Free practice
Continental Math League Practice Problems
Ten original practice problems for every grade from 2 to 8, two complete practice meets in the real CML format, and links to CML's official sample tests, with a worked solution for every question. Free to read, no sign-up needed.
CML full prep and practice guides: 2 3 4 5 6
Free practice by grade: 2 3 4 5 6 7 8
Printable PDF: a free 10-problem set for each grade, 2 to 8, with every answer explained.
On this page · 10 sections
Free CML practice problems by grade
CML writes a separate paper for each grade, so each grade has its own free page: ten original problems, ramped from warm-ups to two Challenges, then every answer explained with the most common wrong answer. From grade 4 up, the sets are pitched at the harder Pythagorean division or a notch above it. No sign-up to read them, and each set also comes as a printable PDF.
Grade 2
A made-up symbol, letters for digits, a calendar count and a coin purse.
10 practice problems →
Grade 3
A number machine, a balance puzzle, a letter sum and an age problem.
10 practice problems →
Grade 4
Pythagorean level: timetables, hidden digits, a cut-up square and runners on a track.
10 practice problems →
Grade 5
Pythagorean level: remainders, a shared gift, a bus timetable and counting ways.
10 practice problems →
Grade 6
Above Pythagorean level: factor counts, clock hands, triangle areas and remainders.
10 practice problems →
Grade 7
Above Pythagorean level: powers, polygon angles, dice probability and circles.
10 practice problems →
Grade 8
Above Pythagorean level: modular arithmetic, rates, a paper fold and probability.
10 practice problems →
Official CML sample tests (PDF)
CML posts one free sample test for each grade and division on its 2026–27 registration pages. Each one is a real past meet: six questions on the first page and CML's solutions on the second. Grades 2 and 3 each have their own sample. From grade 4, there is one for the Euclidean division and one for the harder Pythagorean division (which division fits your child?).
| Grade 2 | Sample test |
| Grade 3 | Sample test |
| Grade 4 | Euclidean · Pythagorean |
| Grade 5 | Euclidean · Pythagorean |
| Grade 6 | Euclidean · Pythagorean |
| Grade 7 | Euclidean · Pythagorean |
| Grade 8 | Euclidean · Pythagorean |
One sample per grade goes fast. Have your child sit it under the 30-minute timer to see the real level, then move on to the ten problems for their grade above.
How a CML meet works
Practice is far less stressful when the format isn't a surprise. Here's the shape of a real meet:
- 6 problems per meet.
- 30 minutes, about 5 minutes a problem.
- Written answers, not multiple choice.
- 3 or 5 meets a season (3 for grades 2–3).
CML 2026–27 meet dates
As of October 2026, from cmleague.com. Schools may give a meet on the nearest possible day.
| Meet | Date (a Thursday) | Grades 2–3 | Grades 4–9 |
|---|---|---|---|
| Meet 1 | Nov 5, 2026 | — | ✓ |
| Meet 2 | Dec 3, 2026 | — | ✓ |
| Meet 3 | Jan 7, 2027 | ✓ | ✓ |
| Meet 4 | Feb 4, 2027 | ✓ | ✓ |
| Meet 5 | Mar 4, 2027 | ✓ | ✓ |
Schools register teams: $90 per team for grades 2–3, $100 for a first team at grades 4–9. A child whose school doesn't take part can enter as a Home School student for $25.
Read the full parent's guide: scoring, fees and how to enter →
How to use these
Set a timer for 30 minutes and let your child work through all six problems on paper, writing each answer down plainly. Then go through the solutions together. That second part is where the learning actually happens.
Practice meet: Grades 2–3
CML writes a separate paper for grade 2 and grade 3; this practice meet is pitched to suit both. 6 problems, 30 minutes. Write your answer for each.
Problem 1
How much larger is (8 + 5 + 7) than (30 − 4 − 9)?
Problem 1
Answer: 3
(8 + 5 + 7) = 20. (30 − 4 − 9) = 17. The difference is 20 − 17 = 3. Work out each bracket completely before you compare them — that is where most points get lost.
Problem 2
The symbol ★ means: double the first number, then add the second. For example, 4 ★ 3 = (4 × 2) + 3 = 11. What is 7 ★ 5?
Problem 2
Answer: 19
Follow the rule exactly as written: (7 × 2) + 5 = 14 + 5 = 19. A made-up symbol looks alarming, but it is only ever a set of instructions. Read them twice, then do exactly what they say.
Problem 3
Ben has 6 coins that are all nickels and dimes. Together they are worth 45¢. How many dimes does he have?
Problem 3
Answer: 3
If all 6 coins were nickels, that would be 30¢ — 15¢ short. Swapping a nickel for a dime adds 5¢ each time, so you need 15 ÷ 5 = 3 swaps. That gives 3 dimes and 3 nickels: 30 + 15 = 45¢. ✓
Problem 4
Mrs. Diaz lines up 12 chairs in a row. She puts a red cushion on every 2nd chair and a blue cushion on every 3rd chair. How many chairs get both cushions?
Problem 4
Answer: 2
A chair needs to be both a 2nd chair and a 3rd chair, so it has to be a 6th chair. In a row of 12, that is chair 6 and chair 12 — 2 chairs.
Problem 5
There are 9 birds on a fence. Some fly away, then 4 more land. Now there are 8 birds. How many flew away?
Problem 5
Answer: 5
Start with 9 and end with 8, having gained 4 along the way. Without the 4 landing, there would be 8 − 4 = 4 birds left. So 9 − 4 = 5 flew away.
Problem 6
In the addition below, A and B stand for two different digits. What is A + B?
| A5 | |
| + | 2B |
| 71 |
Problem 6
Answer: 10
The ones column gives 5 + B ending in 1, so B = 6 and 1 carries. The tens column is then A + 2 + 1 = 7, so A = 4. Check: 45 + 26 = 71. ✓ A + B = 4 + 6 = 10.
Practice meet: Grades 4–6
From grade 4 up CML splits into Euclidean and Pythagorean divisions. This meet sits around the Euclidean level. 6 problems, 30 minutes. Write your answer for each.
Problem 1
What is the smallest whole number greater than 1 that leaves a remainder of 1 when divided by 2, by 3, and by 4?
Problem 1
Answer: 13
A number leaving remainder 1 each time is 1 more than a common multiple of 2, 3 and 4. The smallest number divisible by all three is 12, so the answer is 12 + 1 = 13. Check: 13 ÷ 2, ÷ 3 and ÷ 4 all leave 1. ✓
Problem 2
For any two numbers, a ⊗ b means (a × b) − (a + b). What is 6 ⊗ 4?
Problem 2
Answer: 14
(6 × 4) − (6 + 4) = 24 − 10 = 14. Do each bracket first, then subtract. Rushing straight to a single line is how this one goes wrong.
Problem 3
Nina has 9 coins worth $1.20 altogether. She has only nickels, dimes, and quarters, and at least one of each. How many nickels does she have?
Problem 3
Answer: 3
Let n, d, q be the counts. Then n + d + q = 9 and 5n + 10d + 25q = 120. Divide the second by 5: n + 2d + 5q = 24. Subtract the first: d + 4q = 15. Now test quarters. q = 1 gives d = 11 — too many coins. q = 2 gives d = 7, leaving 0 nickels, which breaks "at least one of each". q = 3 gives d = 3 and n = 3. ✓ Check: 15 + 30 + 75 = 120¢ in 9 coins.
Problem 4
A rectangle is 3 times as long as it is wide. Its perimeter is 64 cm. What is its area, in square centimeters?
Problem 4
Answer: 192
Half the perimeter is one length plus one width: 32 cm. If the width is w, the length is 3w, so 4w = 32 and w = 8. The length is 24. Area = 8 × 24 = 192 cm². Check the perimeter: 2 × (8 + 24) = 64. ✓
Problem 5
Tom thinks of a number. He doubles it, subtracts 7, then multiplies the result by 3. He ends up with 39. What number did he think of?
Problem 5
Answer: 10
Undo each step backwards. Before multiplying by 3: 39 ÷ 3 = 13. Before subtracting 7: 13 + 7 = 20. Before doubling: 20 ÷ 2 = 10. Check forwards: 10 → 20 → 13 → 39. ✓
Problem 6
In a class of 30 students, 18 play soccer and 15 play basketball. Every student plays at least one of the two sports. How many play both?
Problem 6
Answer: 3
Adding 18 and 15 gives 33, which is 3 more than the 30 students in the class. Those 3 extra are the students counted twice — the ones who play both. Check: 15 play soccer only, 12 basketball only, 3 both, and 15 + 12 + 3 = 30. ✓
Ready for a full season of practice?
Our per-grade CML prep guides for Grade 2, Grade 3, Grade 4, Grade 5 and Grade 6 are each written for that grade's meet: 14–16 chapters, 140–160 original write-in problems, six full practice meets, and every answer explained.
See the CML Grade 2 guide ($20.00) →
See the CML Grade 3 guide ($20.00) →
See the CML Grade 4 guide ($20.00) →
See the CML Grade 5 guide ($20.00) →
See the CML Grade 6 guide ($20.00) →
CML practice: common questions
Are there free Continental Math League practice problems?
Yes. Every grade from 2 to 8 has its own free page of ten original practice problems, and this page has two full practice meets. All come with step-by-step solutions and are free to read with no sign-up. Every problem is original, written in the style and difficulty of a real CML meet. Each grade's set is also a free printable PDF by email.
Where can I find CML sample tests or past papers?
CML posts one free sample test for each grade and division on its registration pages at cmleague.com (the "Download Sample Test" link). Each is a real past meet: six questions on one page and the solutions on the next. The links are collected on this page. Beyond that, every grade from 2 to 8 has ten free original problems here, also as a printable PDF.
How many questions are on a CML math meet, and what is the time limit?
Every elementary CML math meet has 6 questions with a 30-minute time limit, about five minutes per problem. CML is not multiple choice: students write each answer in an answer column on the paper.
How many CML meets are there in a season?
It depends on the grade. Grades 2 and 3 have 3 meets, running January to March. Grades 4 through 9 have 5 meets, starting in November. Scores are cumulative across the season, so an early meet never decides anything on its own.
What grade levels does Continental Math League cover?
CML runs math contests from grade 2 up through grade 9, plus a Calculus league. Grades 2 and 3 each have their own paper, with no divisions. From grade 4 upward, schools choose between the Euclidean and Pythagorean divisions.
Can my child do CML if their school does not take part?
Yes. CML has a Home School option, which covers any student whose school does not participate, not only children educated at home. The fee is reduced to $25 per child per league, and you register an account first, buy nothing, then email support@cmleague.com to have the pricing applied.
Keep exploring
- CML parent's guide
- Which competition is right for your child?
- All elementary math competitions
- Free Problem of the Week
All problems on this page are original, written in the style and difficulty of CML meets. This guide is not affiliated with or endorsed by Continental Mathematics League. “Continental Mathematics League” is a trademark of its respective owners.