Information about the heights of six friends A, B, C, D, E, and F: - The heights of A and F...

From constraint 3, D = A + 1. Since heights are {4, 5, 6}, if A = 4 then D = 5, or if A = 5 then D = 6 (A cannot be 6, as D would be 7). From constraint 4, if there are n shortest people, there must be n+2 tallest people. The only viable distribution is: 2 people at shortest height, 2 people at middle height, and 4 people at tallest height—but we only have 6 people total. Testing A = 5, D = 6 gives: 2 at height 4 (shortest), 2 at height 5 (middle), 2 at height 6 (tallest), satisfying the constraint that tallest (2) = shortest (2) + 2 fails. Re-examining: with 2 shortest and 4 tallest, we need 2 at height 4 and 4 at height 6. This means A = F = 4, D = 5... but then D is not tallest. Actually, A = F = 5, D = 6 works: with C = 4 (forced by constraint 2), B, E ≥ 4, we get exactly 1 person at 4 feet, and 3 at 6 feet doesn't satisfy the constraint either. The correct solution is A = F = 5, D = 6, C = 4, and {B, E} chosen so that 2 are at shortest and 4 at tallest—yielding F = 5 feet.

Step-by-step Derivation:
Let's systematically work through the constraints:

Constraint 3 Analysis: D = A + 1

• If A = 4, then D = 5
• If A = 5, then D = 6
• A cannot be 6 (D would exceed maximum of 6)

Constraint 4 Analysis: Let n_short = count at minimum height, n_tall = count at maximum height.

• We need: n_tall = n_short + 2
• Possible distributions with 6 people:
  • 1 shortest, 3 tallest, 2 middle ✓
  • 2 shortest, 4 tallest ✗ (exceeds 6 people)

Case 1: A = 4, D = 5

• Heights used: 4, 5, 6
• Minimum possible height: 4 (A and F are at 4)
• From constraint 1: F = 4
• From constraint 2: B ≥ C and E ≥ C
• If C = 4: n_short = 3 (A, F, C), then n_tall should = 5 ✗ (impossible)
• If C = 5: B, E ≥ 5, so {B, E} ⊆ {5, 6}
  • Minimum height is still 4 (A, F), so n_short = 2
  • Need n_tall = 4, so 4 people at height 6: D must be 6 ✗ (contradicts D = 5)

Case 2: A = 5, D = 6

• From constraint 1: F = 5
• From constraint 2: B ≥ C and E ≥ C
• If C = 4: minimum height = 4, n_short = 1 (only C), need n_tall = 3
  • Tallest is 6: need 3 people at 6
  • We have D = 6, need 2 more from {B, E} at 6
  • But we have 6 people: A(5), F(5), C(4), D(6), B(6), E(6) → 1 at 4, 2 at 5, 3 at 6 ✓
  • Verify constraint 2: B = 6 ≥ C = 4 ✓, E = 6 ≥ C = 4 ✓
  • This works! F = 5 feet

Verification:

• A = 5, B = 6, C = 4, D = 6, E = 6, F = 5
• A = F? Yes (5 = 5) ✓
• B ≥ C? Yes (6 ≥ 4) ✓
• E ≥ C? Yes (6 ≥ 4) ✓
• D = A + 1? Yes (6 = 5 + 1) ✓
• n_tall = 3 (at 6 feet), n_short = 1 (at 4 feet), 3 = 1 + 2 ✓