Exams¶

Format¶

The exam is five hours (08:00–13:00), no aids allowed, and consists of nine problems worth four points each, split into three parts of three:

Part	Problems	Character
A	1–3	Standard techniques. Bonus points are added here.
B	4–6	Multi-step, several parts per problem.
C	7–9	Proof-heavy; primarily for the higher grades.

Bonus points from the course are added to your Part A result, but Part A caps at 12 points — bonus points cannot carry you past it.

Grade boundaries¶

Grade	A	B	C	D	E	Fx
Total credit	27	24	21	18	16	15
of which from Part C	6	3	–	–	–	–

The Part C requirement is the part people miss. An A needs 6 points from Part C and a B needs 3, no matter how well Parts A and B went. Scoring 27 with nothing from Part C is a C, not an A. If you are aiming high, budget real time for two Part C problems rather than polishing Part A.

Presentation is graded¶

Every paper carries this instruction, and it is not boilerplate:

For full marks on a problem, the solution must be well-presented and easy to follow. This means in particular that all introduced variables are explained, that the logical structure is clearly presented in words or symbols, and that the reasoning is well-motivated and clearly written. Solutions that seriously fall short in these respects are awarded at most two points.

So a correct answer, written badly, scores 2/4. Concretely:

Define your notation. If you write \(G_1 \cong G_2\), say what the map is.
Say which theorem you are invoking, by name, before you use it. "By Lagrange's theorem, \(\lvert H \rvert\) divides \(\lvert G \rvert\)" costs you four words and buys the mark.
Mark the induction hypothesis where it is used. See Induction.
Answer the question asked. "Does it have a Hamiltonian cycle?" wants yes/no and the cycle, or yes/no and an argument.

What gets examined¶

Across the recent papers, the pattern is stable. Part A is nearly always a diophantine equation, a graph, and RSA:

Problem	May 2024	May 2023
1	Diophantine \(15x + 17y = 20\)	Euler path & isomorphism
2	Planarity, Euler's formula, Euler/Hamilton	Diophantine \(23x + 31y = 1000\), non-negative solutions
3	RSA with \(p=5\), \(q=7\)	Relation on divisors of 100
4	Counting telephone numbers	—
5	Linear codes, distance, decoding	—
6	Subgroups of \(S_6\), Lagrange	—
7	Induction: \(19 \mid 3^{3n-2} + 2^{3n+1}\)	—
8	Countability of a set of decimals	—
9	Homomorphisms \(S_n \to \mathbb{Z}_2\)	—

Part C leans on the same three things year after year: induction, countability, and a structural group-theory proof. Those are the pages to reread last.

Worked solutions to the problems above are on the topic pages:

Past papers¶

Available under course-pdfs/ in this repository:

Paper	Solutions
May 29, 2024 (`june_2024.pdf`)	Included in the same file
August 2024 (`tentamen_Augusti_2024.pdf`)	`August_2024_solutions.pdf`
May 31, 2023 (`SF1610 Tentamen 230531.pdf`)	—
August 16, 2023 (`SF1610 Tentamen 230816.pdf`)	—
May 27, 2025	`2025_05_27_facit.pdf`
August retakes (`aug14`, `aug20A`, `aug20B`, `aug21`)	`…s.pdf` variants

Solution sheets contain errors

At least one is confirmed: the May 2024 RSA solution mislabels the exponent and a power of two, though its final answer is correct — see the note on that page. Check the arithmetic yourself rather than assuming a disagreement means you are wrong.

Practice material¶

Seminars 1–4 (Seminar*.pdf) — the discussion problems, bilingual Swedish/English.
Diskussionsuppgifter Sem 1–5 (SF1610 Diskussionsuppgifter…) — the 2023 versions.
Extra exercises (Extra exercises.pdf, Extra extra exercises.pdf, extra_exercises_groups-1.pdf) — drill sets, heaviest on induction, modular arithmetic, and groups.
Lectures 1–20 (Lecture N.pdf) — handwritten lecture notes.