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How many different triangles are there in the figure?
Problems
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
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

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, or

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

Problems
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
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
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 , where C is the circumference, but please do not make a mistake. The diameter of the log is not .

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
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