🔒 Level 9 Clearance Required

StreamWave
Crisis Protocol

Extension Task · For those who cracked the algorithm early
Harder. Stranger. The CEO needs answers.

Encrypted message from
V. Hartley · CEO, StreamWave
📩
You finished early. Good. That means you're ready for what the others aren't.

Three crises hit the platform today. Our infrastructure team made a rounding error that could cost us millions. Our network expansion is failing geometry checks. And someone needs to verify — algebraically, formally — that our growth model actually holds.

These aren't practice questions. These are the real problems. No hints. No worked examples. Just you, the maths, and what the business needs.
CLASSIFIED · LEVEL 9 · DO NOT DISTRIBUTE · SW-CRISIS-01

StreamWave — Crisis Protocol

Level 9 Extension · Mr. Goddard's Maths · Show all working · Interpret every answer in context

🔒 CLASSIFIED
⚠ Mission parameters
These three tasks cover material at the very top of the IGCSE specification. Show every line of working. The CEO doesn't just want the answer — she wants to see the mathematics that proves it. Each task is independent; tackle them in any order.
X1

The Growth Proof

Algebraic proof · Odd/even integers · Formal mathematical argument

Crisis 1
Maths used
Algebraic proof
Integer expressions
Formal argument
Proof by counter-example

StreamWave's data scientists claim their monthly growth model has a special mathematical property. They need it formally verified before presenting to investors.

The model states that for any integer n, the expression:
n³ − n
represents the net change in users (thousands) over a growth cycle.
(a) Show algebraically that n³ − n is the product of three consecutive integers. Write it in fully factorised form.
(b) Hence prove that n³ − n is always divisible by 6 for any positive integer n. You must show a complete, formal proof — the investors have mathematicians on their board.
(c) A junior analyst claims: "n³ − n is always divisible by 12." Prove or disprove this claim. If you disprove it, a single counter-example is sufficient.
(d) The analyst then claims: "The sum of any three consecutive integers is always divisible by 3." Prove this algebraically. What does this mean in context for StreamWave's 3-month rolling averages?
Working & Answer
X2

The Network Geometry

Vectors · Collinearity · Ratio along a line segment

Crisis 2
Maths used
Column vectors
Proving collinearity
Dividing a line in ratio
Vector paths

StreamWave is planning three server hub locations across a city grid. The infrastructure team modelled them as position vectors from the city centre O.

OA = 2a + b Hub Alpha
OB = 5a − 2b Hub Beta
OC = 8a − 5b Hub Gamma
(a) Find vector AB in terms of a and b.
(b) Find vector AC in terms of a and b.
(c) Prove that A, B and C are collinear. What does this mean for StreamWave's hub placement strategy?
(d) The engineers want to place a signal booster at point M, which divides AC in the ratio 2 : 1. Find the position vector of M in terms of a and b.
(e) A rival company places a hub at point D where OD = 14a − 11b. Is D on the same straight line as A, B and C? Show your reasoning fully.
Working & Answer
X3

The Rounding Crisis

Upper & lower bounds · Compound calculations · Error interval

Crisis 3
Maths used
Upper & lower bounds
Compound measure bounds
Maximum/minimum values
Percentage error

StreamWave's infrastructure team rounded key figures when calculating server bandwidth. The CEO needs to know the worst-case financial exposure from these rounding errors.

Revenue per GB = Total Revenue ÷ Data Transferred
Used to price StreamWave's wholesale data contracts

The team recorded:

Total Revenue = £2 400 000 Rounded to the nearest £100 000
Data Transferred = 3 800 GB Rounded to the nearest 100 GB
(a) Write down the upper and lower bounds for Total Revenue and for Data Transferred.
(b) Calculate the upper bound for Revenue per GB. Show your working clearly, explaining which combination of bounds you are using and why.
(c) Calculate the lower bound for Revenue per GB.
(d) StreamWave signed a contract guaranteeing clients £620 per GB. Using your bounds, determine whether StreamWave can definitely honour this contract, definitely cannot honour it, or whether it is uncertain. Justify your answer fully.
(e) The percentage error in a rounded value is defined as: (bound − rounded value) / rounded value × 100. Calculate the maximum percentage error in the Revenue per GB figure. Give your answer to 3 significant figures.
Working & Answer
📩

Final Message from the CEO

Encrypted · StreamWave Internal

"If you've made it here, you're not a student anymore. You're a data scientist.

The work you've done today keeps 280 million people listening. Present your findings — clearly, confidently, with the maths to back every word.

StreamWave is counting on you."

— V. Hartley · CEO, StreamWave