In a public bathroom there are taps numbered 1, 2, 3, ..., n. Tap 1 and Tap 2 each take the same time to fill a tank when they run alone. Tap 3 takes half the time taken by Tap 2, Tap 4 takes half the time taken by Tap 3, and in general, every subsequent tap takes half the time taken by the previous tap. If Tap 10 takes exactly 2 hours to fill the tank when it runs alone, what is the ratio of the efficiency of Tap 8 to that of Tap 12?

Introduction / Context:
This puzzle style question is about relative efficiency of taps that fill a tank. The times taken by successive taps form a geometric progression, and we are given the filling time for the tenth tap. The objective is to compare the efficiencies of the eighth and twelfth taps, where efficiency means the rate of filling the tank.

Given Data / Assumptions:

• Tap 1 and Tap 2 each take the same time to fill the tank.
• Tap 3 takes half the time taken by Tap 2.
• Tap 4 takes half the time taken by Tap 3, and so on.
• Each Kth tap takes half the time taken by the (K - 1)th tap.
• Tap 10 alone takes 2 hours to fill the tank.
• Efficiency of a tap is defined as work rate, equal to 1 divided by the time taken to fill the tank.

Concept / Approach:
If time decreases by a factor of 2 from one tap to the next, then going forward through the taps halves the time, while going backward doubles the time. First we find the times for Tap 8 and Tap 12 by stepping back or forward from Tap 10. Then we convert these times into rates and form the ratio of efficiencies (rates) of Tap 8 to Tap 12.

Step-by-Step Solution:
Tap 10 takes 2 hours. Working backward: Tap 9 takes twice the time of Tap 10, so Tap 9 takes 2 * 2 = 4 hours. Tap 8 takes twice the time of Tap 9, so Tap 8 takes 2 * 4 = 8 hours. Working forward: Tap 11 takes half the time of Tap 10, so Tap 11 takes 2 / 2 = 1 hour. Tap 12 takes half the time of Tap 11, so Tap 12 takes 1 / 2 = 0.5 hours. Efficiency is rate = 1 / time. Efficiency of Tap 8 = 1 / 8. Efficiency of Tap 12 = 1 / 0.5 = 2. Ratio of efficiency of Tap 8 to Tap 12 = (1 / 8) : 2 = 1 : 16.

Verification / Alternative check:
We can express all times relative to Tap 10. Since each step multiplies time by 2 or 1/2, Tap 8 is two steps back, so time factor is 2^2 = 4, giving 2 * 4 = 8 hours. Tap 12 is two steps forward, so factor is (1 / 2)^2 = 1 / 4, giving 2 * 1 / 4 = 0.5 hours. The ratio of times is 8 : 0.5 = 16 : 1, so the ratio of efficiencies is the inverse, 1 : 16. This matches our earlier result.

Why Other Options Are Wrong:
4:1 and 16:1 both put Tap 8 as more efficient than Tap 12, which is impossible since Tap 12 fills the tank much faster. 5:3 does not follow from the geometric halving pattern of times. 8:1 also implies Tap 8 is faster than Tap 12 and is inconsistent with the computed times.

Common Pitfalls:
The main trap is going in the wrong direction with the halving rule, for example dividing times when moving to earlier taps instead of multiplying by 2. Another mistake is to compare times directly instead of rates. Remember that efficiency is inversely proportional to time, so the ratio of efficiencies is the inverse of the ratio of times.

Final Answer:
The ratio of the efficiency of Tap 8 to Tap 12 is 1 : 16.