Cambridge IGCSE International Mathematics 0607/52 is the May/June 2026 Paper 5 (Investigation, Core), variant 2. It lasts 1 hour 15 minutes, is worth 40 marks and is sat by every Core candidate, alongside Papers 1 and 3. The whole paper was one investigation into the game of Nim, split into seven questions, and the first carried 12 marks. This guide goes through the paper question by question and shows what the official mark scheme gave marks for, including the communication marks. The paper and its mark scheme are free on the Tutopiya Learning Portal.

For an overview of all six 0607 papers from this series, see IGCSE International Maths 0607 May/June 2026 past papers: what came up and how it was marked.

Paper 6 (Extended) used the same Nim investigation for its Section A, so the 0607 Paper 62 guide is extra practice on the same ideas.

0607/52 at a glance

In the game, two players take turns removing sticks from a pile, and the player who takes the last stick wins. Questions 1 to 4 have your friend moving first; Questions 5 to 7 have you moving first.

Question	Topic	Marks
1	Taking 1 or 2 sticks: game tables for 3, 6, 9 and 18 sticks, multiples of 3	12
2	Total number of moves to win, working forwards and backwards	4
3	Taking 1, 2 or 3 sticks: why 4 sticks wins, moves for 8 sticks, multiples of 4	5
4	Taking up to k sticks: sequences and nth terms, the values of k for 30 sticks	9
5	You start, taking 1 or 2: first moves and starting piles	4
6	You start, taking 1, 2 or 3 and first taking 1: the sequence and its nth term	4
7	You start, taking 1 to 4 and first taking 3: the nth term	2
	Total	40

How 0607/52 is marked

The cover says marks are given for full reasons, examples and steps that communicate your mathematics clearly. These are the C marks (communication marks), shown as C1 or C2 in the mark scheme. There were 7 of them on this paper, in Questions 2(a), 2(b), 3(b), 4(b) and 6(a). A correct final answer with no supporting work loses them.

The other marks are answer marks and B marks (marks for a correct answer or statement, given independently). The scheme also uses oe (or equivalent: another correct form is accepted), SC (special case: a mark for a specific partial answer named in the scheme), soi (seen or implied) and BOD (benefit of the doubt). This paper’s scheme has no M (method) or A (accuracy) marks: the method credit comes through the C marks.

Question-by-question walkthrough

Question 1: taking 1 or 2 sticks (12 marks)

Your friend moves first and each player takes 1 or 2 sticks.

• 3 sticks, friend takes 2 (1 mark): 1 stick is left and you take 1, leaving 0.
• 6 sticks, two ways to win (2 marks, B1 for each table): friend 1, you 2, friend 1, you 2 (piles 5, 3, 2, 0); or friend 2, you 1, friend 2, you 1 (piles 4, 3, 1, 0). The idea is that you always leave 3 after your move.
• 9 sticks (2 marks): friend 1 (8 left), you 2 (6), friend 2 (4), you 1 (3), friend 2 (1), you 1 (0). B1 for moves 1 to 3 and B1 for moves 4 to 6.
• 3, 6 and 9 are multiples of 3 (1 mark): “times tables” or “divisible by 3” were accepted. “A sequence in 3s” and “factors” were not.
• The next two terms (1 mark): 12 and 15. Extra numbers had to be correct.
• Why you win from 18 on your friend’s move (1 mark): 18 is a multiple of 3. “In the 3 times table”, “3 is a factor of 18” and 3 × 6 = 18 were all accepted.
• Sticks to take from 20 on your move (1 mark): 2, which leaves 18.
• The 18-stick game table (3 marks): after each of your moves the pile is 15, 12, 9, 6, 3 and 0. Your moves are 2, 1, 1, 2, 1, 2, each one taking the pile back to a multiple of 3. B1 for the piles 15, 12, 6 and 3; B1 for your moves 2, 1, 1 that lead to 9; B1 for your moves 2, 1, 2 that lead to the win.

Question 2: total number of moves (4 marks)

A table showed the total moves for you to win from 3, 6, 9 and 18 sticks: 2, 4, 6 and 12. The total moves are two-thirds of the number of sticks, because each pair of moves (your friend’s and yours) removes exactly 3.

• Moves from 24 sticks (2 marks): 16 (1 mark), plus C1 for showing how: two differences of 2, continuing the sequence 8, 10, 12, 14, 16, multiplying 24 by ⅔ or dividing by 1.5, or 24 − (24 ÷ 3). Writing 24 = 3 × 8 and then 24 − 8 = 16 did not earn the C mark.
• Sticks for 40 moves (2 marks): 60 (1 mark), plus C1 for 40 ÷ 2 × 3 or an equivalent, which may be in stages (40 ÷ 2 = 20 rounds, 20 × 3 = 60), or for continuing the sequence of sticks from 24 up to 60.

Question 3: taking 1, 2 or 3 sticks (5 marks)

• Why 4 sticks always wins (1 mark): whatever your friend takes, you take the rest. Listing the three cases (friend 1, you 3; friend 2, you 2; friend 3, you 1) was accepted.
• Total moves from 8 sticks (3 marks): 4 (1 mark), with C2 for a fully correct table of exactly four rows in which your first move leaves 4 sticks, for example friend 1 (7 left), you 3 (4), friend 1 (3), you 3 (0). C1 was available for at least 2 correct rows out of 4, or for a less efficient game that still ends with you winning: four rows in which your first move doesn’t leave 4, or a six-row game. A game in which your friend could have won did not count.
• Two possible starting piles above 12 (1 mark): any two multiples of 4 greater than 12, such as 16 and 20.

Question 4: taking up to k sticks (9 marks)

• The sequences and nth terms (5 marks): taking 1 or 2 gives 3, 6, 9, 12 and 3n; taking 1 to 3 gives 4, 8, 12, 16 and 4n; taking 1 to 4 gives 5, 10, 15, 20 and 5n; taking 1 to 5 gives 6, 12, 18, 24 and 6n; taking up to k gives (k + 1)n. The marks: B1 for the sequences in the first three rows, B1 for 6, 12, 18, 24, B2 for the first four nth terms (B1 for one of them) and B1 for (k + 1)n. kn + n was accepted, but k + 1 × n without brackets was not.
• The seven values of k for 30 sticks (4 marks): 1, 2, 4, 5, 9, 14 and 29. You win after your friend starts if 30 is a multiple of k + 1, so k + 1 is a factor of 30. The answer earned 2 marks (one value could be missing; B1 for four correct; SC1 for 1 ≤ k ≤ 29 if nothing else scored). The working earned C2: listing the factors of 30 (1, 2, 3, 5, 6, 10, 15, 30), or writing four correct equations such as 2n = 30 or 5(k + 1) = 30, or n = 30 ÷ (k + 1) with two examples. C1 was for four correct factors or one correct equation solved. k + 1 = 30 did not count as an equation.

Question 5: you start, taking 1 or 2 (4 marks)

• 5 sticks (1 mark): you take 2 (3 left), friend takes 2 (1 left), you take 1. The scheme also accepted you 1, friend 2, you 2.
• First move from 4 sticks (1 mark): take 1, leaving 3.
• Why take 1 from 19 (1 mark): it leaves 18, a multiple of 3. “18 is a multiple of 6” was also accepted, as was showing the piles after each of your turns.
• Starting piles if your first move leaves 12 (1 mark): 13 and 14.

Question 6: you start, taking 1, 2 or 3, first taking 1 (4 marks)

• The missing terms (2 marks): 1, 5, 9, 13, 17, 21, 25, 29 (1 mark), plus C1 for showing a difference of 4, or for listing the multiples of 4 from 4 to 28.
• The nth term (2 marks): 4n − 3. B1 for kn − 3 with k ≠ 0, or for 4n + j. Writing Tn = 4n − 3 was accepted, but n = 4n − 3 scored only B1.

Question 7: you start, taking 1 to 4, first taking 3 (2 marks)

• The nth term: 5n − 2. Taking 3 must leave a multiple of 5, so the starting piles are 3, 8, 13 and so on. B1 for kn − 2 with k ≠ 0, or for 5n + j; n = 5n − 2 again scored only B1.

Where the marks are on 0607/52

• Sequences and nth terms carried 11 marks: the table in 4(a) (5), Question 6 (4) and Question 7 (2).
• Playing out games in tables carried 12 marks: 1(a), 1(b), 1(c), 1(e), 3(b) and 5(a). Most of the marks are for the piles you leave and the number you take.
• Short reasons and multiples carried 9 marks: 1(d)(i) to (iv), 3(a), 3(c) and 5(b) to 5(d). Each is one mark for a clear reason, usually that the pile left is a multiple of k + 1.
• Counting moves carried 4 marks: Question 2.
• Factors of 30 carried 4 marks: Question 4(b), the hardest part of the paper.
• 7 of the 40 marks were C marks, so writing the reasoning down matters.

How to self-mark 0607/52 with the mark scheme

1. Separate answer marks from C marks. For 2(a), 2(b), 3(b), 4(b) and 6(a), give yourself the answer mark first, then check whether your working matches one of the listed C-mark methods.
2. Check every pile in your tables, not just the last row. In 1(e), the scheme gives B1 for each block of three of your moves, so one wrong pile costs a mark.
3. Count rows in 3(b). C2 needs exactly four rows with your first move leaving 4. A longer correct game scores C1 at most.
4. Look at the nth term form. 4n − 3 and Tn = 4n − 3 score 2; n = 4n − 3 scores 1. In 4(a), (k + 1)n needs the brackets.
5. Mark lists strictly. In 4(b), the 2 marks need the values of k with no extras (one missing value is condoned), and C2 needs the factors of 30 with no extras. The partial marks are kinder: B1 for four correct values ignores extras, and C1 allows up to four extra factors.
6. Log each lost mark as a wrong table entry, a missing reason, an nth-term slip or missing working, and practise in that order.

Sit 0607/52 as a timed mock

The paper allows 75 minutes for 40 marks, a little under two minutes per mark. A workable plan:

Block	Marks	Time
Question 1	12	20 minutes
Question 2	4	7 minutes
Question 3	5	9 minutes
Question 4	9	17 minutes
Question 5	4	6 minutes
Question 6	4	6 minutes
Question 7	2	4 minutes
Check tables and write any missing reasons	–	6 minutes

Each question builds on the one before, so keep your earlier answers in view. Questions 4(b) and 7 tell you to use earlier parts.

Other May/June 2026 0607 papers

• 0607 Paper 12 (Core Non-calculator)
• 0607 Paper 22 (Extended Non-calculator)
• 0607 Paper 32 (Core Calculator)
• 0607 Paper 42 (Extended Calculator)
• 0607 Paper 62 (Extended Investigation and Modelling)

Practise and download

• Download 0607/52 and its mark scheme on the Learning Portal →
• All 0607 May/June 2026 papers: overview and marking patterns
• Try a free IGCSE International Mathematics quiz

Need help with a 0607 topic? Book a free trial lesson with an IGCSE International Mathematics tutor →

Frequently asked questions

What was the 0607 Paper 52 investigation in May/June 2026? +

The game of Nim. Two players take turns removing sticks from a pile and whoever takes the last stick wins. The seven questions looked at which starting numbers let you win, how many moves a win takes, and the nth terms of the winning numbers when players can take up to k sticks.

How many marks is 0607/52 and how long is it? +

It is out of 40 marks and lasts 1 hour 15 minutes. Question 1, playing out games with 3, 6, 9 and 18 sticks, is the biggest at 12 marks.

What were the values of k in 0607/52 Question 4(b)? +

1, 2, 4, 5, 9, 14 and 29. You win after your friend starts when the pile is a multiple of k + 1, so k + 1 must be a factor of 30 (other than 1). The values earned 2 marks and listing the factors of 30 earned 2 communication marks.

What are the C marks on 0607 Paper 5? +

Communication marks. The cover says marks are given for full reasons, examples and steps. On 0607/52 there were 7 of them, for example C1 for showing differences of 2 in Question 2(a) and C2 for a correct four-row game table in Question 3(b).

Is 0607/52 the same investigation as 0607/62? +

Both used Nim, and several parts overlap: the 6-stick and 18-stick tables, the general nth term (k + 1)n, the values of k for 30 sticks and the reason for taking 1 stick from 19. Paper 6 is the Extended version and adds a modelling section on regular investments.

Where can I download 0607/52 May/June 2026 and its mark scheme? +

Both are free to download and print on the Tutopiya Learning Portal, on the Cambridge IGCSE International Mathematics 0607 past papers page.

Ready to Excel in Your Studies?

Get personalised help from Tutopiya's expert tutors. Whether it's IGCSE, IB, A-Levels, or any other curriculum — we match you with the perfect tutor and your first session is free.

Book Your Free Trial