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2026 Gauss Contest
Grade 7

May 11 to May 22, 2026
(in North America and South America)

May 11 to May 22, 2026
(outside of North American and South America)

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©2026 University of Waterloo

Instructions

Time: 1 hour

Calculating devices are allowed, provided that they do not have any of the following features: (i) internet access, (ii) the ability to communicate with other devices, (iii) information previously stored by students (such as formulas, programs, notes, etc.), (iv) a computer algebra system, (v) dynamic geometry software.

  1. Do not open the Contest booklet until you are told to do so.
  2. You may use rulers, compasses and paper for rough work.
  3. Be sure that you understand the coding system for your answer sheet. If you are not sure, ask your teacher to explain it.
  4. This is a multiple-choice test. Each question is followed by five possible answers marked A, B, C, D, and E. Only one of these is correct. When you have made your choice, enter the appropriate letter for that question on your answer sheet.
  5. Scoring:
    1. Each correct answer is worth 5 in Part A, 6 in Part B, and 8 in Part C.
    2. There is no penalty for an incorrect answer.
    3. Each unanswered question is worth 2, to a maximum of 10 unanswered questions.
  6. Diagrams are not drawn to scale. They are intended as aids only.
  7. When your supervisor instructs you to start, you will have sixty minutes of working time.
The name, school and location of some top-scoring students will be published on the Web site, cemc.uwaterloo.ca. On this website, you will also be able to find copies of past Contests and excellent resources for enrichment, problem solving and contest preparation.
Scoring:
  1. There is no penalty for an incorrect answer.
  2. Each unanswered question is worth 2, to a maximum of 10 unanswered questions.

Part A: Each correct answer is worth 5.

  1. In the diagram, how many of the shapes are triangles?

    Two rows of shapes. Both rows have the following five shapes: triangle, circle, triangle, circle, triangle.

    1. \(4\)
    2. \(5\)
    3. \(6\)
    4. \(7\)
    5. \(8\)

  2. The integer that must replace the square so that \(\square - 3 = 5\) is

    1. \(6\)
    2. \(8\)
    3. \(10\)
    4. \(9\)
    5. \(7\)

  3. Which of the following numbers is farthest away from \(0\) on a number line?

    1. \(-10\)
    2. \(-8\)
    3. \(3\)
    4. \(8\)
    5. \(9\)

  4. The graph shows the amount of money that each of five students donated to a charity.

    A bar graph entitled Donations to Charity. The horizontal axis has Student and the vertical axis has Donated amount in dollars. Daia donated 12, Joe donated 6, Bel donated 10, Susie donated 8, Zara donated 2.

    The total amount of money that they donated was

    1. \(\$36\)
    2. \(\$30\)
    3. \(\$28\)
    4. \(\$38\)
    5. \(\$32\)

  5. Which digit is in the thousands place of the number \(12\,635\)?

    1. \(1\)
    2. \(2\)
    3. \(6\)
    4. \(3\)
    5. \(5\)

  6. In the diagram, \(\angle PQR\) is a straight angle.

    Line segment QS divides straight angle PQR into two angles: obtuse angle PQS measures x degrees and angle RQS measures 40 degrees.

    The value of \(x\) is

    1. \(100\)
    2. \(140\)
    3. \(50\)
    4. \(120\)
    5. \(320\)

  7. If a square's side length is a whole number, measured in centimetres, which of the following could be the perimeter of the square?

    1. \(26~\text{cm}\)
    2. \(22~\text{cm}\)
    3. \(31~\text{cm}\)
    4. \(34~\text{cm}\)
    5. \(32~\text{cm}\)

  8. Mauricio goes for a walk every day for five consecutive days. On the first day, he walks for \(11\) minutes. Each day after the first, he walks for \(3\) minutes longer than he walked the day before. In minutes, what is the total time that he walks over the five days?

    1. \(85\)
    2. \(73\)
    3. \(106\)
    4. \(65\)
    5. \(74\)

  9. In the diagram, the shape is created from five squares. The shape is reflected in the horizontal line below it.

    Five identical squares arranged to form a grid with 3 rows and 2 columns with the top-left square missing. The bottom side of the grid is parallel to a horizontal line below the grid.

    The orientation of the shape as a result of the reflection is

    1. Five identical squares arranged to form a grid with 2 rows and 3 columns with the top-right square missing.
    2. Five identical squares arranged to form a grid with 3 rows and 2 columns with the top-right square missing.
    3. Five identical squares arranged to form a grid with 3 rows and 2 columns with the bottom-right square missing.
    4. Five identical squares arranged to form a grid with 3 rows and 2 columns with the bottom-left square missing.
    5. Five identical squares arranged to form a grid with 3 rows and 2 columns with the top-left square missing.

  10. In the sequence \(43,41,39,37,...,3,1\), the first term is \(43\), each term after the first is \(2\) less than the term before it, and the last term is \(1\). How many terms are in the sequence?

    1. \(19\)
    2. \(20\)
    3. \(21\)
    4. \(22\)
    5. \(23\)

Part B: Each correct answer is worth 6.

  1. Lana participates in long jump. She completes \(4\) jumps and achieves the following results: \(1.60~\text{m}\), \(1.65~\text{m}\), \(1.85~\text{m}\), \(1.90~\text{m}\). What is the mean (average) of her results?

    1. \(1.70~\text{m}\)
    2. \(1.85~\text{m}\)
    3. \(1.80~\text{m}\)
    4. \(1.65~\text{m}\)
    5. \(1.75~\text{m}\)

  2. Riley is \(3\) years older than Elsa. The sum of their ages is \(27\). How old is Elsa?

    1. \(10\)
    2. \(11\)
    3. \(12\)
    4. \(13\)
    5. \(14\)

  3. In the diagram, \(\triangle PQR\) has base \(PQ=8~\text{cm}\) and height \(6~\text{cm}\). \(\triangle PST\) has base \(PS=4~\text{cm}\) and height \(3~\text{cm}\).

    In triangle PQR, point S lies on base PQ and point T lies inside the triangle. The region inside triangle PQR but outside triangle PST is shaded.

    What is the area of the shaded region?

    1. \(18~\text{cm}^2\)
    2. \(6~\text{cm}^2\)
    3. \(36~\text{cm}^2\)
    4. \(12~\text{cm}^2\)
    5. \(24~\text{cm}^2\)

  4. How many different four-digit integers greater than \(2000\) can be made using each of the four digits \(2\), \(0\), \(2\), \(6\) exactly once?

    1. \(6\)
    2. \(8\)
    3. \(10\)
    4. \(7\)
    5. \(9\)

  5. The numbers on the opposite faces of a standard six-sided die have a sum of \(7\). Which of the following is not the net of a standard six-sided die?

    1. Four squares form the middle row of the net. These squares, from left to right, contain the following numbers of dots: 1, 3, 6, and 4. Below the square with 3 dots is a square with 2 dots. Above the square with 6 dots is a square with 5 dots.
    2. Two squares form the top row of the net. These squares, from left to right, contain 3 and 6. Below the 6 is a square with 2. To the right of 2 is a square with 4. Below the 4 is a square with 1. To the right of 1 is a square with 5.
    3. Four squares form the middle row of the net. These squares, from left to right, contain 5, 4, 3, and 1. Above the 4 is a square with 6. Below the 3 is a square with 2.
    4. Two squares form the bottom row of the net. These squares, from left to right, contain 2 and 6. Above the 6 is a square with 3. To the right of 3 is a square with 5. Above the 5 is a square with 1. To the right of 1 is a square with 4.
    5. Four squares form the middle row of the net. These squares, from left to right, contain 2, 1, 5, and 6. Above the 2 is a square with 4. Below the 2 is a square with 3.

  6. In the game shown, Riah begins on the start square. She rolls two standard six-sided dice and moves forward the number of squares equal to the sum rolled. She continues in this way, rolling two dice and moving forward until she lands on or passes the finish square. If she lands on a shaded square, she follows the instructions shown. If Riah begins with \(0\) points, what is the greatest number of points that she can finish this game with?

    There are 18 squares between the start square and the finish square numbered 1 through 18. Eight numbered squares have instructions. #1: add 3 points. #3: add 6 points. #6: add 4 points. #8: add 5 points. #11: add 7 points. #12: add 8 points. #16: add 10 points. #17: double your points.

    1. \(43\)
    2. \(46\)
    3. \(52\)
    4. \(56\)
    5. \(86\)

  7. Asha is asked to cut a \(36~\text{m}\) length of ribbon into smaller pieces so that each piece has equal length, and the ribbon width does not change. The length of each of the pieces must be a whole number of metres. If she must make at least \(1\) cut, how many possibilities are there for the length of the smaller ribbons?

    1. \(10\)
    2. \(7\)
    3. \(6\)
    4. \(5\)
    5. \(8\)

  8. Let \(0.ABC\) represent a decimal number. When \(0.ABC\) is rounded to the nearest tenth, the result is \(0.024\) greater than \(0.ABC\). What is the value of \(B+C\)?

    1. \(14\)
    2. \(12\)
    3. \(13\)
    4. \(9\)
    5. \(11\)

  9. Emil chooses a positive integer so that

    How many different such integers could Emil choose?

    1. \(20\)
    2. \(19\)
    3. \(22\)
    4. \(23\)
    5. \(21\)

  10. The Dollop sells ice cream cones. Each cone sold is exactly one of three different types: chocolate or vanilla or twist (chocolate and vanilla combined). Of all cones sold, \(\frac{11}{17}\) contained some chocolate and \(\frac{10}{17}\) contained some vanilla. If \(85\) ice cream cones were sold in total, how many were twist cones?

    1. \(15\)
    2. \(4\)
    3. \(25\)
    4. \(17\)
    5. \(20\)

Part C: Each correct answer is worth 8.

  1. In the diagram, \(PQRS\) is a square with \(RS=30~\text{cm}\). \(TUVW\) is a square with \(U\) and \(V\) on \(PS\) so that \(PU=UV=VS\).

    Square PQRS with top side PS, vertex Q below P, and vertex R below S. Square TUVW is placed on top with side UV along side PS, vertex T above U, and vertex W above V.

    Point \(X\) is on \(QR\) so that \(TX\) is parallel to \(WR\). The area of parallelogram \(TXRW\) is

    1. \(\text{200 cm}^2\)
    2. \(\text{400 cm}^2\)
    3. \(\text{600 cm}^2\)
    4. \(\text{300 cm}^2\)
    5. \(\text{412.3 cm}^2\)

  2. Beverly creates a sequence of integers. She begins by setting \(\text{sum}=0\) and \(\text{a}=1\). She then writes down the value of \(\text{sum}\) as the first term of the sequence. Beverly continues the sequence by applying the following steps:

    What is the last term in Beverly's sequence?

    1. \(36\)
    2. \(81\)
    3. \(16\)
    4. \(9\)
    5. \(25\)

  3. Two bowls each contain both some blueberries and some raspberries. In the first bowl, the ratio of the number of blueberries to the number of raspberries is \(3 : 7\). In the second bowl, the ratio of the number of blueberries to the number of raspberries is \(2 : 5\). If there are \(89\) raspberries in total, the smallest possible number of blueberries is

    1. \(36\)
    2. \(38\)
    3. \(35\)
    4. \(37\)
    5. \(34\)

  4. A Sorrol number is a five-digit positive integer \(abcde\) for which

    The positive difference between the largest Sorrol number and the smallest Sorrol number is

    1. \(84\,978\)
    2. \(83\,989\)
    3. \(85\,015\)
    4. \(84\,038\)
    5. \(85\,968\)

  5. The letters \(A\), \(B\), \(C\), and \(D\) are placed into a \(5\) by \(5\) grid, as shown.

    A is in the square in the 2nd row from the top and the 2nd column from the left. B is in the 2nd row and 4th column. C is in the 4th row and 2nd column. D is in the 4th row and 4th column.

    Four or fewer of the empty \(1\) by \(1\) squares are shaded in, so that

    For example, Figure 1 and Figure 2, below, each show a possible way to shade the grid. Figures 3, 4 and 5 are not permitted ways to shade the grid.

    The following three squares in the grid are shaded: 1st row and 4th column, 3rd row and 2nd column, 4th row and 5th column.
    Figure 1
    The following four squares in the grid are shaded: 1st row and 5th column, 2nd row and 3rd column, 4th row and 1st column, 4th row and 5th column.
    Figure 2
    The following three squares in the grid are shaded: 2nd row and 3rd column, 4th row and 3rd column, 4th row and 5th column.
    Figure 3
    The following three squares in the grid are shaded: 1st row and 1st column, 3rd row and 4th column, 5th row and 2nd column.
    Figure 4
    The following four squares in the grid are shaded: 1st row and 5th column, 2nd row and 5th column, 3rd row and 2nd column, 5th row and 4th column.
    Figure 5

    How many different ways can the grid be shaded?

    1. \(110\)
    2. \(112\)
    3. \(116\)
    4. \(122\)
    5. \(128\)