The Shelmikedmu are an elusive and nomadic tribe whose members are unusually heterogeneous in respect of hair and eye color, and skull shape. A persistent anthropologist establishes the following facts:

1. 75% have dark hair, the rest have fair hair
2. 80% have brown eyes, the rest have blue eyes
3. No narrow-headed person has fair hair and blue eyes
4. The proportion of blue-eyed broad-headed tribespeople is the same as the proportion of blue-eyed narrow-headed tribespeople
5. Those who are blue-eyed and broad-headed are fair-haired or dark-haired in equal proportion
6. Half the tribe is dark-haired and broad-headed
7. The proportion who are browned-eyed, fair-haired, and broad-headed is equal to the proportion who are brown-eyed, dark-haired, and narrow-headed

What is the proportion of the tribe who are narrow-headed ?

(Stop scrolling down if you want to give this a shot :))

Solution:

The challenge is how to represent these various facts in a form that can be manipulated via simple arithmetic. First, observe that the entire sample space consists of only three underlying attributes: Hair color, Eye color and Head shape, each with only 2 possible values, giving a total of 2^3 possible outcomes:

Hair: {Dark, Fair}
Eye: {Brown, Blue}
Head: {Narrow, Board}

If we use A, B, and C to represent Hair, Eye and Head, and {A1, A2}, {B1, B2}, {C1, C2} to represent the respective values, this would give:

A: {A1, A2}
B: {B1, B2}
C: {C1, C2}

The 8 possible outcomes are:

A1,B1,C1
A1,B1,C2
A1,B2,C1
A1,B2,C2
A2,B1,C1
A2,B1,C2
A2,B2,C1
A2,B2,C2

Suppose the total number of tribespeople is 100. Let's go through the given facts above from most specific to least specific:

• "No narrow-headed person has fair hair and blue eyes" => |{A2,B2,C1}| = 0
• "The proportion who are browned-eyed, fair-haired, and broad-headed is equal to the proportion who are brown-eyed, dark-haired, and narrow-headed" => x = |{A2,B1,C2}| = |{A1,B1,C1}|
• "Those who are blue-eyed and broad-headed are fair-haired or dark-haired in equal proportion" => y = |{A2,B2,C2}| = |A1,B2,C2}|

This gives us:

          Size
          ====
A1,B1,C1    x
A1,B1,C2    
A1,B2,C1
A1,B2,C2    y
A2,B1,C1
A2,B1,C2    x
A2,B2,C1    0
A2,B2,C2    y

Continuing,

• "The proportion of blue-eyed broad-headed tribespeople is the same as the proportion of blue-eyed narrow-headed tribespeople" => |{*,B2,C2}| = |{*,B2,C1}| => |{A1,B2,C1}| = 2y

Now let z = |{A1,B1,C2}|, we get:

          Size
          ====
A1,B1,C1    x
A1,B1,C2    z
A1,B2,C1   2y
A1,B2,C2    y
A2,B1,C1
A2,B1,C2    x
A2,B2,C1    0
A2,B2,C2    y

Given the assumed total population is 100, |{A2,B1,C1}| = 100 - 2x - 4y - z, or:

          Size
          ====
A1,B1,C1    x
A1,B1,C2    z
A1,B2,C1   2y
A1,B2,C2    y
A2,B1,C1    100 - 2x - 4y - z
A2,B1,C2    x
A2,B2,C1    0
A2,B2,C2    y

Moving on,

• "75% have dark hair, the rest have fair hair" => 75 = x + z + 2y + y = x + z + 3y
• "80% have brown eyes, the rest have blue eyes" => 20% have blue eyes => 20 = 2y + y + y = 4y, so y = 5
• "Half the tribe is dark-haired and broad-headed" => 50 = z + y, so z = 45

This means:

• 75 = x + z + 3y = x + 45 + 15, so x = 15
• Number of narrow-headed => x + 2y + (100 - 2x - 4y - z) = 15 + 10 + (100 - 30 - 20 - 45) = 30

The answer is therefore 30% :)