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Brilliant.org

@brilliantorg

8 months

Think you can cut it? Test your sharpness at Halfsies, a new game from Brilliant.

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Brilliant.org

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4 years

There are many proofs to this famous theorem — here's one that we particularly enjoy.

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Brilliant.org

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6 years

You’ll never forget the difference of squares once you’ve seen this visual representation.

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Brilliant.org

@brilliantorg

4 years

Although the difference of squares identity is usually proven algebraically, there is a simple geometric proof that involves... well... a difference of squares. You just cut out a square from another square, rearrange the pieces, and the identity becomes apparent.

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Brilliant.org

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5 years

Behold the beauty of geometry! All polygons of equal area can be dissected into smaller pieces that can be "hinged" to form any other polygon. 📐 #brilliantgifs #geometry

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Brilliant.org

@brilliantorg

4 years

Angles of the same color have the same measure. What is the measure of the green angle?

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Brilliant.org

@brilliantorg

5 years

If you sum consecutive odd integers starting with 1, then the result is always a perfect square. There are many proofs of this fact, but in this proof without words, we only need to arrange dots in a certain pattern.

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Brilliant.org

@brilliantorg

5 years

Putting these powers into a calculator would be a mess, but is there a simpler way? What is this expression equal to? A. -11/3 B. 13/3 C. 9/5 D. 27/5

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Brilliant.org

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4 years

Brilliant's 100 day challenge has begun! Day 1: In this diagram, the grey angles have the same measure. How are x, y, and z related? A. x = y = z B. x + y + z = 180 C. y + z = x D. 2z = x+y

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Brilliant.org

@brilliantorg

4 years

A rectangle is inscribed in a right triangle. What is the area of the rectangle?

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Brilliant.org

@brilliantorg

2 years

@NASASpaceflight 🙌

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Brilliant.org

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4 years

These two squares are arranged side-by-side. What is the area of the shaded region?

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Brilliant.org

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5 years

If you’re thinking about squaring both sides of the equation — hold up! There’s another way! 😲

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Brilliant.org

@brilliantorg

5 years

The radius of the circle is 1 m. What is the area of the shaded region? A. π/2 m² B. 2π/3 m² C. 3π/4 m² D. π m²

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Brilliant.org

@brilliantorg

4 years

Every angle in the figure is a right angle, and the colored rectangle has an area of 48. What is x?

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Brilliant.org

@brilliantorg

4 years

Which is larger? A. 100! × 100! B. 99! × 101! C. They are equal

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Brilliant.org

@brilliantorg

4 years

The colored triangles are equilateral. What is the value of z? A. 30 B. 35 C. 40 D. 45

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Brilliant.org

@brilliantorg

5 years

When the parameters of the Lorenz System of differential equations are chosen just right, all solutions are attracted towards a very strange-looking set that's neither an equilibrium nor a cycle. The animation shows the progress of two such solutions.

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Brilliant.org

@brilliantorg

4 years

Every rectangle is similar. What is the missing length?

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Brilliant.org

@brilliantorg

5 years

Pythagoras’ famous theorem relating the lengths of a right triangle is taught in every introductory geometry class. Proving this algebraically is not so simple, but there are several ways to visualize the theorem. This proof without words relies on comparing areas of squares.

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Brilliant.org

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5 years

When part of a problem can change while the rest remains constant, we call the part that stays the same “invariant”. What is the invariant area of this changing shape?

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7 years

The Reuleaux Triangle can be rotated inside a square and remain in constant contact with all four sides of the square... Mesmerizing, right?

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Brilliant.org

@brilliantorg

4 years

The three larger squares meet at the center of the small square. How much of the small square is shaded orange? A. 1/6 B. 1/4 C. 1/3 D. 3/8

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Brilliant.org

@brilliantorg

4 years

Each number inside a colored region is the area of that region. What is the value of x?

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Brilliant.org

@brilliantorg

5 years

Polar graphs often have curves with beautiful symmetry, like the graph of r = sin(2θ), which looks like a flower with four petals. In this animation, the Cartesian graph y = sin(2x) transforms into the polar graph r = sin(2θ).

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Brilliant.org

@brilliantorg

4 years

Angles of the same color are identical. How much larger is a blue angle than a yellow angle? A. 2° B. 5° C. 10° D. The angles are equal

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Brilliant.org

@brilliantorg

4 years

What happens when two math wizards are tasked with summing the numbers in this grid? They each come up with their own ingenious way of computing the sum! It may seem like magic, but anyone can do it. Which is your favorite method? Can you come up with your own method?

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Brilliant.org

@brilliantorg

4 years

An isosceles triangle is a triangle with two sides that are the same length. Isosceles triangles also have two angles with the same measure — the angles opposite the equal sides. In the image, all the orange segments are the same length. What is the measure of the blue angle?

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Brilliant.org

@brilliantorg

4 years

These two parallelograms have the same bases and heights. The parallelogram on the left is a rectangle that fits three circles snugly. Will the same three circles fit in the parallelogram on the right without overlap?

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Brilliant.org

@brilliantorg

4 years

While skulking through a dark alley, you find a note with 100 statements written on it. Is Statement #1 true or false?

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Brilliant.org

@brilliantorg

5 years

All angles are right angles, and the orange region has an area of 21. What is the value of x? A. 2 B. 2.5 C. 3 D. 3.5

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Brilliant.org

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4 years

Which of these two lattice triangles has the larger area? A. A has the larger area. B. B has the larger area. C. A and B have the same area.

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Brilliant.org

@brilliantorg

4 years

How many triangles are there in this figure? A. 10 B. 20 C. 25 D. 35

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Brilliant.org

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4 years

What is the missing length? A. 115 B. 120 C. 125 D. 130

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Brilliant.org

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4 years

Out of these options, what could balance three diamonds? A. 🔵 B. 🟩🔵 C. 🟩 D. 🟩🟩

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Brilliant.org

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4 years

One side of the triangle is half the side of the regular hexagon. What fraction of the hexagon is shaded? A. 1/4 B. 1/5 C. 1/6 D. 1/8

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Brilliant.org

@brilliantorg

5 years

You have a crystal in the shape of a regular octahedron. Which of these kinds of cross-sections can you get by slicing through the interior of the crystal? Choose one or more: A. Triangle B. Square C. Pentagon D. Hexagon E. Octagon F. Decagon

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Brilliant.org

@brilliantorg

4 years

All angles in the figure are right angles. What is the total area of the square? A. 94 B. 96 C. 98 D. 100

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Brilliant.org

@brilliantorg

5 years

Each square is double the area of the one to its left. Which color has a greater area? Hint: 1000 > 999 A. Yellow B. Blue C. The colors have equal areas

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Brilliant.org

@brilliantorg

4 years

How many triangles are there in this figure? A. 12 B. 15 C. 18 D. 20

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Brilliant.org

@brilliantorg

6 years

Brush off your geometry skills to solve this tricky area problem.

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Brilliant.org

@brilliantorg

4 years

Given that every letter corresponds to a different digit, what is the value of H?

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Brilliant.org

@brilliantorg

4 years

If the ball travels in a straight line and stops only if it strikes a pocket precisely at its center, what eventually happens to it?

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Brilliant.org

@brilliantorg

4 years

The 100×96 rectangle is cut up into four triangles. Three of the triangles have perimeters of 300, 224, and 168, as indicated by the numbers inside them. What is the perimeter of the remaining orange triangle? A. 280 B. 300 C. 320 D. 340

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Brilliant.org

@brilliantorg

5 years

Which formula describes the number of dots in this pattern? A. 3(2n-1) B. 3n(n+1)/2 C. 3^n D. 3+6(n-1)

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Brilliant.org

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4 years

What is the value of A + B + C? A. 8 B. 10 C. 12 D. 14

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Brilliant.org

@brilliantorg

5 years

In the Bridges of Königsberg problem, the goal is to cross each bridge exactly once (no swimming allowed!). You could try every possible path, but you'll never find one that solves the problem. There's an easier way to prove this without trying every path. Can you find it?

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Brilliant.org

@brilliantorg

5 years

There is more than one way to solve this challenge. Get creative! 🧠⚡

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Brilliant.org

@brilliantorg

5 years

A star in the distance leads you to a geometry puzzle. Will you heed its call? What is the sum of the three pink angles? A. 90° B. 100° C. 110° D. 120°

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Brilliant.org

@brilliantorg

4 years

A simple loop with a single twist, the Möbius strip has many surprising properties. For example, if you trace a line down the strip, you’ll eventually end up where you started with both “sides” covered.

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Brilliant.org

@brilliantorg

5 years

Alice and Bob are both logical and truthful, and each has complete knowledge that the other is logical and truthful. They are each given a distinct one-digit positive number in secret, and then they make these statements. What is Bob’s number? A. 1 B. 3 C. 5 D. 7 E. 9

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Brilliant.org

@brilliantorg

4 years

This matchstick arrangement has five squares. What is the minimum number of matchsticks that must be removed from this arrangement to leave exactly two squares? A. 1 B. 2 C. 3 D. 4

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Brilliant.org

@brilliantorg

5 years

The Sieve of Eratosthenes was the official prime number generating algorithm of the ancient Greeks, and the only one allowed on the Acropolis (just kidding). It works by removing multiples of prime numbers that we already know.

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Brilliant.org

@brilliantorg

5 years

The distance of a circle’s radius is measured out along its edge, and a dot is drawn. This process repeats again and again around the circumference of the circle. Will one of the dots ever line up perfectly with another?

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Brilliant.org

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5 years

Can you solve for N to make these two quantities equal?

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Brilliant.org

@brilliantorg

4 years

Put away the calculator! If you know the difference of squares identity, you’ll be able to do this in a jiffy. A. 1001×999999 B. 10001×99999 C. 100001×9999 D. 1000001×999

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Brilliant.org

@brilliantorg

4 years

Happy New Year to all of our STEM-loving friends! Let’s celebrate with an algebra puzzle! Which is greater, x or y?

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Brilliant.org

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5 years

Which has a greater area, a square inscribed in a semi-circle or two adjacent squares inscribed in the same semi-circle? Hint: Every point on a circle is the same distance from the center.

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Brilliant.org

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6 years

1 1 1 1 = 5 Can you make this equation true?

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Brilliant.org

@brilliantorg

5 years

Thrilled to collaborate with @MSFTQuantum and @Theteamatx to bring you this new interactive course where you can construct a quantum algorithm from the ground up! 👇

Learn Quantum Computing on Brilliant

Learn to build quantum algorithms from the ground up with a quantum computer simulated in your browser in this course, created in collaboration with quantum researchers and practitioners from...

brilliant.org

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Brilliant.org

@brilliantorg

5 years

All the triangles shown are isosceles triangles. What is the sum of the pink angles? A. 120° B. 140° C. 160° D. 180° It cannot be determined from the given information.

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Brilliant.org

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5 years

This star was created by placing equilateral triangles along each edge of a regular nonagon (nine-sided polygon). What is the sum of the measures of the pink angles? A) 540° B) 810° C) 900° D) 990°

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Brilliant.org

@brilliantorg

5 years

In the Tower of Hanoi puzzle, you can only move one disk at a time, and you can't put a larger disk on a smaller one. How many moves does it take to move the stack of three to a different peg? Can you use recursive thinking to calculate how many moves are needed for 64 disks?

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Brilliant.org

@brilliantorg

5 years

An incorrect proof is shown. On which step was the mistake made? A. Step 1 B. Step 2 C. Step 3 D. Step 4 E. Step 5

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Brilliant.org

@brilliantorg

4 years

Given that every letter corresponds to a different digit, which equation is true? 1. A + B = C 2. A + C = D 3. B + D = C 4. C + D = B

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Brilliant.org

@brilliantorg

4 years

How many different ways are there to get from point A to point D, assuming no path ever traces back over any part of itself? A. 8 B. 10 C. 12 D. 14

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Brilliant.org

@brilliantorg

4 years

There’s something special about four-sided figures packed into circles. Add up their opposite angles, and the sum is always 180°. Can you use this fact to find the measure of the missing angle? A. 10° B. 15° C. 20° D. 25°

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Brilliant.org

@brilliantorg

5 years

The Reuleaux Triangle can be rotated inside a square and remain in constant contact with all four sides of the square... Mesmerizing, right? 😮

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Brilliant.org

@brilliantorg

4 years

The four triangles' areas are labeled. There are two rectangles in the figure, and every triangle side length is an integer. What is the value of x? A. 6 B. 7 C. 8 D. 9

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Brilliant.org

@brilliantorg

5 years

This square is segmented in a peculiar fashion. Can you find the area in the middle?

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Brilliant.org

@brilliantorg

4 years

Assuming both players play perfectly, who will win the game? A. The first player will win. B. The second player will win. C. There is not enough information to know who will win.

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Brilliant.org

@brilliantorg

4 years

Every time it moves, this robot either hops exactly one dot to the left or one dot to the right. If the robot starts in the center of this pattern, what is the probability that, after four hops, it's positioned on top of a blue dot? A. 1/2 B. 1/3 C. 1/4 D. 2/9

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Brilliant.org

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5 years

What’s the length of this infinitely zig-zagging red line? Hint: There’s a simple trick for measuring it. 📏

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Brilliant.org

@brilliantorg

7 years

Learn the math behind the beauty here:

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Brilliant.org

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4 years

Angles of the same color are identical. How many yellow angles does it take to make the orange angle? A. 5.5 B. 6 C. 6.5 D. 7

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Brilliant.org

@brilliantorg

5 years

Just a little bit of similarity, a dash of algebra, and you can solve this puzzle! What is the height of the rectangle in terms of x, y, and z? A. y+1 B. √(z²-x²) C. yz D. xyz

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Brilliant.org

@brilliantorg

5 years

This fixed center gear has 30 teeth, and the smaller gear has 10 teeth. How many times will the arrow flip over during one circuit? (It’s not 3.) Gear systems like this are an essential component of power transfer in hybrid cars.

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Brilliant.org

@brilliantorg

4 years

• All circles in the dartboard have the same center. • Each circle has half the radius of the previous circle. • The colors alternate orange-white-orange-white-etc. forever. How much of the dartboard is colored orange?

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Brilliant.org

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4 years

Day 3: The silver gear is initially driven to rotate at 10 rpm, which turns a large, gold gear. You add a second silver gear to the system, and its teeth make contact with the other two. At what rate does the newly-added gear rotate? A. 0 rpm B. 2 rpm C. 4 rpm D. 10 rpm

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Brilliant.org

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4 years

Is it possible to arrange the five square tiles numbered 1, 2, 3, 4, and 5 into a "plus" so that the sum of the three-tile column and the sum of the three-tile row are both equal to 9?

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Brilliant.org

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5 years

Chaos theory tells us that simple non-linear equations can have complex solutions. Each frame of this animation is a solution trajectory of a unique quadratic map. Most head towards one point, or off to infinity. But a few are chaotic, and end up orbiting in a strange attractor.

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Brilliant.org

@brilliantorg

5 years

a, b, and c are real numbers. Can you find a-c? Hint: Explore what a^2 + b^2 can be.

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Brilliant.org

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4 years

The figure is made from a square, an equilateral triangle, and three semicircles. The blue semicircle has an area of 3. What is the area of the lune? A. 4 B. 8/π C. 4 - 8/π D. 8 - 16/π

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Brilliant.org

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5 years

Using computer science, can you always win at 20 questions? The right strategy is to ask questions that split the field mostly in half. With this strategy, a well-designed algorithm can easily guess correctly any of the ~200,000 actors on Wikipedia.

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Brilliant.org

@brilliantorg

7 years

Keep your mind sharp with our summer challenge - #100problems in 100 days. Join in and track your progress:

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Brilliant.org

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5 years

The blue figure is composed of arcs whose radii are half the side length of the square. How do the shaded and unshaded areas compare? A. The blue-shaded area is larger B. The unshaded area is larger C. The areas are equal

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Brilliant.org

@brilliantorg

5 years

Which formula describes the number of dots in this pattern? A. (n+1)² B. (n+1)(n+2)/2 + n C. 4n D. n(n+1)/2 + n

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Brilliant.org

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8 years

The number of American teens who excel at advanced math has surged. Why? Great piece by @pegtyre at The Atlantic

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Brilliant.org

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4 years

If you shuffle the four cards and then deal them out in a row on a table, what is the probability that the two queens are dealt out side by side? A. 1/4 B. 1/3 C. 1/2 D. 2/3

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Brilliant.org

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4 years

Ant A makes two complete laps around the base of the can. Ant B goes to the point at the top of the can opposite of the starting point, and then back to the starting point. Which ant finishes first?

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4 years

Which combination would balance with three squares? A. One triangle and one circle B. One triangle and two circles C. Two triangles and one circle D. Two triangles and two circles

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Brilliant.org

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4 years

The star shapes are made by extending the sides of a regular pentagon and a regular hexagon, respectively. Which shape has more of its area shaded?⁠ ⁠ A. Shape A⁠ B. Shape B⁠ C. The shapes have equal proportions of their areas shaded⁠

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Brilliant.org

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6 years

How would you approach calculating an infinite sum? Before breaking out your calculator, check out our geometric depictions.

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Brilliant.org

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4 years

What is the largest number of statements that can be true at the same time, with no logical contradictions? A. None of the statements can be true without contradictions. B. 1 C. 2 D. 3

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Brilliant.org

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5 years

A cuboctahedron is a solid with square and equilateral triangle faces. It can be formed by slicing triangular pyramids from a cube. If the square faces are blue and the triangular faces are purple, then which area is greater, blue or purple?

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4 years

Each outer curve is part of a circle identical to the inner circle, with the curves touching at midpoints of the sides of the regular hexagon. Which fraction is the closest to the portion of the hexagon that is shaded blue? A. 1/10 B. 1/9 C. 1/8 D. 1/6

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Brilliant.org

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5 years

You have a 4x5 chessboard and five tetronimos. Is it possible to fully cover the chessboard with the tetronimos?

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Brilliant.org

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4 years

Dashed arrows mean addition, while solid arrows mean multiplication. What number goes in the place of the question mark? A. 3 B. 4 C. 5 D. 6

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5 years

To draw this recursive figure, the smaller triangle's midpoints are connected over and over and over again (for best results, repeat an infinite number of times).⁠ ⁠ What fraction of the figure would be colored orange if the pattern continued forever?⁠

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4 years

The figure is made of regular octagons and squares. Which triangles have the greater total area? A. The four orange triangles B. The eight blue triangles C. The two sets of triangles have equal areas.

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5 years

The 16 x 24 rectangle above is cut up into 4 triangles, 3 of which have perimeters of 48, 36, and 56. What is the perimeter of the remaining orange triangle? Hint: Add up the perimeters of the other triangles!

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