Noetic Math Contest Practice Problems (Grades 2-6)

Good Noetic Math Contest practice problems are hard to find. The Noetic Learning Math Contest is one of the biggest elementary contests in the country, yet the organizer sells its past tests ($9 per grade per contest), gives out only a small free sample set by email, and keeps its larger library of sample questions behind a team-leader login. Free practice in the contest’s style is scarce.

This page fixes that with ten original practice problems: five for grades 2-4 and five for grades 5-6, with a full worked solution for each. None of them comes from a real Noetic test. We wrote them to match the contest’s four topic areas and the kind of thinking it rewards.

On contest day your child gets 20 problems in 45 minutes (50 minutes for the online version), with no calculator. For the full picture of how the contest works, see our Noetic Learning Math Contest preparation guide.

What a Noetic problem looks like

Noetic Learning lists four topic areas for the contest:

• Computation and number properties
• Pattern and algebra
• Geometry and measurement
• Probability and statistics

On the online version of the test, most problems ask for a number answer: your child types a number (like 14 or 6.5), and only a few problems are multiple choice. That changes how to practice. On most problems there are no answer choices to eliminate or check against, so the only safety net is your child’s own check at the end.

Twenty problems in 45 minutes works out to a little over two minutes per problem (our arithmetic). Each problem below is short but has one step that punishes rushing, and each solution points out where that step is.

How to use these practice problems

• First pass: no timer. Let your child work each problem on paper, as long as it takes. Speed comes later.
• Write only the number. Practice the habit of a clean final answer, because that is all a free-response box on the online test accepts.
• Check before you look. Ask “how could you check that?” before turning to the solution. Several of these problems have a trap answer that a quick check catches.
• Two or three problems per sitting is plenty for a second or third grader. A fifth or sixth grader can try all five in their band in one go, then time a second attempt a week later at about 2 minutes per problem.

Each problem is labeled with its topic area and a suggested grade. Grade 4 students who find the first set easy should try the grades 5-6 set too.

Grades 2-4: five practice problems

Problem 1: Friendly numbers (Computation, grade 2)

What is 38 + 47 + 62 + 53?

Problem 2: Toothpick squares (Pattern and algebra, grade 3)

Leo builds squares in a row out of toothpicks. Neighboring squares share a side.

• 1 square uses 4 toothpicks.
• 2 squares in a row use 7 toothpicks.
• 3 squares in a row use 10 toothpicks.

How many toothpicks does Leo need to build 10 squares in a row?

Problem 3: Soccer practice (Geometry and measurement, grade 3)

Soccer practice starts at 3:40 PM and ends at 5:15 PM. How many minutes long is practice?

Problem 4: Outfits (Probability and statistics, grade 3)

Sam has 3 T-shirts, 2 pairs of pants, and 2 caps. An outfit is one T-shirt, one pair of pants, and one cap. How many different outfits can Sam make?

Problem 5: Leftover one (Number properties, grade 4)

What is the smallest whole number greater than 1 that leaves a remainder of 1 when it is divided by 2, by 3, by 4, and by 5?

Grades 5-6: five practice problems

Problem 6: Pairs (Computation, grade 5)

What is the value of 100 − 99 + 98 − 97 + 96 − 95 + … + 4 − 3 + 2 − 1?

Problem 7: Three even numbers (Pattern and algebra, grade 5)

The sum of three consecutive even numbers is 126. What is the largest of the three numbers?

Problem 8: Cutting a square (Geometry and measurement, grade 5)

A square has a perimeter of 36 cm. It is cut into two identical rectangles with one straight cut. What is the perimeter of one of the rectangles, in centimeters?

Problem 9: Two dice (Probability and statistics, grade 6)

Two ordinary six-sided dice, one red and one blue, are rolled. There are 36 possible outcomes (6 for the red die times 6 for the blue). In how many of those outcomes do the two numbers add up to 8?

Problem 10: The recipe (Pattern and algebra, grade 6)

A muffin recipe uses flour and sugar in the ratio 5 : 2. Maya makes a big batch and uses 6 more cups of flour than sugar. How many cups of flour does she use?

Solutions: grades 2-4

Problem 1: Answer 200

Adding left to right works, but it’s slow and invites carrying mistakes. Look for pairs that make round numbers instead:

• 38 + 62 = 100
• 47 + 53 = 100

So the total is 100 + 100 = 200.

The habit: before adding a long list, scan it for pairs that end in digits adding to 10 (8 and 2, 7 and 3). Contest computation problems are often built so a smart grouping turns a long sum into a quick one.

Problem 2: Answer 31

The first square needs 4 toothpicks. Every square after that shares one side with the square before it, so it only needs 3 new toothpicks.

• 1 square: 4
• 2 squares: 4 + 3 = 7
• 3 squares: 4 + 3 + 3 = 10

For 10 squares: the first square takes 4, and the other 9 squares take 3 each, so 4 + 9 × 3 = 4 + 27 = 31.

The trap: 10 squares × 4 toothpicks = 40 counts every shared side twice. Having your child build 3 or 4 squares with real toothpicks (or draw them) shows why the pattern grows by 3, not 4.

Problem 3: Answer 95

Break the time into easy jumps instead of subtracting clock times:

• 3:40 to 4:00 is 20 minutes.
• 4:00 to 5:00 is 60 minutes.
• 5:00 to 5:15 is 15 minutes.

Total: 20 + 60 + 15 = 95 minutes.

The trap: subtracting 5:15 − 3:40 like ordinary numbers (515 − 340 = 175) gives nonsense, because an hour has 60 minutes, not 100. A number line with 4:00 and 5:00 marked makes elapsed-time problems easy at this age.

Problem 4: Answer 12

Pick the T-shirt first: 3 choices. For each T-shirt there are 2 choices of pants, which makes 3 × 2 = 6 shirt-and-pants combinations. Each of those 6 can go with either cap, so 6 × 2 = 12 outfits.

If multiplying feels too abstract, list them. Call the shirts A, B, C, the pants 1 and 2, and the caps x and y: A1x, A1y, A2x, A2y, and so on. Each shirt gives 4 outfits, and 3 shirts give 12.

The trap: adding (3 + 2 + 2 = 7) instead of multiplying. Ask your child to list a few outfits; they’ll see quickly that there are more than 7.

Problem 5: Answer 61

If a number leaves remainder 1 when divided by 2, 3, 4, and 5, then one less than the number divides evenly by all four. So look for the smallest number that 2, 3, 4, and 5 all go into.

• Multiples of 4 and 5 together: 20, 40, 60, …
• 20 isn’t divisible by 3. 40 isn’t either. 60 is (60 ÷ 3 = 20), and it’s even.

So 60 is the smallest, and the answer is 60 + 1 = 61.

Check: 61 ÷ 2 = 30 remainder 1; 61 ÷ 3 = 20 r 1; 61 ÷ 4 = 15 r 1; 61 ÷ 5 = 12 r 1. ✓

The trap: multiplying 2 × 3 × 4 × 5 = 120 and answering 121. It works, but it isn’t the smallest, because 4 already contains a 2.

Solutions: grades 5-6

Problem 6: Answer 50

Group the terms in pairs, starting from the left:

(100 − 99) + (98 − 97) + (96 − 95) + … + (2 − 1)

Every pair equals 1. The numbers run from 100 down to 1, which is 100 numbers, so there are 50 pairs. The total is 50 × 1 = 50.

The habit: when a long expression alternates plus and minus, look for a grouping that makes every group the same. Counting the groups correctly (100 numbers make 50 pairs, not 100) is where most mistakes happen.

Problem 7: Answer 44

Three consecutive even numbers are evenly spaced, so the middle one is their average: 126 ÷ 3 = 42. The three numbers are 40, 42, and 44, and the largest is 44.

Check: 40 + 42 + 44 = 126. ✓

The trap: stopping at 42 because that’s the number the division produces. The question asks for the largest. Rereading the question before typing the answer is the cheapest point on the test.

Problem 8: Answer 27

A perimeter of 36 cm means each side of the square is 36 ÷ 4 = 9 cm. Cutting the square in half gives two rectangles, each 9 cm long and 9 ÷ 2 = 4.5 cm wide.

Perimeter of one rectangle: 9 + 4.5 + 9 + 4.5 = 27 cm.

The trap: answering 18, half of 36. Cutting the square doesn’t halve the perimeter, because the cut creates a new side on each rectangle. A quick sketch with the side lengths written on it prevents this.

The final answer is a whole number, but the width along the way is 4.5. The online test accepts decimal answers (like 6.5) when a problem needs one.

Problem 9: Answer 5

List every pair (red, blue) that adds to 8, in order:

A 1 on either die can’t work, because the other die would need a 7. So there are 5 outcomes.

The trap: counting (2, 6) and (6, 2) as the same roll and answering 3. The dice are different colors, so red 2 with blue 6 is a different outcome from red 6 with blue 2. That’s why the problem says there are 36 outcomes in all.

Problem 10: Answer 10

Think of the recipe in parts. Flour is 5 parts and sugar is 2 parts, so flour is 3 parts more than sugar.

Those 3 extra parts are the 6 extra cups, so 1 part = 6 ÷ 3 = 2 cups. Flour is 5 parts: 5 × 2 = 10 cups. (Sugar is 2 × 2 = 4 cups.)

Check: 10 − 4 = 6 more cups of flour, and 10 : 4 simplifies to 5 : 2. ✓

The habit: in ratio problems, find what one part is worth first. A bar drawing (5 boxes over 2 boxes) makes the “3 parts more” visible, and your child doesn’t need any algebra.

Dates and deadlines

As of September 2026. Always confirm on Noetic Learning’s contest details page before registering.

Schools and after-school programs register teams. If your child’s school doesn’t offer the contest, the At-Home edition is the way in; our preparation guide covers its separate dates and cost.

More Noetic Math Contest practice

If these ten problems went well, the next step is a steady diet of problems in the same style. A few options:

• Our Noetic Grades 2-4 guide and Noetic Grades 5-6 guide: strategy chapters, original practice problems, a full practice contest, and kid-friendly step-by-step solutions. Each has a free sample.
• Our free Problem of the Week: one original problem by email every week, with the solution.
• Noetic Learning’s free samples: the contest’s main page offers a small set of official sample problems, emailed to you for free.
• Registration practice tests: Noetic Learning includes three practice tests with a team’s registration, so it’s worth asking your child’s team leader whether they share them.

For the format, the awards, and a four-week prep plan, read the full Noetic Learning Math Contest preparation guide.

This guide is not affiliated with or endorsed by Noetic Learning. “Noetic Learning Math Contest” is a trademark of its respective owners.