Twelve disks, each with holes at some of six positions, are to be rotated and stacked so that the pegs pass through them.
combinatorics/disks.on.pegs New
New entry, edited from material collected for the archive and not yet in a released version.
Problem
Twelve disks, each with holes at some of six positions, are to be rotated and stacked so that the pegs pass through them. In how many ways can it be done? [ Editor: the puzzle, from a December 1996 rec.puzzles thread "13 pegs, 12 disks, one brain", is not stated in the collected material; this statement is the editor's, inferred from the description of the solving program in the answer, and the rules (how the pegs and holes must line up) are not known. ]
Show solution
This problem was solved by computer using a program which does the following: 1. Read in the data (can be used for other disks/holes/pegs configurations as well). 2. Repeat steps 3 and 4 until second disk returns to original position. 3. Count the holes at each position. If all sums are divisible by peg length proceed to step 5. 4. Rotate last disk by one position. If back to original state (or first equivalent state in case of symmetrical hole arrangement), rotate before last disk by one position, and so on down to second disk (first disk is not rotated). 5. Repeat steps 6 and 7 until disks restored to original stacking order (disks are not rotated; only stacking order is changed). 6. Position a disk (loop through all disks). Position another disk (loop through remaining disks) and check whether rules satisfied. If yes, position another disk, and so on. 7. If all disks successfully positioned, output result if number of first disk is less than that of last disk (to avoid counting reversed stacking as a different solution). 8. Return to step 4. This cuts down drastically the amount of computation needed in comparison with a simple-minded brute force method. On a 166 MHz PC this took just over 10 minutes. 66 solutions were found, and they are listed below (*'s denote holes): Solution No. 1: Disk 1: 3 - * . * * . * Disk 2: 9 - * * * * . * Disk 3: 10 - * * * * * * Disk 4: 4 - . * . . * . Disk 5: 8 - * . * . * . Disk 6: 2 - * . * . . . Disk 7: 1 - * * * . . . Disk 8: 5 - . * . * * . Disk 9: 11 - . * . * * * Disk 10: 6 - * . . * * * Disk 11: 7 - * . . . . * Disk 12: 12 - * . . . . . Solution No. 2: Disk 1: 3 - * . * * . * Disk 2: 9 - * * * * . * Disk 3: 10 - * * * * * * Disk 4: 4 - . * . . * . Disk 5: 8 - * . * . * . Disk 6: 2 - * . * . . . Disk 7: 5 - * . * * . . Disk 8: 7 - . . . * * . Disk 9: 6 - . * * * * . Disk 10: 11 - * * * . * . Disk 11: 1 - * * * . . . Disk 12: 12 - * . . . . . Solution No. 3: Disk 1: 3 - * . * * . * Disk 2: 9 - * . * * * * Disk 3: 10 - * * * * * * Disk 4: 4 - . * . . * . Disk 5: 7 - . * * . . . Disk 6: 8 - * . * . * . Disk 7: 11 - * . * . * * Disk 8: 6 - * * . . * * Disk 9: 5 - . * * . . * Disk 10: 1 - * * * . . . Disk 11: 2 - * . * . . . Disk 12: 12 - * . . . . . Solution No. 4: Disk 1: 3 - * . * * . * Disk 2: 9 - * . * * * * Disk 3: 10 - * * * * * * Disk 4: 4 - . * . . * . Disk 5: 7 - * * . . . . Disk 6: 2 - * . * . . . Disk 7: 1 - * * * . . . Disk 8: 5 - . * * . . * Disk 9: 6 - * * . . * * Disk 10: 11 - * . * . * * Disk 11: 8 - * . * . * . Disk 12: 12 - . . * . . . Solution No. 5: Disk 1: 3 - * . * * . * Disk 2: 9 - * . * * * * Disk 3: 10 - * * * * * * Disk 4: 4 - . * . . * . Disk 5: 1 - * * * . . . Disk 6: 2 - * . * . . . Disk 7: 8 - * . * . * . Disk 8: 7 - . . . * * . Disk 9: 6 - . . * * * * Disk 10: 11 - . * * * . * Disk 11: 5 - . * * . . * Disk 12: 12 - . * . . . . Solution No. 6: Disk 1: 3 - * . * * . * Disk 2: 9 - * . * * * * Disk 3: 10 - * * * * * * Disk 4: 4 - . * . . * . Disk 5: 1 - * * * . . . Disk 6: 2 - * . * . . . Disk 7: 8 - * . * . * . Disk 8: 7 - . . . * * . Disk 9: 6 - . * * * * . Disk 10: 11 - . * * * . * Disk 11: 5 - . * * . . * Disk 12: 12 - . . . . . * Solution No. 7: Disk 1: 2 - * . * . . . Disk 2: 9 - * * * * * . Disk 3: 10 - * * * * * * Disk 4: 11 - . * . * * * Disk 5: 4 - . . * . . * Disk 6: 7 - . * * . . . Disk 7: 5 - . * * . . * Disk 8: 8 - . * . * . * Disk 9: 3 - * . * * . * Disk 10: 6 - * * * * . . Disk 11: 1 - * * * . . . Disk 12: 12 - . * . . . . Solution No. 8: Disk 1: 3 - . * * . * * Disk 2: 9 - . * * * * * Disk 3: 10 - * * * * * * Disk 4: 4 - * . . * . . Disk 5: 2 - * . * . . . Disk 6: 7 - . * * . . . Disk 7: 1 - * * * . . . Disk 8: 5 - * * . . * . Disk 9: 6 - * . . * * * Disk 10: 11 - . * . * * * Disk 11: 8 - . * . * . * Disk 12: 12 - . * . . . . Solution No. 9: Disk 1: 4 - * . . * . . Disk 2: 9 - * * * * * . Disk 3: 10 - * * * * * * Disk 4: 3 - . * * . * * Disk 5: 7 - * . . . . * Disk 6: 2 - * . * . . . Disk 7: 1 - * * * . . . Disk 8: 5 - . * * . . * Disk 9: 6 - * * . . * * Disk 10: 11 - * . * . * * Disk 11: 8 - * . * . * . Disk 12: 12 - . . * . . . Solution No. 10: Disk 1: 3 - . * * . * * Disk 2: 9 - * * * . * * Disk 3: 10 - * * * * * * Disk 4: 4 - * . . * . . Disk 5: 7 - . . * * . . Disk 6: 8 - * . * . * . Disk 7: 11 - * . * . * * Disk 8: 6 - * * . . * * Disk 9: 5 - . * * . . * Disk 10: 1 - * * * . . . Disk 11: 2 - * . * . . . Disk 12: 12 - * . . . . . Solution No. 11: Disk 1: 4 - * . . * . . Disk 2: 9 - * . * * * * Disk 3: 10 - * * * * * * Disk 4: 3 - . * * . * * Disk 5: 7 - * * . . . . Disk 6: 2 - * . * . . . Disk 7: 1 - * * * . . . Disk 8: 5 - . * * . . * Disk 9: 6 - * * . . * * Disk 10: 11 - * . * . * * Disk 11: 8 - * . * . * . Disk 12: 12 - . . * . . . Solution No. 12: Disk 1: 6 - * * * * . . Disk 2: 9 - * * * * . * Disk 3: 10 - * * * * * * Disk 4: 7 - . . . . * * Disk 5: 8 - * . * . * . Disk 6: 2 - * . * . . . Disk 7: 1 - * * * . . . Disk 8: 4 - . * . . * . Disk 9: 3 - * * . * * . Disk 10: 11 - * . * * * . Disk 11: 5 - * . * * . . Disk 12: 12 - . . * . . . Solution No. 13: Disk 1: 6 - * * * . . * Disk 2: 9 - * * * * . * Disk 3: 10 - * * * * * * Disk 4: 7 - . . . * * . Disk 5: 8 - * . * . * . Disk 6: 2 - * . * . . . Disk 7: 1 - * * * . . . Disk 8: 4 - . * . . * . Disk 9: 3 - * * . * * . Disk 10: 11 - * . * * * . Disk 11: 5 - * . * * . . Disk 12: 12 - . . * . . . Solution No. 14: Disk 1: 4 - . . * . . * Disk 2: 9 - . * * * * * Disk 3: 10 - * * * * * * Disk 4: 3 - * * . * * . Disk 5: 2 - * . * . . . Disk 6: 7 - . * * . . . Disk 7: 1 - * * * . . . Disk 8: 5 - * * . . * . Disk 9: 6 - * . . * * * Disk 10: 11 - . * . * * * Disk 11: 8 - . * . * . * Disk 12: 12 - . * . . . . Solution No. 15: Disk 1: 3 - * * . * * . Disk 2: 9 - * * * * * . Disk 3: 10 - * * * * * * Disk 4: 4 - . . * . . * Disk 5: 7 - * . . . . * Disk 6: 2 - * . * . . . Disk 7: 1 - * * * . . . Disk 8: 5 - . * * . . * Disk 9: 6 - * * . . * * Disk 10: 11 - * . * . * * Disk 11: 8 - * . * . * . Disk 12: 12 - . . * . . . Solution No. 16: Disk 1: 4 - . . * . . * Disk 2: 9 - * * * . * * Disk 3: 10 - * * * * * * Disk 4: 3 - * * . * * . Disk 5: 7 - . . * * . . Disk 6: 8 - * . * . * . Disk 7: 11 - * . * . * * Disk 8: 6 - * * . . * * Disk 9: 5 - . * * . . * Disk 10: 1 - * * * . . . Disk 11: 2 - * . * . . . Disk 12: 12 - * . . . . . Solution No. 17: Disk 1: 4 - . . * . . * Disk 2: 9 - * . * * * * Disk 3: 10 - * * * * * * Disk 4: 3 - * * . * * . Disk 5: 7 - . * * . . . Disk 6: 8 - * . * . * . Disk 7: 11 - * . * . * * Disk 8: 6 - * * . . * * Disk 9: 5 - . * * . . * Disk 10: 1 - * * * . . . Disk 11: 2 - * . * . . . Disk 12: 12 - * . . . . . Solution No. 18: Disk 1: 11 - * . * * * . Disk 2: 9 - * * * * * . Disk 3: 10 - * * * * * * Disk 4: 2 - . * . . . * Disk 5: 7 - . . . . * * Disk 6: 4 - . * . . * . Disk 7: 5 - . * . * * . Disk 8: 8 - . * . * . * Disk 9: 3 - * . * * . * Disk 10: 6 - * * * . . * Disk 11: 1 - * * * . . . Disk 12: 12 - . * . . . . Solution No. 19: Disk 1: 7 - . * * . . . Disk 2: 9 - * * * * . * Disk 3: 10 - * * * * * * Disk 4: 6 - * . . * * * Disk 5: 4 - . * . . * . Disk 6: 8 - . * . * . * Disk 7: 11 - * * . * . * Disk 8: 3 - * . * * . * Disk 9: 1 - * * * . . . Disk 10: 5 - . * * . . * Disk 11: 2 - . * . . . * Disk 12: 12 - . . . . . * Solution No. 20: Disk 1: 7 - * * . . . . Disk 2: 9 - * * * * . * Disk 3: 10 - * * * * * * Disk 4: 6 - . . * * * * Disk 5: 4 - . * . . * . Disk 6: 8 - . * . * . * Disk 7: 11 - * * . * . * Disk 8: 3 - * . * * . * Disk 9: 1 - * * * . . . Disk 10: 5 - . * * . . * Disk 11: 2 - . * . . . * Disk 12: 12 - . . . . . * Solution No. 21: Disk 1: 6 - * * * * . . Disk 2: 9 - * * * * . * Disk 3: 10 - * * * * * * Disk 4: 7 - . . . . * * Disk 5: 4 - . * . . * . Disk 6: 8 - . * . * . * Disk 7: 11 - * * . * . * Disk 8: 3 - * . * * . * Disk 9: 1 - * * * . . . Disk 10: 5 - . * * . . * Disk 11: 2 - . * . . . * Disk 12: 12 - . . . . . * Solution No. 22: Disk 1: 2 - . * . . . * Disk 2: 9 - * * * * . * Disk 3: 10 - * * * * * * Disk 4: 11 - * . * * * . Disk 5: 4 - . * . . * . Disk 6: 1 - * * * . . . Disk 7: 6 - * * * * . . Disk 8: 3 - * . * * . * Disk 9: 8 - . * . * . * Disk 10: 5 - . * * . . * Disk 11: 7 - . * * . . . Disk 12: 12 - . . * . . . Solution No. 23: Disk 1: 6 - * * * . . * Disk 2: 9 - * * * * . * Disk 3: 10 - * * * * * * Disk 4: 7 - . . . * * . Disk 5: 4 - . * . . * . Disk 6: 8 - . * . * . * Disk 7: 11 - * * . * . * Disk 8: 3 - * . * * . * Disk 9: 1 - * * * . . . Disk 10: 5 - . * * . . * Disk 11: 2 - . * . . . * Disk 12: 12 - . . . . . * Solution No. 24: Disk 1: 6 - * * . . * * Disk 2: 9 - * * . * * * Disk 3: 10 - * * * * * * Disk 4: 7 - . . * * . . Disk 5: 4 - . . * . . * Disk 6: 2 - . * . . . * Disk 7: 5 - . * * . . * Disk 8: 1 - * * * . . . Disk 9: 3 - * . * * . * Disk 10: 11 - * * . * . * Disk 11: 8 - . * . * . * Disk 12: 12 - . * . . . . Solution No. 25: Disk 1: 6 - * . . * * * Disk 2: 9 - * * . * * * Disk 3: 10 - * * * * * * Disk 4: 7 - . * * . . . Disk 5: 4 - . . * . . * Disk 6: 2 - . * . . . * Disk 7: 5 - . * * . . * Disk 8: 1 - * * * . . . Disk 9: 3 - * . * * . * Disk 10: 11 - * * . * . * Disk 11: 8 - . * . * . * Disk 12: 12 - . * . . . . Solution No. 26: Disk 1: 7 - * . . . . * Disk 2: 9 - * * . * * * Disk 3: 10 - * * * * * * Disk 4: 6 - . * * * * . Disk 5: 4 - . . * . . * Disk 6: 2 - . * . . . * Disk 7: 5 - . * * . . * Disk 8: 1 - * * * . . . Disk 9: 3 - * . * * . * Disk 10: 11 - * * . * . * Disk 11: 8 - . * . * . * Disk 12: 12 - . * . . . . Solution No. 27: Disk 1: 7 - . . . . * * Disk 2: 9 - * * . * * * Disk 3: 10 - * * * * * * Disk 4: 6 - * * * * . . Disk 5: 4 - . . * . . * Disk 6: 2 - . * . . . * Disk 7: 5 - . * * . . * Disk 8: 1 - * * * . . . Disk 9: 3 - * . * * . * Disk 10: 11 - * * . * . * Disk 11: 8 - . * . * . * Disk 12: 12 - . * . . . . Solution No. 28: Disk 1: 11 - * . * * * . Disk 2: 9 - * * * * * . Disk 3: 10 - * * * * * * Disk 4: 2 - . * . . . * Disk 5: 4 - . . * . . * Disk 6: 7 - . * * . . . Disk 7: 5 - . * * . . * Disk 8: 8 - . * . * . * Disk 9: 3 - * . * * . * Disk 10: 6 - * * * * . . Disk 11: 1 - * * * . . . Disk 12: 12 - . * . . . . Solution No. 29: Disk 1: 2 - . * . . . * Disk 2: 9 - * * * * . * Disk 3: 10 - * * * * * * Disk 4: 11 - * . * * * . Disk 5: 4 - . * . . * . Disk 6: 7 - . * * . . . Disk 7: 1 - * * * . . . Disk 8: 8 - * . * . * . Disk 9: 6 - * * . . * * Disk 10: 3 - . * * . * * Disk 11: 5 - . * * . . * Disk 12: 12 - . . * . . . Solution No. 30: Disk 1: 6 - . . * * * * Disk 2: 9 - . * * * * * Disk 3: 10 - * * * * * * Disk 4: 7 - * * . . . . Disk 5: 2 - * . . . * . Disk 6: 4 - . * . . * . Disk 7: 5 - * * . . * . Disk 8: 1 - * * * . . . Disk 9: 3 - * . * * . * Disk 10: 11 - . * * * . * Disk 11: 8 - . * . * . * Disk 12: 12 - . * . . . . Solution No. 31: Disk 1: 6 - . * * * * . Disk 2: 9 - . * * * * * Disk 3: 10 - * * * * * * Disk 4: 7 - * . . . . * Disk 5: 2 - * . . . * . Disk 6: 4 - . * . . * . Disk 7: 5 - * * . . * . Disk 8: 1 - * * * . . . Disk 9: 3 - * . * * . * Disk 10: 11 - . * * * . * Disk 11: 8 - . * . * . * Disk 12: 12 - . * . . . . Solution No. 32: Disk 1: 7 - . . . . * * Disk 2: 9 - . * * * * * Disk 3: 10 - * * * * * * Disk 4: 6 - * * * * . . Disk 5: 2 - * . . . * . Disk 6: 4 - . * . . * . Disk 7: 5 - * * . . * . Disk 8: 1 - * * * . . . Disk 9: 3 - * . * * . * Disk 10: 11 - . * * * . * Disk 11: 8 - . * . * . * Disk 12: 12 - . * . . . . Solution No. 33: Disk 1: 7 - . . . * * . Disk 2: 9 - . * * * * * Disk 3: 10 - * * * * * * Disk 4: 6 - * * * . . * Disk 5: 2 - * . . . * . Disk 6: 4 - . * . . * . Disk 7: 5 - * * . . * . Disk 8: 1 - * * * . . . Disk 9: 3 - * . * * . * Disk 10: 11 - . * * * . * Disk 11: 8 - . * . * . * Disk 12: 12 - . * . . . . Solution No. 34: Disk 1: 3 - * . * * . * Disk 2: 9 - * * * * . * Disk 3: 10 - * * * * * * Disk 4: 4 - . * . . * . Disk 5: 6 - * . . * * * Disk 6: 5 - * . . * . * Disk 7: 11 - * * . * . * Disk 8: 7 - . * * . . . Disk 9: 1 - * * * . . . Disk 10: 8 - * . * . * . Disk 11: 2 - * . . . * . Disk 12: 12 - . . . . * . Solution No. 35: Disk 1: 2 - * . . . * . Disk 2: 9 - * * * * * . Disk 3: 10 - * * * * * * Disk 4: 11 - . * * * . * Disk 5: 7 - . . . . * * Disk 6: 4 - . * . . * . Disk 7: 5 - . * . * * . Disk 8: 8 - . * . * . * Disk 9: 3 - * . * * . * Disk 10: 6 - * * * . . * Disk 11: 1 - * * * . . . Disk 12: 12 - . * . . . . Solution No. 36: Disk 1: 2 - * . . . * . Disk 2: 9 - * * * . * * Disk 3: 10 - * * * * * * Disk 4: 11 - . * * * . * Disk 5: 7 - . . . * * . Disk 6: 4 - . * . . * . Disk 7: 5 - . * . * * . Disk 8: 8 - . * . * . * Disk 9: 3 - * . * * . * Disk 10: 6 - * * * . . * Disk 11: 1 - * * * . . . Disk 12: 12 - . * . . . . Solution No. 37: Disk 1: 11 - . * * * . * Disk 2: 9 - * * * * . * Disk 3: 10 - * * * * * * Disk 4: 2 - * . . . * . Disk 5: 4 - . * . . * . Disk 6: 1 - * * * . . . Disk 7: 6 - * * * * . . Disk 8: 3 - * . * * . * Disk 9: 8 - . * . * . * Disk 10: 5 - . * * . . * Disk 11: 7 - . * * . . . Disk 12: 12 - . . * . . . Solution No. 38: Disk 1: 11 - . * * * . * Disk 2: 9 - * * * * . * Disk 3: 10 - * * * * * * Disk 4: 2 - * . . . * . Disk 5: 4 - . * . . * . Disk 6: 7 - . * * . . . Disk 7: 1 - * * * . . . Disk 8: 8 - * . * . * . Disk 9: 6 - * * . . * * Disk 10: 3 - . * * . * * Disk 11: 5 - . * * . . * Disk 12: 12 - . . * . . . Solution No. 39: Disk 1: 3 - * . * * . * Disk 2: 9 - * . * * * * Disk 3: 10 - * * * * * * Disk 4: 4 - . * . . * . Disk 5: 8 - . * . * . * Disk 6: 2 - . . . * . * Disk 7: 5 - * . . * . * Disk 8: 1 - * * * . . . Disk 9: 11 - * * * . * . Disk 10: 6 - . * * * * . Disk 11: 7 - . . . * * . Disk 12: 12 - . . . * . . Solution No. 40: Disk 1: 3 - * . * * . * Disk 2: 9 - * . * * * * Disk 3: 10 - * * * * * * Disk 4: 4 - . * . . * . Disk 5: 8 - . * . * . * Disk 6: 2 - . . . * . * Disk 7: 11 - * * . * . * Disk 8: 1 - * * * . . . Disk 9: 6 - * * * . . * Disk 10: 5 - . . * . * * Disk 11: 7 - . . . . * * Disk 12: 12 - . . . . * . Solution No. 41: Disk 1: 11 - * * * . * . Disk 2: 9 - * * * * * . Disk 3: 10 - * * * * * * Disk 4: 2 - . . . * . * Disk 5: 7 - . . . . * * Disk 6: 4 - . * . . * . Disk 7: 5 - . * . * * . Disk 8: 8 - . * . * . * Disk 9: 3 - * . * * . * Disk 10: 6 - * * * . . * Disk 11: 1 - * * * . . . Disk 12: 12 - . * . . . . Solution No. 42: Disk 1: 11 - * * * . * . Disk 2: 9 - * * * . * * Disk 3: 10 - * * * * * * Disk 4: 2 - . . . * . * Disk 5: 7 - . . . * * . Disk 6: 4 - . * . . * . Disk 7: 5 - . * . * * . Disk 8: 8 - . * . * . * Disk 9: 3 - * . * * . * Disk 10: 6 - * * * . . * Disk 11: 1 - * * * . . . Disk 12: 12 - . * . . . . Solution No. 43: Disk 1: 11 - * * * . * . Disk 2: 9 - * * * * * . Disk 3: 10 - * * * * * * Disk 4: 2 - . . . * . * Disk 5: 4 - . . * . . * Disk 6: 7 - . * * . . . Disk 7: 5 - . * * . . * Disk 8: 8 - . * . * . * Disk 9: 3 - * . * * . * Disk 10: 6 - * * * * . . Disk 11: 1 - * * * . . . Disk 12: 12 - . * . . . . Solution No. 44: Disk 1: 3 - . * * . * * Disk 2: 9 - . * * * * * Disk 3: 10 - * * * * * * Disk 4: 4 - * . . * . . Disk 5: 1 - * * * . . . Disk 6: 7 - . * * . . . Disk 7: 5 - . * * . . * Disk 8: 2 - . . . * . * Disk 9: 6 - * . . * * * Disk 10: 11 - * . * * * . Disk 11: 8 - * . * . * . Disk 12: 12 - . . * . . . Solution No. 45: Disk 1: 3 - . * * . * * Disk 2: 9 - . * * * * * Disk 3: 10 - * * * * * * Disk 4: 4 - * . . * . . Disk 5: 1 - * * * . . . Disk 6: 7 - . * * . . . Disk 7: 5 - . * * . . * Disk 8: 2 - . . . * . * Disk 9: 6 - . . * * * * Disk 10: 11 - * . * * * . Disk 11: 8 - * . * . * . Disk 12: 12 - * . . . . . Solution No. 46: Disk 1: 6 - * * . . * * Disk 2: 9 - * * . * * * Disk 3: 10 - * * * * * * Disk 4: 7 - . . * * . . Disk 5: 2 - . . * . * . Disk 6: 4 - . * . . * . Disk 7: 5 - * * . . * . Disk 8: 1 - * * * . . . Disk 9: 3 - * . * * . * Disk 10: 11 - . * * * . * Disk 11: 8 - . * . * . * Disk 12: 12 - . * . . . . Solution No. 47: Disk 1: 3 - * . * * . * Disk 2: 9 - * * * * . * Disk 3: 10 - * * * * * * Disk 4: 4 - . * . . * . Disk 5: 2 - . . * . * . Disk 6: 7 - . * * . . . Disk 7: 1 - * * * . . . Disk 8: 5 - * * . . * . Disk 9: 6 - * . . * * * Disk 10: 11 - . * . * * * Disk 11: 8 - . * . * . * Disk 12: 12 - . * . . . . Solution No. 48: Disk 1: 6 - * . . * * * Disk 2: 9 - * * . * * * Disk 3: 10 - * * * * * * Disk 4: 7 - . * * . . . Disk 5: 2 - . . * . * . Disk 6: 4 - . * . . * . Disk 7: 5 - * * . . * . Disk 8: 1 - * * * . . . Disk 9: 3 - * . * * . * Disk 10: 11 - . * * * . * Disk 11: 8 - . * . * . * Disk 12: 12 - . * . . . . Solution No. 49: Disk 1: 7 - . . . . * * Disk 2: 9 - * * . * * * Disk 3: 10 - * * * * * * Disk 4: 6 - * * * * . . Disk 5: 2 - . . * . * . Disk 6: 4 - . * . . * . Disk 7: 5 - * * . . * . Disk 8: 1 - * * * . . . Disk 9: 3 - * . * * . * Disk 10: 11 - . * * * . * Disk 11: 8 - . * . * . * Disk 12: 12 - . * . . . . Solution No. 50: Disk 1: 7 - . . . * * . Disk 2: 9 - * * . * * * Disk 3: 10 - * * * * * * Disk 4: 6 - * * * . . * Disk 5: 2 - . . * . * . Disk 6: 4 - . * . . * . Disk 7: 5 - * * . . * . Disk 8: 1 - * * * . . . Disk 9: 3 - * . * * . * Disk 10: 11 - . * * * . * Disk 11: 8 - . * . * . * Disk 12: 12 - . * . . . . Solution No. 51: Disk 1: 3 - * . * * . * Disk 2: 9 - * . * * * * Disk 3: 10 - * * * * * * Disk 4: 4 - . * . . * . Disk 5: 11 - * * . * . * Disk 6: 5 - * . . * . * Disk 7: 6 - * . . * * * Disk 8: 2 - . . * . * . Disk 9: 8 - * . * . * . Disk 10: 1 - * * * . . . Disk 11: 7 - * * . . . . Disk 12: 12 - . * . . . . Solution No. 52: Disk 1: 2 - . . * . * . Disk 2: 9 - * * * * * . Disk 3: 10 - * * * * * * Disk 4: 11 - * * . * . * Disk 5: 7 - . . . . * * Disk 6: 4 - . * . . * . Disk 7: 5 - . * . * * . Disk 8: 8 - . * . * . * Disk 9: 3 - * . * * . * Disk 10: 6 - * * * . . * Disk 11: 1 - * * * . . . Disk 12: 12 - . * . . . . Solution No. 53: Disk 1: 2 - . . * . * . Disk 2: 9 - * * * . * * Disk 3: 10 - * * * * * * Disk 4: 11 - * * . * . * Disk 5: 7 - . . . * * . Disk 6: 4 - . * . . * . Disk 7: 5 - . * . * * . Disk 8: 8 - . * . * . * Disk 9: 3 - * . * * . * Disk 10: 6 - * * * . . * Disk 11: 1 - * * * . . . Disk 12: 12 - . * . . . . Solution No. 54: Disk 1: 11 - * * . * . * Disk 2: 9 - * * * * . * Disk 3: 10 - * * * * * * Disk 4: 2 - . . * . * . Disk 5: 4 - . * . . * . Disk 6: 1 - * * * . . . Disk 7: 6 - * * * * . . Disk 8: 3 - * . * * . * Disk 9: 8 - . * . * . * Disk 10: 5 - . * * . . * Disk 11: 7 - . * * . . . Disk 12: 12 - . . * . . . Solution No. 55: Disk 1: 2 - . . * . * . Disk 2: 9 - * * * * * . Disk 3: 10 - * * * * * * Disk 4: 11 - * * . * . * Disk 5: 4 - . . * . . * Disk 6: 7 - . * * . . . Disk 7: 5 - . * * . . * Disk 8: 8 - . * . * . * Disk 9: 3 - * . * * . * Disk 10: 6 - * * * * . . Disk 11: 1 - * * * . . . Disk 12: 12 - . * . . . . Solution No. 56: Disk 1: 11 - * * . * . * Disk 2: 9 - * * * * . * Disk 3: 10 - * * * * * * Disk 4: 2 - . . * . * . Disk 5: 4 - . * . . * . Disk 6: 7 - . * * . . . Disk 7: 1 - * * * . . . Disk 8: 8 - * . * . * . Disk 9: 6 - * * . . * * Disk 10: 3 - . * * . * * Disk 11: 5 - . * * . . * Disk 12: 12 - . . * . . . Solution No. 57: Disk 1: 3 - . * * . * * Disk 2: 9 - . * * * * * Disk 3: 10 - * * * * * * Disk 4: 4 - * . . * . . Disk 5: 1 - * * * . . . Disk 6: 7 - . * * . . . Disk 7: 11 - . * * * . * Disk 8: 5 - * . . * . * Disk 9: 6 - * . . * * * Disk 10: 8 - * . * . * . Disk 11: 2 - . . * . * . Disk 12: 12 - . . * . . . Solution No. 58: Disk 1: 3 - . * * . * * Disk 2: 9 - . * * * * * Disk 3: 10 - * * * * * * Disk 4: 4 - * . . * . . Disk 5: 8 - * . * . * . Disk 6: 2 - . . * . * . Disk 7: 6 - . . * * * * Disk 8: 5 - * . . * . * Disk 9: 11 - * * . * . * Disk 10: 1 - * * * . . . Disk 11: 7 - . * * . . . Disk 12: 12 - . . * . . . Solution No. 59: Disk 1: 4 - . . * . . * Disk 2: 9 - * * * * . * Disk 3: 10 - * * * * * * Disk 4: 3 - * * . * * . Disk 5: 2 - . . * . * . Disk 6: 7 - . * * . . . Disk 7: 1 - * * * . . . Disk 8: 5 - * * . . * . Disk 9: 6 - * . . * * * Disk 10: 11 - . * . * * * Disk 11: 8 - . * . * . * Disk 12: 12 - . * . . . . Solution No. 60: Disk 1: 3 - * * . * * . Disk 2: 9 - * * . * * * Disk 3: 10 - * * * * * * Disk 4: 4 - . . * . . * Disk 5: 1 - * * * . . . Disk 6: 7 - * * . . . . Disk 7: 5 - * * . . * . Disk 8: 2 - . . * . * . Disk 9: 6 - . . * * * * Disk 10: 11 - . * * * . * Disk 11: 8 - . * . * . * Disk 12: 12 - . * . . . . Solution No. 61: Disk 1: 3 - * * . * * . Disk 2: 9 - * * . * * * Disk 3: 10 - * * * * * * Disk 4: 4 - . . * . . * Disk 5: 1 - * * * . . . Disk 6: 7 - * * . . . . Disk 7: 5 - * * . . * . Disk 8: 2 - . . * . * . Disk 9: 6 - . * * * * . Disk 10: 11 - . * * * . * Disk 11: 8 - . * . * . * Disk 12: 12 - . . . . . * Solution No. 62: Disk 1: 11 - * . * . * * Disk 2: 9 - * * * . * * Disk 3: 10 - * * * * * * Disk 4: 2 - . * . * . . Disk 5: 7 - . . . * * . Disk 6: 4 - . * . . * . Disk 7: 5 - . * . * * . Disk 8: 8 - . * . * . * Disk 9: 3 - * . * * . * Disk 10: 6 - * * * . . * Disk 11: 1 - * * * . . . Disk 12: 12 - . * . . . . Solution No. 63: Disk 1: 2 - . * . * . . Disk 2: 9 - * * * * . * Disk 3: 10 - * * * * * * Disk 4: 11 - * . * . * * Disk 5: 4 - . * . . * . Disk 6: 1 - * * * . . . Disk 7: 6 - * * * * . . Disk 8: 3 - * . * * . * Disk 9: 8 - . * . * . * Disk 10: 5 - . * * . . * Disk 11: 7 - . * * . . . Disk 12: 12 - . . * . . . Solution No. 64: Disk 1: 2 - . * . * . . Disk 2: 9 - * * * * . * Disk 3: 10 - * * * * * * Disk 4: 11 - * . * . * * Disk 5: 4 - . * . . * . Disk 6: 7 - . * * . . . Disk 7: 1 - * * * . . . Disk 8: 8 - * . * . * . Disk 9: 6 - * * . . * * Disk 10: 3 - . * * . * * Disk 11: 5 - . * * . . * Disk 12: 12 - . . * . . . Solution No. 65: Disk 1: 3 - * * . * * . Disk 2: 9 - * * . * * * Disk 3: 10 - * * * * * * Disk 4: 4 - . . * . . * Disk 5: 1 - * * * . . . Disk 6: 7 - * * . . . . Disk 7: 11 - * * * . * . Disk 8: 5 - . . * . * * Disk 9: 6 - . . * * * * Disk 10: 8 - . * . * . * Disk 11: 2 - . * . * . . Disk 12: 12 - . * . . . . Solution No. 66: Disk 1: 3 - * * . * * . Disk 2: 9 - * * * * * . Disk 3: 10 - * * * * * * Disk 4: 4 - . . * . . * Disk 5: 8 - . * . * . * Disk 6: 2 - . * . * . . Disk 7: 5 - . * . * * . Disk 8: 7 - . . . . * * Disk 9: 11 - * . * . * * Disk 10: 6 - * * * . . * Disk 11: 1 - * * * . . . Disk 12: 12 - . * . . . . -- Ilan Mayer <ilmayer@aol.com> (December 1996)