Ask The Wizard #410

While there are nine available starting positions for dropping the puck, we only need to focus on five since the layout of the prizes and pegs is symmetrical. Let's number these positions, beginning with 1 at the leftmost edge and going to 5.

Additionally, there are nine potential ending spots which we will also number from 1 to 5, starting from the left.

The table below indicates the likelihood of landing in each location (shown in the left column) based on the starting position (indicated in the top row).

Landing Spot	Drop 1 or 9	Drop 2 or 8	Drop 3 or 7	Drop 4 or 6	Drop 5
1 or 9	0.225586	0.193359	0.121094	0.056641	0.032227
2 or 8	0.386719	0.346680	0.250000	0.153320	0.113281
3 or 7	0.242188	0.250000	0.257813	0.250000	0.242188
4 or 6	0.113281	0.153320	0.250000	0.346680	0.386719
5	0.032227	0.056641	0.121094	0.193359	0.225586
Total	1.000000	1.000000	1.000000	1.000000	1.000000

The subsequent table presents the expected values for each puck drop position during the Big Money Week you mentioned. As anticipated, players should aim to drop the puck directly over the highest prize.

Drop Location	Expected Win
1 or 9	$7,550.29
2 or 8	$12,396.48
3 or 7	$25,173.83
4 or 6	$39,478.52
5	$45,852.05

Start by examining the curve represented by the equation y=1/x for values of x ranging from 1 to infinity. The graph for these values, particularly up to 17, will be illustrated as follows.

Now, envision this curve being rotated around the x-axis. This rotation results in a shape resembling a horn that extends infinitely at one end, a phenomenon mathematicians refer to as Gabriel's Horn.

Image source: Soul of Mathematics .

Let's take a moment to reflect on the inquiries surrounding Gabriel's Horn:

• What is the volume?
• What is the surface area?

Here are the answers:

You can find my work detailing these solutions here (PDF).

The essence of the paradox lies in questioning how it is possible for the volume to be finite while the surface area remains infinite.

This apparent contradiction arises from confusing dimensions: we must avoid equating two-dimensional and three-dimensional concepts. For example, think about Circle A with a radius of 1 and Circle B with a radius of 1.1, where Circle A is enclosed within Circle B.

Circle A has a circumference of 2*pi == 6.283185 and circle B has an area of of 1.1^2 * pi = 3.801327. Note that circle B is larger than circle A, yet it's area is less than the circumference of B. It is not that the circumference of B is part of the area of A. The area and circumference are different measurements in different dimensions and shouldn't be compared.

Another perspective is if we were to create a record from Circle A, incorporating an infinite number of grooves, the total length of those grooves would be infinite.

Returning to the concept of Gabriel's Horn, consider a simplified version involving circles that have radii of 1, 1/2, 1/3, 1/4, and continuing in this manner.

The cumulative volume of these circles can be expressed as pi times the sum of (1/1 + 1/4 + 1/9 + 1/16 + ...). There is a famous infinite series that states:

1/1 + 1/4 + 1/9 + 1/16 + ... = pi^2/6 = 1.644934.

Adding another pi term, the sum of the areas of the circles is pi^3/6 = 5.167713.

Meanwhile, the total length of the circumferences translates to 2*pi times the sum of (1/1 + 1/2 + 1/3 + 1/4 + ...).

This series is also a well-renowned infinite series in mathematics, known as the Harmonic Series, which states:

1/1 + 1/2 + 1/3 + 1/4 + .. = ∞ = infinity.

2*pi*∞ still equals ∞.

By extending this reasoning to infinitely small intervals between the circles, we construct Gabriel's Horn.