) <> hist(mno - 1) THEN
SWAP cube(currx, curry, currz), cube(tox, curry, currz)
prex = currx
currx = tox
ok = checksol(mno)
IF ok = 0 AND mno < maxmove THEN
makeMove mno + 1
END IF
currx = prex
SWAP cube(currx, curry, currz), cube(tox, curry, currz)
END IF
' y-axis
toy = 1 - curry
hist(mno) = cube(currx, toy, currz)
IF hist(mno) <> hist(mno - 1) THEN
SWAP cube(currx, curry, currz), cube(currx, toy, currz)
prey = curry
curry = toy
ok = checksol(mno)
IF ok = 0 AND mno < maxmove THEN
makeMove mno + 1
END IF
curry = prey
SWAP cube(currx, curry, currz), cube(currx, toy, currz)
END IF
' z-axis
toz = 1 - currz
hist(mno) = cube(currx, curry, toz)
IF hist(mno) <> hist(mno - 1) THEN
SWAP cube(currx, curry, currz), cube(currx, curry, toz)
prez = currz
currz = toz
ok = checksol(mno)
IF ok = 0 AND mno < maxmove THEN
makeMove mno + 1
END IF
currz = prez
SWAP cube(currx, curry, currz), cube(currx, curry, toz)
END IF
END SUB
finding
5 1 4 3 2 6 7 5 3 2
5 1 4 3 7 6 2 4 3 7
5 1 2 6 7 3 4 2 6 4
5 6 7 3 2 1 4 2 3 5
5 6 7 3 4 1 2 4 3 7
5 6 2 3 7 5 6 1 4 7
5 6 2 3 7 5 6 2 5
5 6 2 1 6 5 7 3 4 7
4 1 5 6 7 3 2 5 6 7
4 1 5 6 2 3 7 4 6 2
4 1 2 3 7 6 5 2 3 5
4 3 2 6 7 4 3 1 5 7
4 3 2 6 7 4 3 2 4
4 3 2 1 3 4 7 6 5 7
4 3 7 6 2 1 5 2 6 4
4 3 7 6 5 1 2 5 6 7
7 6 2 3 4 1 3 2
7 6 2 3 4 7 5 1 3 4
7 6 2 3 4 7 6
7 6 5 7 4 3 2 5 6
7 6 5 7 4 3 2 1 3 4
7 6 5 1 4 7 6 5 2 3
7 6 5 1 2 5 6 7 4 3
7 3 2 6 5 7 4 1 6 5
7 3 2 6 5 7 3
7 3 2 6 5 1 6 2
7 3 4 1 5 7 3 4 2 6
7 3 4 1 2 4 3 7 5 6
7 3 4 7 5 6 2 4 3
7 3 4 7 5 6 2 1 6 5
There are two 7-move solutions shown:
7 6 2 3 4 7 6
7 3 2 6 5 7 3
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Posted by Charlie
on 2013-04-18 17:51:54 |
parts 2 and 3
