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Several Bounds for the K-Tower of Hanoi Puzzle

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Abstract

We consider special cases of a modified version of the Tower of Hanoi puzzle and demonstrate how to find upper bounds on the minimum number of moves that it takes to complete these cases.

Figures (30)

Figure 1: Upper Bound for k = (n—3). Whenever n = 4, the disk labelled n—2 coincides with the disk labelled 2. Next, we flip the n*® and (n—1)* largest disks onto the second peg, which saves 1 move, getting us to the midpoint. Then we move the largest disk onto the third peg. Next, we reverse the steps prior to moving the largest disk, which saves 1 more move and completes the puzzle. See Figure HI
Figure 1: Upper Bound for k = (n—3). Whenever n = 4, the disk labelled n—2 coincides with the disk labelled 2. Next, we flip the n*® and (n—1)* largest disks onto the second peg, which saves 1 move, getting us to the midpoint. Then we move the largest disk onto the third peg. Next, we reverse the steps prior to moving the largest disk, which saves 1 more move and completes the puzzle. See Figure HI
Figure 2: Building a Stack of 3 Disks
Figure 2: Building a Stack of 3 Disks
Figure 3: Building a Stack of 4 Disks. Whenever n = 7, the disk labelled n — 4 coincides with the disk labelled 3.
Figure 3: Building a Stack of 4 Disks. Whenever n = 7, the disk labelled n — 4 coincides with the disk labelled 3.
Figure 4: Overview of Building Stacks of 4, 5, ---, (n — 3) Disks. Notice that the pegs for the final configuration are fixed and are labelled accordingly.
Figure 4: Overview of Building Stacks of 4, 5, ---, (n — 3) Disks. Notice that the pegs for the final configuration are fixed and are labelled accordingly.
Then we flip the n** and (n — 1)* largest disks onto the third peg. This fli saves 1 more move and completes the stack of (n — 2). See Figure Figure 5: Overview of Building a Stack of (n — 2) Disks
Then we flip the n** and (n — 1)* largest disks onto the third peg. This fli saves 1 more move and completes the stack of (n — 2). See Figure Figure 5: Overview of Building a Stack of (n — 2) Disks
Figure 7: Overview of Building a Stack of n Disks Thus, to build this stack it takes
Figure 7: Overview of Building a Stack of n Disks Thus, to build this stack it takes
Figure 8: Building a Stack of 3 Disks
Figure 8: Building a Stack of 3 Disks
Figure 9: Building a Stack of 4 Disks Next, we build a stack of 4 in the same manner that we built a stack of 4 in the bound for k = (n — 4). This takes 6 moves, as before. See Figure
Figure 9: Building a Stack of 4 Disks Next, we build a stack of 4 in the same manner that we built a stack of 4 in the bound for k = (n — 4). This takes 6 moves, as before. See Figure
Figure 10: Building a Stack of 5 Disks building) from the intermediate peg onto the destination peg. We then flip 3 disks from the source peg onto the destination peg, which saves 4 more moves. We have now completed the stack of 5 disks and we have saved 8 moves. See Figure
Figure 10: Building a Stack of 5 Disks building) from the intermediate peg onto the destination peg. We then flip 3 disks from the source peg onto the destination peg, which saves 4 more moves. We have now completed the stack of 5 disks and we have saved 8 moves. See Figure
Figure 11: Overview of Building Stacks of 5,6,--- ,(m — 4) Disks. Wheneve n = 9, the disk labelled n — 6 coincides with the disk labelled 3.
Figure 11: Overview of Building Stacks of 5,6,--- ,(m — 4) Disks. Wheneve n = 9, the disk labelled n — 6 coincides with the disk labelled 3.
Figure 12: Overview of Building a Stack of (n — 3) Disks. Whenever n = 9, the disk labelled n — 6 coincides with the disk labelled 3. Then we flip the n*® and (n—1)* largest disks onto the second peg, which saves 1 more move.See Figure [J]
Figure 12: Overview of Building a Stack of (n — 3) Disks. Whenever n = 9, the disk labelled n — 6 coincides with the disk labelled 3. Then we flip the n*® and (n—1)* largest disks onto the second peg, which saves 1 more move.See Figure [J]
Figure 13: Overview of Building a Stack of (n — 2) Disks. Whenever n = 9, the disk labelled n — 6 coincides with the disk labelled 3. Recreational Mathematics Magazine, Number 7, pp. 79-118
Figure 13: Overview of Building a Stack of (n — 2) Disks. Whenever n = 9, the disk labelled n — 6 coincides with the disk labelled 3. Recreational Mathematics Magazine, Number 7, pp. 79-118
We then flip the n™, (n — 1)** and (n — 2)"¢ largest disks onto the first peg and save 4 more moves. Then we move the 3" largest disk from the third peg onto the second peg.
We then flip the n™, (n — 1)** and (n — 2)"¢ largest disks onto the first peg and save 4 more moves. Then we move the 3" largest disk from the third peg onto the second peg.
This concludes the first part of this step. Now we begin the second part of this step.
This concludes the first part of this step. Now we begin the second part of this step.
Next, we make a similar sequence of moves.
Next, we make a similar sequence of moves.
Next, we flip the n*® largest and the (n — 1)** largest disks onto the first peg, which saves 1 more move. See Figure Figure 19: Moves made for 4 < j < (n— 4) whenever n is even. The disks on the second peg are uniquely determined by the disks depicted on the other two pegs.
Next, we flip the n*® largest and the (n — 1)** largest disks onto the first peg, which saves 1 more move. See Figure Figure 19: Moves made for 4 < j < (n— 4) whenever n is even. The disks on the second peg are uniquely determined by the disks depicted on the other two pegs.
Figure 20: Moves made for 4 < 7 < (n — 4) whenever n is odd. The disks on the second peg are uniquely determined by the disks depicted on the other two pegs.
Figure 20: Moves made for 4 < 7 < (n — 4) whenever n is odd. The disks on the second peg are uniquely determined by the disks depicted on the other two pegs.
Figure 23: Overview of Building a Stack of n Disks moves to build this stack. See Figure Therefore, it takes
Figure 23: Overview of Building a Stack of n Disks moves to build this stack. See Figure Therefore, it takes
Figure 24: Overview of Building a Stack of 4 Disks moves to build a stack of 4 disks. See Figure 24 We repeat this same process to build the next stack of 5 disks. We begin by moving the (n — 4)*® largest disk from the source peg onto the destination peg. We then flip all disks, except the (n — 3)"4 largest disk, from the intermediate peg onto the source peg. This takes 3 moves. We then move the (n—3)*¢ largest disk onto the destination peg. This takes 1 move. Next, we flip the same disks that were flipped onto the source peg, onto the destination peg. This takes another 3 moves, and gives us a stack of 5. Thus, it takes
Figure 24: Overview of Building a Stack of 4 Disks moves to build a stack of 4 disks. See Figure 24 We repeat this same process to build the next stack of 5 disks. We begin by moving the (n — 4)*® largest disk from the source peg onto the destination peg. We then flip all disks, except the (n — 3)"4 largest disk, from the intermediate peg onto the source peg. This takes 3 moves. We then move the (n—3)*¢ largest disk onto the destination peg. This takes 1 move. Next, we flip the same disks that were flipped onto the source peg, onto the destination peg. This takes another 3 moves, and gives us a stack of 5. Thus, it takes
Figure 25: Overview of Building a Stack of 5 Disks
Figure 25: Overview of Building a Stack of 5 Disks
Figure 26: Overview of Building a Stack of x Disks. Stacks of an even number of disks will be on the intermediate peg and stacks of an odd number of disks will be on the destination peg. The final stack of (n — 4) disks will be on the third peg. In general, to build a stack of x disks, for 4 < « < (n — 4), it takes 1 move to relocate the largest disk of the stack we are building, (a — 2) moves to flip disks, 1 move to put the 24 largest disk of the stack we are building onto the largest disk of the stack we are building, and (x — 2) more moves to flip the remaining disks onto the stack we are building. That is, it takes
Figure 26: Overview of Building a Stack of x Disks. Stacks of an even number of disks will be on the intermediate peg and stacks of an odd number of disks will be on the destination peg. The final stack of (n — 4) disks will be on the third peg. In general, to build a stack of x disks, for 4 < « < (n — 4), it takes 1 move to relocate the largest disk of the stack we are building, (a — 2) moves to flip disks, 1 move to put the 24 largest disk of the stack we are building onto the largest disk of the stack we are building, and (x — 2) more moves to flip the remaining disks onto the stack we are building. That is, it takes
Figure 27: Overview of Building a Stack of (n — 3) Disks. Notice that we make a counterintuitive move by taking out the 7'* largest disk. Doing this will save many moves in the long term. After building a stack of (n — 4) disks, we reach the configuration where the the second peg. 1 largest disk from we move the 6** move. We then fl [his takes 1 move. Next, we flip the top (n — 5) disks argest disk from the first peg onto the second peg. T onto the second peg, giving us a stack of (n — 3) disks. See Figure 27. first peg contains the 4'?, 3°4, 2°¢, and largest disks; the second peg contains no disks; and the third peg contains the remainder of the disks. We now need to build a stack of (n — 3) disks, and we start by moving the 4” largest d isk onto from the third peg onto the first peg. This takes (n — 5) moves. Then we move the 5‘ the third peg onto the second peg. This takes 1 move. Next, his takes | move. Then we move the 7‘ largest disk onto the third peg. This takes 1 ip (n — 7) disks from the first peg onto the second peg, which takes (n — 7) moves. Lastly, we move the 7‘ largest disk from the t hird peg
Figure 27: Overview of Building a Stack of (n — 3) Disks. Notice that we make a counterintuitive move by taking out the 7'* largest disk. Doing this will save many moves in the long term. After building a stack of (n — 4) disks, we reach the configuration where the the second peg. 1 largest disk from we move the 6** move. We then fl [his takes 1 move. Next, we flip the top (n — 5) disks argest disk from the first peg onto the second peg. T onto the second peg, giving us a stack of (n — 3) disks. See Figure 27. first peg contains the 4'?, 3°4, 2°¢, and largest disks; the second peg contains no disks; and the third peg contains the remainder of the disks. We now need to build a stack of (n — 3) disks, and we start by moving the 4” largest d isk onto from the third peg onto the first peg. This takes (n — 5) moves. Then we move the 5‘ the third peg onto the second peg. This takes 1 move. Next, his takes | move. Then we move the 7‘ largest disk onto the third peg. This takes 1 ip (n — 7) disks from the first peg onto the second peg, which takes (n — 7) moves. Lastly, we move the 7‘ largest disk from the t hird peg
Figure 28: Overview of Building a Stack of (n — 2) Disks. Whenever n = 9, the disk labelled n coincides with the disk labelled 9.
Figure 28: Overview of Building a Stack of (n — 2) Disks. Whenever n = 9, the disk labelled n coincides with the disk labelled 9.
ove the 8°" largest disk from the third peg onto the first peg. This takes 1 move. Move the 6" largest disk from the second peg onto the third peg. This takes 1 move. Flip (n — 7) disks from the first peg onto the second peg. This takes (n — 7) moves. Move the 7 largest disk from the first peg onto the third peg. This takes 1 move. Flip (n — 7) disks from the second peg onto the third peg. This takes (n —7) moves. Move the 4*” largest disk from the first peg onto the second peg. This takes 1 move. Flip (n — 4) disks from the third peg onto the second peg. This takes (n — 4) moves. See Figure BI 14+2+1+(n-8)+1+1+(n-7)+1+4 (n-7)+1+(n-7)+141+4 (n-6)+1+4 (n-7)+14 (n-7)+1 +(n-7)+2+1+4+1+1+(n-7)+1+4+1+1+4+14 (n-8)+141+4 (n-7)41-+ (n-7)+14 (n-4)=13n-63
ove the 8°" largest disk from the third peg onto the first peg. This takes 1 move. Move the 6" largest disk from the second peg onto the third peg. This takes 1 move. Flip (n — 7) disks from the first peg onto the second peg. This takes (n — 7) moves. Move the 7 largest disk from the first peg onto the third peg. This takes 1 move. Flip (n — 7) disks from the second peg onto the third peg. This takes (n —7) moves. Move the 4*” largest disk from the first peg onto the second peg. This takes 1 move. Flip (n — 4) disks from the third peg onto the second peg. This takes (n — 4) moves. See Figure BI 14+2+1+(n-8)+1+1+(n-7)+1+4 (n-7)+1+(n-7)+141+4 (n-6)+1+4 (n-7)+14 (n-7)+1 +(n-7)+2+1+4+1+1+(n-7)+1+4+1+1+4+14 (n-8)+141+4 (n-7)41-+ (n-7)+14 (n-4)=13n-63

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