Tower of Hanoi Trivia Quiz

Answer: Édouard Lucas

Apart from the Tower of Hanoi, Lucas was also known for his work on the famous Fibonacci sequence. Another version of the sequence, namely the Lucas sequence was named after him.

Answer: No

You can only move one disk at a time during the entire game. Moreover, no other disk can be put on the smallest disk. In other words, a smaller disk must be placed on a larger one. If you put a larger disk on top of a smaller one, the "tower" will collapse.

Answer: 7

The game can be represented by the mathematical notation T(3,3). T stands for tower. The first 3 denotes the 3 pegs and the second 3 means there are 3 circular disks.

Now, we can move the 3 circular disks to either peg two or peg three, which are both empty. Suppose that we want to move the disks to the third peg.

Step 1: We make our first move by taking the first disk (the smallest one) from peg one and move it to peg three.

Step 2: Then, we take the middle-sized disk from peg one and move it to peg two. Notice that we cannot place it at peg three since this violates the rule of the game, namely by placing a larger disk on a smaller one.

Step 3: Next, we take the smallest disk from peg three and put it on the top of the middle-sized disk at peg two.

Step 4: Move the largest disk from peg one to the empty peg three.

Step 5: Move the smallest disk from peg two to the empty peg one.

Step 6: Move the middle-sized disk from peg two to peg three.

Step 7: Move the smallest disk from peg one to peg three.

Answer: 2m + 1

Of course, a similar Tower of Hanoi with an additional disk will require more steps.

Let's look into a specific case where initially you have 3 pegs and 3 disks. You need a total of 7 steps to complete the game.

Now, if there are 3 pegs and 3 + 1 = 4 disks, how many moves are needed?

To solve this puzzle with 4 disks, we can use or manipulate the result that we obtain from the previous puzzle with 3 disks.

We argue this by first moving the top 3 disks from peg one to peg three. Hence, a total of m moves are needed. After this m steps, the largest disk will remain unmoved at peg 1. We move this peg to peg two, which is empty. Next, we will move back all the 3 disks from peg three back to peg two. Again, a total of m steps are needed.

So, in total, you will need m + 1 + m = (2m + 1) steps to solve the puzzle.

Answer: (2^n) - 1

The general formula is number of steps = (2^n) - 1.

We can verify this easily. Taking n = 1, (2^1) - 1 = 1 step.

When n = 2, (2^2) - 1 = 3 steps.

When n = 3, (2^3) - 1 = 7 steps.

When n = 4, (2^4) - 1 = 15 steps.

Now let's see how the equation (2^n) - 1 is obtained.

We start from the recursive equation an = 2a(n-1) + 1, where an means the nth term and a(n-1) means the (n-1)th term. The initial condition is a1 = 1.

We have an = 2a(n-1) + 1. We also have a(n-1) = 2a(n-2) + 1.

We arrive at an = 2[2a(n-2)+1] + 1 = 2^2 x a(n-2) + 2 + 1 = 2^2 x [2a(n-3) + 1] + 2 + 1 = 2^3 x a(n-3) + 2^2 + 2 + 1.

In general, an = 2^(n-1) x a1 + 2^(n-2) + ... + 2 + 1.

The initial condition is given as a1 = 1.

So, an = 2^(n-1) x 1 + 2^(n-2) + ... + 2 + 1 = 2^(n-1) + 2^(n-2) + ... + 2 + 1.

Now, an = 2^(n-1) + 2^(n-2) + ... + 2 + 1. This is a geometric series.

By using the summation formula of the geometric series, we arrive at an = [(2^n) - 1]/[2-1] = (2^n) - 1.

Answer: The smallest disk

One of the most useful strategies to solve the puzzle is that your moves should always alternate between the smallest disk and the remaining disk. In the Tower of Hanoi with 3 pegs and 10 disks, a total of (2^10) - 1 = 1023 moves are needed. Out of these 1023 moves, 1023/2 +0.5 = 511.5 + 0.5 = 512 moves are performed on the smallest disk.

Answer: Tower A with 4 pegs and 3 disks.

You need (2^3) - 1 = 7 steps to solve the puzzle with 3 pegs and 3 disks.

On the other hand, you need (2^4) - 1 = 15 steps to solve the puzzle with 3 pegs and 4 disks.

If more empty pegs are given, the number of steps required to complete the puzzle will decrease. This is because more empty pegs are available for you to put your disks on.

You only need 5 steps to solve the puzzle with 4 pegs and 3 disks.

You need 9 steps to complete the game with 4 pegs and 4 disks.

Answer: Recurrence relation

Given a Tower of Hanoi game with 3 pegs and 1 disk. The solution is obvious and simple - only 1 move is required to solve the puzzle.

For a game with 3 pegs and 2 disks, 3 moves are needed.

For a game with 3 pegs and 3 disks, 7 moves are required.

We can show that for the puzzles with 3 pegs and 4 disks, 15 moves are needed.

Now, the sequence is 1, 3, 7, 15.

Let the first term, a1 = 1. Also, a2 = 3, a3 = 7 and a4 = 15.

Notice that each term of the sequence can be defined as a function of its previous term.

Now, our job is to find the function and we can use the useful recurrence relation to do this.

The equation that we have at the end is an = 2a(n-1) + 1. Here, an means the nth term and a(n-1) means the (n-1)th term. The initial condition is a1 = 1.

So, by using the formula, we can easily compute a2 = 2a1 + 1 = 2(1) + 1 = 3. Also, a3 = 2a2 + 1 = 2(3) + 1 = 7 and a4 = 2a3 + 1 = 2(7) + 1 = 15.

Answer: All of these

The Tower of Hanoi is a widely-used psychological test to examine one's problem solving skills.

It is also used in the data backup rotation scheme to minimize the time taken and improve efficiency.

The recursive property of the solution of the Tower of Hanoi is studied and applied in computer programming and algorithms. A well-written source code or programming language takes less time when it executes a task or program.

Answer: Vietnam

The 48th International Mathematical Olympiad (IMO) was held in Hanoi, Vietnam from 19 to 31 July, 2007. A total of 93 countries participated in this competition. The first place went to the Russian team. The China team emerged as the first runner-up while the second runner-up went to the Vietnam team.