Free worked solutions · 2026 Edition

IB Maths HL Paper 3 Chapter 18: Final strategy and checklists worked solutions

Worked solutions to the practice questions in Chapter 18 of Photon Guidebook: Mathematics HL Paper 3 for the IB Diploma (2026 Edition). Each solution shows every step with the mark it earns, the way an IB mark scheme does.

Chapter 18 of 18
2 practice questions
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Drills

Short drills from the chapter, with full worked solutions.

Question 1

[4 marks]

Drill 1 (AA): Paths on a grid. Read only what is printed here, then write a four-line plan. This question asks you to count shortest paths on a grid, including the paths that stay on one side of a line.

(a) Count the shortest paths from (0,0) to (2,2) moving only right or up.

[2]

…

(f) Prove that the number of shortest paths from (0,0) to (n,n) that never pass above the line y=x is \dfrac{1}{n+1}\dbinom{2n}{n}.

[7]

Four parts are hidden between (a) and (f). Plan, one mark per line: the tools, where the marks are, what the printed results give you, a check.

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Model plan. Tools: small cases, a table, binomial coefficients, a recurrence or a reflection argument. Marks: the 7-mark proof is the largest block. Printed results: a closed form or a recurrence in an earlier part, which the proof builds on. Check: n=1,2,3 give 1,2,5 restricted paths, against \binom{2n}{n}=2,6,20 in total. (1 mark per line.)

Final answer

Plan: small cases, table, \binom{2n}{n}, recurrence or reflection; check 1, 2, 5.

Question 2

[4 marks]

Drill 2 (AI): Ice on a pond. Read only what is printed here, then write a four-line plan. This question asks you to model the thickness of ice on a pond during a cold spell.

(a) Plot the data of ice thickness against time and describe the trend.

[2]

…

(g) Use your model to decide whether the ice will be thick enough to walk on after 10 days, and comment on the reliability of your answer.

[4]

Five parts are hidden between (a) and (g). Plan, one mark per line: the tools, where the marks are, what the printed results give you, a check.

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Model plan. Tools: scatter plot, a model that slows as the ice thickens (square root or logarithm), linearise, regression. Marks: the prediction and the criticism are worth 4 together. Printed results: any parameter values given in a middle part. Check: the fitted curve against the first and last data points. (1 mark per line.)

Final answer

Plan: \sqrt t or \ln model, linearise, regress, predict, criticise the extrapolation.

From the book

Photon Guidebook: Mathematics HL Paper 3 for the IB Diploma

Chapter 18 teaches final strategy and checklists: how the IB builds this kind of Paper 3 question, the techniques, marked worked investigations, and practice investigations like these.

About the book