 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ib/paper-3/19h/d/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Orient every edge of the feasible convex polytope in the direction of increasing objective. Write

O = 000, A = (1/2, 1, 1), B = (1, 1/2, 1), C = (1, 1, 1/2) .

The possible improving simplex method paths from

O

A

are

O \to 010 \to 011 \to A, O \to 100 \to 110 \to C \to A, O \to 100 \to 110 \to C \to B \to A, O \to 100 \to 101 \to B \to A, O \to 001 \to 011 \to A, O \to 010 \to 110 \to C \to A, O \to 010 \to 110 \to C \to B \to A, O \to 001 \to 101 \to B \to A .

These exhaust the directed edge graph: the ordinary cube edges remain except those incident to the removed vertex

111

; each such edge ends at one of

A, B, C

; and the cutting face contributes the triangle with edges

A B, A C, BC

The first two paths use three pivots, while the two paths through both

C

and

B

use five. Therefore the smallest and largest possible numbers of simplex steps are

3 and 5,

and the total number of distinct outcomes is

8 .

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