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    How many different triangles are there in the figure?
    Problems
    Grades: 3rd to 5th
    Geometry
    Classify two-dimensional figures into categories based on their properties.
    5.G.B.4

    It’s not too hard to form the number 9 using three 3’s and any of the four standard mathematical operations +, –, × and ÷. But can you come up with four different solutions, each of which uses only one of the four operations? (Other standard mathematical symbols can be used as needed.)

    9 = 3 + 3 + 3  

    Problems
    Grades: 3rd to 5th
    Algebraic Thinking
    Write and interpret numerical expressions.
    Multiply and divide within 100.
    3.OA.C.7, 5.OA.A.1
    When the ends of the rope at left are pulled in opposite directions, how many knots will be formed along the rope's length?
    Problems
    Write 2014 with the first four prime numbers, with the aid of the operations addition, multiplication and exponentiation.
    Problems
    Grades: 6th to 8th, 3rd to 5th
    Expression/Equation
    Algebraic Thinking
    Apply and extend previous understandings of arithmetic to algebraic expressions.
    Gain familiarity with factors and multiples.
    Multiply and divide within 100.
    3.OA.C.7, 4.OA.B.4, 6.EE.A.1

    Equations to solve in your head:

    \begin{array}{l}
 6,751x + 3,249y = 26,751 \\ 
 3,249x + 6,751y = 23,249 \\ 
 \end{array}

    Is this a joke? Not if you can multiply the first equation by 6,751 and the second by 3,249 in your head, and not if you use a second, simpler method.

    Problems
    Grades: 9th to 12th, 6th to 8th
    Algebra
    Expression/Equation
    Reasoning with Equations and Inequalities
    Analyze and solve linear equations and pairs of simultaneous linear equations.
    8.EE.C.8b, HSA-REI.C.6, HSA-REI.C.5

    Which is bigger, \sqrt {10}  + \sqrt {29} or \sqrt {73}

    Don’t even think about using a calculator for this one.

    Problems
    Grades: 6th to 8th
    Expression/Equation
    The Number System
    Work with radicals and integer exponents.
    Know that there are numbers that are not rational, and approximate them by rational numbers.
    8.NS.A.2, 8.EE.A.2
    There are 7 number cards, each with a number from 1 to 7, in a box.  Henry took out 3 cards and Mathew took out 2 cards. Henry looked at his 3 cards and said to Mathew, “The sum of the numbers on your 2 cards must be an even number.”   Mathew thought for a minute and said, “Then I know the sum of the numbers on your 3 cards.” What is the sum of the three numbers on Henry’s cards?
    Problems

    According to Waring’s theorem, any positive integer can be represented as the sum of nine or fewer perfect cubes (not necessarily distinct).

    For instance, 89 can be represented as the sum of four perfect cubes: 27 + 27 + 27 + 8 = 89.

    Can you express 239 as a sum of nine or fewer perfect cubes?

    Problems

    There are 29 students in Miss Spelling’s class. As a special holiday gift, she bought each of them chocolate letters with which they can spell their names. Unfortunately, some letters cost more than others — for instance, the letter A, which is in high demand, is rather pricey; whereas the letter Q, which almost no one wants, is relatively inexpensive.

    The price of the chocolate letters for each student in her class is shown in the table below.

    AIDEN – 386

    ARI – 209

    ARIEL – 376

    BLAIRE – 390

    CHARLES – 457

    CLARE – 334

    DEAN – 317

    EARL – 307

    FRIDA – 273

    GABRIEL – 410

    IVY – 97

    KOLE – 249

    LEIA – 317

    LEO – 242

    MAVIS – 246

    NADINE – 453

    NED – 236

    PAUL – 167

    QASIM – 238

    RACHEL – 394

    RAFI – 231

    SAM – 168

    TIRA – 299

    ULA – 148

    VERA – 276

    VIJAY – 179

    WOLKE – 272

    XAVIER – 346

    ZERACH – 355

     

    How much would it cost to buy the letters in your name?

    Problems
    Grades: High School, 6th to 8th, 3rd to 5th
    Num & Ops Base Ten
    Use place value understanding and properties of operations to perform multi-digit arithmetic.
    3.NBT.A.2, 4.NBT.B.4

    A prisoner was thrown into a medieval dungeon with 145 doors. Nine, shown by black bars, are locked, but each one will open if before you reach it you pass through exactly 8 open doors. You don’t have to go through every open door but you do have to go through every cell and all 9 locked doors. If you enter a cell or go through a door a second time, the doors clang shut, trapping you.

    The prisoner (in the lower right corner cell) had a drawing of the dungeon. He thought a long time before he set out. He went through all the locked doors and escaped through the last, upper left corner one.

    What was his route?

    Problems

    Ask a friend to pick a number from 1 through 1,000. After asking him ten questions that can be answered yes or no, you tell him the number.

    What kind of Questions?

     

    Problems
    Grades: 3rd to 5th
    Num & Ops Base Ten
    Generalize place value understanding for multi-digit whole numbers.
    4.NBT.A.2
    A regular octagon is inscribed inside a square. Another square is inscribed inside the octagon. What is the ratio of the area of the smaller square to the area of the larger square?
    Problems
    Grades: 9th to 12th, 3rd to 5th, 6th to 8th
    Geometry
    Measurement & Data
    Ratio & Proportion
    Circles
    Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
    Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
    Understand ratio concepts and use ratio reasoning to solve problems.
    6.RP.A.1, 3.MD.C.7b, 4.MD.A.3, HSG-C.A.3

    What is the sum of the following?

    432 + 432 + 432 + 432 + 432 + 432 + 864 + 864

    Problems
    Grades: 3rd to 5th
    Num & Ops Base Ten
    Use place value understanding and properties of operations to perform multi-digit arithmetic.
    3.NBT.A.2, 4.NBT.B.4
    In 2008, September and December both began on a Monday. But every year, there are two months that do not begin on the same day of the week as any other month. What are those two months?
    Problems
    Two chess players compete in a best-of-five match. If Chekmatova has a 60% chance of winning any particular game, what is the likelihood that she will win the match?
    Problems
    Grades: 9th to 12th, 6th to 8th
    Stats & Probability
    Conditional Probability and the Rules of Probability
    Investigate chance processes and develop, use, and evaluate probability models.
    7.SP.C.5, 7.SP.C.7a, 7.SP.C.8a, 7.SP.C.8b, HSS-CP.B.8, HSS-CP.B.9

    Tom was born on Thanksgiving Day.

    On his seventh birthday, he noticed that Thanksgiving had never fallen on his birthday. How old will he be when he finally has a Thanksgiving birthday?

    Problems
    Assuming that the circumference of each circle below passes through the centers of the other two, and that the radius of each circle is 1, what is the total gray area?
    Problems
    Grades: 9th to 12th, 6th to 8th
    Geometry
    Geometric Measurement and Dimension
    Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
    7.G.B.4, HSG-GMD.A.1

    A plywood sheet is 45 by 45 inches. What is the approximate diameter of the log the sheet was made from?

     

    The diameter d of a circle equals \frac{C}{\pi }, where C is the circumference, but please do not make a mistake. The diameter of the log is not \frac{{45}}{\pi }.

    Problems
    Grades: 9th to 12th, 6th to 8th
    Geometry
    Geometric Measurement and Dimension
    Circles
    Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
    7.G.B.4, HSG-C.A.3, HSG-GMD.A.1
    The following isosceles trapezoid is composed of 7 matches. Modify the position of three matches in order to obtain two equilateral triangles.
    Problems
    Grades: 6th to 8th
    Geometry
    Draw construct, and describe geometrical figures and describe the relationships between them.
    7.G.A.2
    In a four‑digit number, the sum of the digits is 10. All the digits are different. What is the largest such four‑digit number?
    Problems
    Grades: 3rd to 5th
    Num & Ops Base Ten
    Generalize place value understanding for multi-digit whole numbers.
    4.NBT.A.2
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