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10 Math Kangaroo Practice Problems (Grades 5-6)
Ten original Math Kangaroo-style practice problems for grades 5-6, from 3-point warm-ups to the hardest 5-point kind, with every answer explained.
Math Kangaroo full prep and practice guides: 1–2 3–4 5–6 7–8
Free practice, other grades: 1–2 3–4 7–8
Other contests, grades 5–6: Noetic 5 Noetic 6 MOEMS Div E MOEMS Div M CML 5 CML 6 AMC 8
Printable PDF: all 10 problems with room to work, then every answer explained.
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Ten original Math Kangaroo practice problems for grades 5–6, each with a worked solution and the most common wrong answer, also as a printable PDF. Problems 1–3 are like the 3-point questions, 4–7 like the 4-point ones and 8–10 like the hardest 5-point ones: try each on your own before you look at the answers.
The real Levels 5–6 paper has 30 multiple-choice questions in 75 minutes, five answer choices each: ten 3-point, ten 4-point and ten 5-point questions, so the top score is 120, and a wrong answer costs nothing. Here your child writes each answer in instead of picking from five, so there is nothing to guess from. Math Kangaroo is held on the third Thursday in March; see the Math Kangaroo 2027 date and registration timeline for fees and how to sign up.
For levels and scoring in full, see our Math Kangaroo format guide. For the official sample questions and past papers, see where to find Math Kangaroo past papers.
The 10 problems
Problem 1
A die starts on the square marked start, with 1 on top, 2 facing you and 3 on the right. It rolls over its edges along the path, one square at a time: right, away from you, right, away, right. Which number is on top when the die reaches the star?
Fact: On a die, opposite faces add up to 7: 1 and 6, 2 and 5, 3 and 4.
Problem 2
The numbers 1, 2, 3, … are written in a table with 4 columns. They snake back and forth, as shown. Which number is written directly below 70?
Problem 3
A digital clock shows hours and minutes, from 00:00 to 23:59. At 20:47 its four digits add up to 2 + 0 + 4 + 7 = 13. How many times in one day do the four digits add up to 23?
Problem 4
Leo lies on Tuesdays, Wednesdays and Thursdays and tells the truth on all other days. Mia lies on Saturdays, Sundays and Mondays and tells the truth on all other days. One day Leo says, “I lied yesterday,” and Mia says, “I will lie tomorrow.” What day of the week is it?
Tip: Test each day of the week. Is the sentence true that day? Is it one of the speaker’s truth days?
Problem 5
How many squares of any size can you find in this shape? Count only squares whose four sides are lines of the drawing.
Problem 6
Put the numbers 2, 3, 4, 5, 6 and 7 in the six circles, one number in each, so that the three numbers along each side of the triangle have the same sum. What is the largest this sum can be?
Problem 7
On one side of a long street, the houses are numbered 1, 3, 5, 7, 9, … with no number skipped. The numbers of six houses standing next to each other add up to 216. What is the largest of these six numbers?
Problem 8
Two lines across and two lines down cut a big rectangle into nine smaller rectangles. The areas of five of them are shown, in cm². The picture is not drawn to scale. What is the area of the big rectangle?
Tip: Rectangles in the same row have the same height, so their areas compare like their widths.
Problem 9
★ Challenge. A building made of equal cubes stands on a 3 × 3 square base. Every cube sits on the base or on top of another cube. The pictures show the building seen from the front and from the right. Find the smallest and the largest number of cubes the building can have. What is their sum?
Problem 10
★ Challenge. Two squares are cut out of a 6 × 6 board, as shown. Dominoes are placed on the board. Each domino covers exactly two squares that share a side, and no two dominoes overlap. What is the largest number of dominoes that can be placed?
Solutions
Problem 1
Problem 1: Answer 5
Roll right: the left face comes up, and it is 7 − 3 = 4. Roll away: the front face comes up. Tops: 1, 4, 2, 6, 3, 5.
Why it works: keep track of three faces, top, front and right, and use “opposite faces add to 7” for the hidden ones. Rolling right, the top goes to the right side and the left face (7 − right) comes up. Rolling away, the top goes to the back and the front comes up, while the old bottom (7 − top) turns to face you. As (top, front, right): (1, 2, 3), (4, 2, 1), (2, 3, 1), (6, 3, 2), (3, 1, 2), (5, 1, 3). Check: the three faces you track never include two that add to 7, because they always meet at one corner. Common wrong answer: 2 (rolling the wrong way each time, so the face on the far side comes up), or 1 (guessing that the rolls cancel out).
Problem 2
Problem 2: Answer 75
4 × 17 = 68, so row 18 holds 69 to 72. Even rows run right to left, so 70 is in the third column. Row 19 runs left to right: 73, 74, 75.
Why it works: two steps: find the row (each row holds 4 numbers), then use the direction of the row. Odd rows run left to right and even rows right to left, so row 18 reads 72, 71, 70, 69 and row 19 reads 73, 74, 75, 76. Check with the picture: the 6 is in an even row and the third column, and below it is 11, which is 6 + 5. In the same place, 70 + 5 = 75. Common wrong answer: 74 (putting 70 in the second column, as if every row ran left to right), or 71 (the next number, not the one below).
Problem 3
Problem 3: Answer 4 times
The biggest sum is 19:59, which makes 24. For 23, take 1 away from the hours or from the minutes: 19:49, 19:58, 09:59 and 18:59. That is 4 times.
Why it works: the hours and minutes add up separately. Hour digits add to at most 10 (only 19), then 9 (09 and 18). Minute digits add to at most 14 (only 59), then 13 (49 and 58). So 23 is 10 + 13 (2 times) or 9 + 14 (2 times). Check: an hour sum of 8 would need minutes adding to 15, which no minute has. Common wrong answer: 2 (finding only 19:49 and 19:58, or only 09:59 and 18:59), or 1 (stopping at 19:59, which adds to 24).
Problem 4
Problem 4: Answer Friday
Leo’s sentence fits only on Tuesday and Friday. Mia’s fits only on Monday and Friday. So it is Friday.
Why it works: on a truth day the sentence must be true, and on a lie day it must be false. Test all seven days for each speaker. Leo: on Friday (a truth day) he did lie on Thursday, so the sentence is true; on Tuesday (a lie day) he told the truth on Monday, so it is false, as a lie must be. Every other day fails. Mia: on Friday she will lie on Saturday (true, on a truth day); on Monday she will tell the truth on Tuesday (false, on a lie day). Check Friday for both once more. Common wrong answer: Tuesday or Monday (checking only one speaker, since each sentence fits one more day).
Problem 5
Problem 5: Answer 23 squares
14 small squares, 7 squares of 2 × 2, and 2 squares of 3 × 3: 14 + 7 + 2 = 23.
Why it works: count by size and use a marker so nothing is counted twice: each 2 × 2 square has a center where four small squares meet, so count those points (7). The 3 × 3 squares are the top left and the bottom right blocks; a 4 × 4 square would need the missing corners. Check: a full 4 × 4 grid has 16 + 9 + 4 + 1 = 30 squares. Each missing corner takes away 1 small square, one 2 × 2 and one 3 × 3, and the 4 × 4 is gone too: 30 − 3 − 3 − 1 = 23. Common wrong answer: 14 (small squares only), or 21 (forgetting the two 3 × 3 squares).
Problem 6
Problem 6: Answer 15
Adding the three sides counts each corner twice: 3 × side = 27 + corners. The biggest corners are 5 + 6 + 7 = 18, so 3 × side = 45 and a side is 15. It works: 7 + 2 + 6, 6 + 4 + 5, 5 + 3 + 7.
Why it works: the numbers add to 2 + 3 + … + 7 = 27. When you add the three side sums, the middle numbers are counted once and the corner numbers twice, so the three sides together make 27 plus the corners. The largest corners give the largest sides, and 45 ÷ 3 = 15. The bound alone isn’t enough: show that 15 is reached, as above. Common wrong answer: 18 (putting 5, 6 and 7 on one side, but then the other two sides can’t match), or 12 (the smallest side sum, with 2, 3 and 4 in the corners).
Problem 7
Problem 7: Answer 41
The average is 216 ÷ 6 = 36. That is even, so it is not a house: it sits between the two middle houses, 35 and 37. Steps of 2: 31, 33, 35, 37, 39, 41.
Why it works: pair from the outside in: the first and last houses average 36, and so do the second and fifth, and the third and fourth, so the six numbers add to 6 × 36. With an even count there is no middle house; the average falls halfway between the two middle ones. Neighbors on this side differ by 2, not 1. Check: 31 + 33 + 35 + 37 + 39 + 41 = 216, and all six are odd. Common wrong answer: 39 (steps of 1: 34 to 39, but half of those are on the other side of the street), or 37 (stopping at a middle house).
Problem 8
Problem 8: Answer 80 cm²
Same row: areas compare like widths. Same column: areas compare like heights. Fill in 20, 2, 6 and 9. The rows make 40 + 10 + 30 = 80.
Why it works: the middle row is the key: 3 and 5 sit side by side, and each also shares a column with a known area. Column 2 gives 12 : 3, so the top row is 4 times as tall as the middle row: top right = 4 × 5 = 20. Column 3 gives 5 : 15, so the bottom row is 3 times as tall: bottom middle = 3 × 3 = 9. The top row gives 8 : 12, so column 1 is ⅔ as wide as column 2: middle left = 2, bottom left = 6. Check: in any four rectangles that form a block, the two diagonal products agree: 8 × 3 = 12 × 2 and 12 × 5 = 20 × 3. Common wrong answer: 43 (adding only the five areas shown), or an answer measured from the picture, which is not to scale.
Problem 9
Problem 9: Answer 24 cubes
Fewest: the front view needs stacks of 2, 3 and 2 in three different places, and the front row needs its own 1: 8. Most: make every place as tall as both views allow: 16. 8 + 16 = 24.
Why it works: every height in a view needs at least one stack of exactly that height, and one stack can serve both views. Fewest: the back-middle 3 fills both 3s, and both front-view 2s can stand in the middle row, which gives the right view its 2. The right view’s 1 is the front row, where no stack is taller than 1, so it needs a cube of its own. Most: each place is as tall as the smaller of its two view heights. Both drawn buildings give exactly these views. Common wrong answer: 23 (taking 7 as the fewest), or 8 or 16 (only one of the two numbers).
Problem 10
Problem 10: Answer 16 dominoes
Color the board like a chessboard. Both cut squares are white, so 18 blue and 16 white are left. Each domino covers one of each: at most 16, and 16 fit.
Why it works: 34 squares suggest 17 dominoes, but hand tries never quite work. Squares that share a side always have different colors, so every domino covers one blue and one white square, and only 16 white squares remain. The picture shows 16 placed. Common wrong answer: 17 (34 ÷ 2, without checking it can be done).
More Math Kangaroo practice for Grades 5-6
Looking for more structured preparation? Our Math Kangaroo Prep Guide for Grades 5–6 covers every major topic area with explanations, practice problems, and test-day strategies matched to the real competition format. There’s a free sample if you’d like to see it first.
You can also browse our Math Kangaroo practice tests and questions page: a free set for every grade band, the format by grade, and where to find the official past papers.
This guide is not affiliated with or endorsed by Math Kangaroo USA or Kangourou sans Frontières.