The two most populous cities and the non-urban region (NUR) of each of three states, Whimshire, Fogglia, and Humbleset, are assigned Pollution Measures (PMs).

These nine PMs are all distinct multiples of 10, ranging from 10 to 90. The six cities in increasing order of their PMs are: Blusterburg, Noodleton, Splutterville, Quackford, Mumpypore, Zingaloo.

The Pollution Index (PI) of a state is a weighted average of the PMs of its NUR and cities, with a weight of 50% for the NUR, and 25% each for its two cities.

There is only one pair of an NUR and a city (considering all cities and all NURs) where the PM of the NUR is greater than that of the city. That NUR and the city both belong to Humbleset.
The PIs of all three states are distinct integers, with Humbleset and Fogglia having the highest and the lowest PI respectively.

What is the PI of Humbleset?

The correct answer is 50.

To find the Pollution Index (PI) of Humbleset, we first need to determine the distribution of the 9 Pollution Measures (PMs) among the 3 Non-Urban Regions (NURs) and the 6 cities. The 9 available PMs are given as distinct multiples of 10 ranging from 10 to 90: 10, 20, 30, 40, 50, 60, 70, 80, and 90.

The problem states that there is exactly one pair of an NUR and a city across all states where the NUR's PM is strictly greater than the city's PM. Let's logically order the cities by their PMs from lowest to highest: $C_1<C_2<C_3<C_4<C_5<C_6$.

If an NUR had a PM greater than $C_2$, it would necessarily also be greater than $C_1$, which would create at least two instances where an NUR is greater than a city. Since the problem explicitly states there is only one such instance, the highest PM among the three NURs must be greater than exactly one city ($C_1$) and less than all the rest. The remaining two NURs must therefore have PMs lower than all cities.

This strict ordering dictates that the two lowest overall PMs must belong to NURs, the third lowest to the city $C_1$, and the fourth lowest to the highest NUR. The remaining 5 highest PMs will naturally belong to the other cities. Assigning the values from our set of PMs gives us:

• NUR 1 = 10
• NUR 2 = 20
• City $C_1$ = 30
• NUR 3 = 40

The single pair where the NUR's PM is greater than the city's PM is 40 > 30. The prompt states this specific NUR and city both belong to Humbleset. Therefore, Humbleset has an NUR with a PM of 40 and one of its cities has a PM of 30.

Next, we calculate the PI of Humbleset. The PI formula is a weighted average: 50% for the NUR and 25% for each of the two cities. Let the unknown second city in Humbleset be $C_h$.

$$\mathrm{PI}=\frac{2\times \text{NUR}+{\text{City}}_1+{\text{City}}_2}{4}$$

Substituting the known values for Humbleset into the equation:

$$\mathrm{PI}=\frac{2\times 40+30+C_h}{4}=\frac{110+C_h}{4}$$

The remaining unassigned cities have PMs of 50, 60, 70, 80, and 90. The problem states that the PIs of all three states are distinct integers. Thus, $110+C_h$ must be evenly divisible by 4. Let's test the available values for $C_h$:

• If $C_h=50$, PI = 160 / 4 = 40
• If $C_h=70$, PI = 180 / 4 = 45
• If $C_h=90$, PI = 200 / 4 = 50

We must check these scenarios against the remaining two states, which have NURs of 10 and 20. Their PIs must also evaluate to distinct integers. For a state with an NUR of 10, the numerator $2\times 10+\text{Sum of Cities}$ requires the sum of its two cities to be a multiple of 20 for the final PI division to yield an integer.

Testing Humbleset PI = 50 (where $C_h=90$):
The remaining cities to distribute are 50, 60, 70, and 80. We can pair them as (50, 70) and (60, 80) since both pairs sum to multiples of 20 (120 and 140, respectively).
- State 1 (NUR=10, Cities=50, 70): PI = (20 + 120) / 4 = 35.
- State 2 (NUR=20, Cities=60, 80): PI = (40 + 140) / 4 = 45.
This results in the PIs 35, 45, and 50. All are distinct integers, with Humbleset having the highest (50) and Fogglia having the lowest (35). This fully satisfies all given conditions.

If we briefly test the other possibilities for Humbleset (PI = 40 or 45), trying to pair the remaining cities to yield integer PIs for the other two states inevitably results in duplicate PIs between two states, violating the rule that all PIs must be distinct.

Therefore, the PI of Humbleset is definitively 50.