Assuming the Pattern Continues What Are the Next Three Terms in the Pattern C E G I K

How Many

Thinking With Words And Visuals

Radius Of The Circle

The radius of the circle is $\large 1 m$ . What is the area of the shaded region

Correct Answer

As suggested by the animation, the leaf shape is really a distorted circle and has exactly the same area of the circle.

Notice that the base of the rectangle is half the circumference of the circle and the height is two time the radius.

Also, recall that the diagonal bisects the rectangle. Thus, the area of the rectangle equals to the area of the leaf shape plus two times the shaded area.

$\large A_{rectangle}=2A_{shaded}+A_{leaf}$

$\large A_{shaded} =frac{1}{2}(A_{rectangle} - A_{leaf})$

$\large =\frac{1}{2}(bh- A_{circle})$

$\large =\frac{1}{2}\left( \left(\frac{1}{2}(2 \pi r)\right)(2r)-\pi r^2\right)$

$\large =\frac{1}{2}(2\pi r^2-\pi r^2)$

$\large =\frac{\pi r^2}{2}$

$\large =\frac{\pi}{2} m^2$

Square In A Square

The outer figure is a square and the lines bisect the opposite sides.

What is the area of the central white square?

Correct Answer

Let the side length of the white square be $x$ .

By Pythagoras' theorem,

$x^2+(x+x)^2=(5+5)^2$
$x^2+(2x)^2=(10)^2$
$x^2+4x^2=100$
$5x^2=100$
$x^2=20$

$\therefore\;\; A =x^2=20$

Answer is 20

Who Won The Race

Four runners compete in a race: Annie, Becca, Carlos, and Dante.

After some confusion at the finish line, it's unclear what the final finishing order was, but the following information is known:

Dante finished before Annie.
Becca wasn't third.
There were two runners between Annie and Carlos.

Who won the race?

Correct Answer

We can use a table to solve this problem:

First, the first clue says that Dante finished before Annie so Dante is definitely not last and Annie is definitely not first.

Next, it says that Becca wasn't third so we can cross that box out.

Finally, it says that there were two runners between Annie and Carlos. That means that both Annie and Carlos couldn't have been $2nd$ or $3rd$ .

Now we can see that Annie must be $4th$ .

That makes Dante $3rd$ and Becca $2nd$ , leaving Carlos as $1st$ .

What Is The Area Of The Purple-shaded Region, If The Sides Of The Smallest Square Are Length 2

What is the area of the purple-shaded region, if the sides of the smallest square are length $\large 2$ ?

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Correct Answer

Partition the diagram into $\large 2\times 2$ triangles, which each have an area of $\large \frac{1}{2}.2.2=2$ .

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The purple-shaded region is made up of $\large 17$ triangles with an area of $\large 2$ , so its area is

$\large 17.2=34$

Area Of A Green Leaf

The green, leaf-shaped area below is the region of overlap between two circles of radius $\large 2$ that are centered, respectively, at the two opposite corners of the $\large 2 \times 2$ square. What is the area of this green region?

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Correct Answer

Correct Answer: $\large 2\pi -4$

Cut the leaf along the square's diagonal, and rotate the bottom section as follows:

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Now the green region is the difference between the semi-circle with a radius of $\large 2$ and a triangle with a base of and a height of $\large 2$ which makes the area

$\large A=\frac{1}{2} \pi2^2-\frac{1}{2}\times 4\times 2 =2\pi -4$

Infinitely-zig-zag

What is the length of this infinitely-zig-zagging red line?

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Correct Answer

Correct Answer: 15

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Define the ratio: $\large 3:2 :: (x+3):(x)$ OR $\large 3$ is to $\large 2$ is proportional to $\large (x+3)$ is to $\large (x)$

Multiply both: $\large 3 \times x = 2\times (x+3)$

Solve further: $\large 3x=2x+6$

$\large x=6$
As shown in diagram, $\large 2$

Length of red line = $\large (x)+(x+3)=(6) +(6+3)=15$

Probability By Outcomes

You are at a charity event and have purchased $\large 1$ ticket for a raffle.

Prior to the drawing, you are told that there are $\large 200$ people at the event (including yourself), and about $\large \frac{3}{4}$ of the people at the event purchased tickets to the raffle. Of those people, equal numbers of people purchased $\large 1, 2,$ and $\large 3$ tickets, respectively.

Based on these estimations, and assuming only one winner, what is the probability that you win the raffle?

Hint: What is the sample size of the tickets sold?

Find The Sum Of All Solutions To The Equation

Is This Triangle Right, Acute, Or Obtuse

If these three segments are connected end-to-end to form a triangle, will the triangle be acute, right, or obtuse?

Correct Answer

Correct answer:Obtuse

Let's start by checking if it's a right triangle. If it were right, then the longest side (length 8) would need to be the hypotenuse, $\LARGE C$ and the other two sides (lengths 5 and 6) would need to be the two legs, $\LARGE a$ and $\LARGE b$ which meet at a right angle.

So we can test if the triangle is right by checking if the Pythagorean identity is true given the lengths of the three sticks: does $\LARGE a^2+b^2=c^2?$

We have $\LARGE a^2+b^2+5^2+6^2=25+36=61$

Therefore , the triangle is not right.

Also, we can see from these calculations that $\LARGE c^2>a^2+b^2.$

This means that the triangle will be obtuse because the "extra" length of the longest side stretches the opposite angle to a measure greater than $\LARGE 90^0$

How Would You Know What These Angles Add Up To

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What is the sum of all of the angles that are shaded green?

Correct Answer

Correct Answer: $\LARGE 720$

There are six separate triangles in the diagram, each with a total of $\LARGE 180^0$

Therefore the total number of degrees between the six triangles is $\LARGE 1080^0$ .

The six unshaded angles in the center of the figure sum to $\LARGE 360^0$ .

All of the green angles must sum to $\LARGE 1080-360=720$ .

Blue X Orange = What

What Is The Length Of The Red Perimeter

What is the length of the red perimeter of this figure?

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Correct Answer

Here is a way to solve this puzzle in $\huge 3$ steps:

Step 1

First, what do we know initially just looking at the diagram in the problem statement?

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Step 2
Draw in a line across the figure that goes through both of the circle centers and then draw in lines from each of the centers up to the top and lower peaks of the red perimeter where circles $\huge A$ and $\huge B$ meet. Extend these lines all the way across the circles:

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Now we can see the following:

Step 3

We can now say that the red perimeter is two thirds of the rim of circle $\huge A$ and two thirds of the rim of circle $\huge B$ .

Therefore, altogether, the perimeter will be $\huge \frac{4}{3}$ the circumference of either circle $\huge A$ or $\huge B$ ,

$\huge \frac{4}{3} \times 6 \pi = 8\pi$

Geometry Warmup - Angles And Lines

How Many Isosceles Triangles

How many isosceles triangles are drawn in this regular pentagon?

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Correct Answer

There are $\huge 5$ isosceles triangles in the diagram given:

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The first $3$ triangles can be shown to be isosceles by the symmetric properties of a regular pentagon. The last $2$ triangles can be shown to be isosceles by finding that the base angles are congruent.

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Using the diagram above, $\huge \fn_phv AB$ and $\huge BC$ are sides of a regular pentagon, so they are congruent, which means $\huge \triangle ABC$ is an isosceles triangle. Likewise, $\huge BC$ and $\huge CD$ are sides of a regular pentagon, so they are also congruent, which means $\huge \triangle BCD$ is an isosceles triangle.

Since $\huge \triangle ABC$ is an isosceles triangle, $\huge \angle BAC =\angle BCA$ , and since the angles of a triangle add up to $\huge 180^0$ and since the interior angle of a regular pentagon is $\huge 108^0$ ,. Likewise, from isosceles $\huge \angle BCD , \angle CBD =\angle CDB=36^0$ , . Since $\huge \angle EBC =\angle ECB =36^0, \; \triangle BCE$ is an isoceles triangle.

Since the angles of $\huge \triangle BCE$ add up to $\huge 180^0$ , $\huge \triangle BEC =180^0-36^0-36^0=108^0$ , which means $\huge \angle AEB =72^0$ and $\huge \angle DEC =72^0$ . Since $\huge \angle ABC = \angle ABE + \angle EBC \;,$ $\huge \angle ABE = 180^0-36^0=72^0.$

Likewise, $\huge \angle DCE=108^0-36^0=72^0.$

Since $\huge \angle ABE = \angle AEB, \triangle ABE$ , is an isosceles triangle, and since ,

$\huge \angle DCE=\angle CED\;, \triangle DCE$ is an isosceles triangle.

Which Shape Has A Greater Perimeter

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Rectangular pieces of square $\huge A$ are removed to obtain shape $\huge \fn_phv B$ . (They both still have the same overall height and width.) Which shape has the greater perimeter? (Note: All angles are right angles.)

Correct Answer

Adding together all of the horizontal line segments in figure $\huge \fn_phv B$ yields the same total length as the two horizontal sides of figure $\huge A$ .

Similarly, adding together all of the vertical line segments in figure $\huge \fn_phv B$ yields the same total length as the two vertical sides of figure $\huge A$ .

Therefore, the perimeters of $\huge A$ and $\huge \fn_phv B$ are the same.

The Quadratic With A Root Of A Root Of A Quadratic

Suppose that the equation $\huge \fn_phv x^2-ax+1=0$ has real solutions.

Let the greater solution of the equation $\huge x^2-ax+1=0$ be $\huge r$ .

Now let $\huge a=\frac{4991}{25},$ and find the value of $\huge b$ such that the greater solution of the equation

$\huge x^2-bx+1=0 \text{ is } \sqrt{2}$ .

Correct Answer

Instead of directly using $\huge x^2-\frac{4991}{25}x+1=0$ , we will simply use $\huge \fn_phv x^2-ax+1=0$ as a generalization (assuming $\huge a > 0$ of course), and then obtain a formula we can just plug in.

$\huge r^2-ar+1=0$

$\huge a=\frac{r^2+1}{r}$

$\huge \sqrt[r-b]{r}+1=0$

$\huge b=\frac{r+1}{\sqrt{r}}$

$\huge b^2=\frac{r^2+2r+1}{r}$

$\huge b^2=a+2$

$\huge b=\boxed{\sqrt{a+2}}$

Plugging in our value of $\huge a$ , we get that $\huge b$ $\huge =\sqrt{\frac{4991}{25}+2}=\frac{71}{5}=\boxed{14.2}$

Is It Possible

Is it possible to fill each square in with an arithmetic operation $\huge (+,-,\times ,\div)$ so that the right side of the equation is $\huge \fn_phv 101?$

Correct Answer

It is possible. For example, $\huge 10\div 10+10 \times 10 = 101.$

We can evaluate this expression using the order of operations. Division and multiplication are evaluated first, from left to right. That gives us $\huge \fn_phv 1+100=101,$ which is clearly true once we perform the addition.

Alternatively, we could also get a total of $\huge 101$ from $\huge 10 \times 10+10\div 10.$

A Tower Of Threes

Playing With Circle's

Playing With Numbers

Can You Logic This Out

Suppose you are visiting an island withknights who always tell the truth,knaves who always lie, andjokers who can do either.

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What must the islander depicted above be?

Correct Answer

A knight cannot lie and say they are a knave.
A knave cannot tell the truth and say they are a knave.
A joker is free to lie and say they are a knave; the islander must be a joker.

Do We Need This Many Guards

An artist has a bird-shaped gallery. To ensure the museum is fully guarded, he places 4 guards at the corners with red dots so that every inner wall is visible by at least one guard.

Can he fully guard the museum with fewer guards?

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Correct Answer

If you want realistic guards, with a limited field of view which they can continuously monitor, placing both in a corner is best, and can be done with two guards, labeled as filled circles. (The yellow guard can see the yellow, orange and green parts, the blue guard can see the green, dark blue and blue parts in the following image.)

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Alternatively, if your guards have a ridiculously large field of view, they could be placed at different locations, labeled as filled stars, so that they cover the largest area twice. (The yellow guard can see the yellow, orange, green and dark blue parts, the blue guard can see the orange, green, dark blue and blue parts in the image.) An additional advantage of these starred locations would be that the guards are nearest to each other (while maintaining full coverage), so in case of a break-in they can assist each other sooner.

PS: Assuming 360 degree vision, even other non-wall locations are possible, for example in the case of the yellow guard on the line between the yellow circle and star.

If Godzilla Keeps Growing

Compare two Godzillas, identical in proportion, but one is $10$ times taller than the other.

If he keeps growing like this indefinitely, what would happen to him?

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Correct Answer

The weight of Godzilla scales differently from his bone strength. In fact, as his overall size grows, his weight grows faster than his bone strength. Therefore, once he is large enough, his leg bones will break under his body weight, effectively immobilizing him.

This Ball Is Half And Half

A ball that's half wood and half iron rolls on a flat surface.

Which position is most likely when it comes to rest?

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Correct Answer

The center of mass of a solid hemisphere of radius $\huge \fn_phv R$ is $\huge \frac{3}{8}$ $\huge \fn_phv R$ from the center of the sphere normal to the flat surface of the hemisphere.

Since the density of iron is $\huge 7.27\;g/cm^3$ and that of wood is $\huge 0.39 \sim 0.60\;g/cm^3$ , the center of mass of the ball is approximately $\huge \frac{3}{8}$ $\huge \fn_phv R$ from its center in the iron hemisphere (shown as red dot in the figure).

The most stable position of the ball is when it has the lowest potential energy mgh where m is the mass of the ball, g the acceleration due to gravity and h the distance from the flat surface. As m and g are constant, the lowest potential energy occurs when h is smallest. of the four positions. $\huge \boxed{B}$ has the smallest h and hence the most stable position.

What Do You Know About Base Angles Of Isosceles Triangles

What can we say about the base angles of any isosceles triangle?

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Correct Answer

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If the vertex (top) angle measures $\huge \fn_phv x^0$ , then the base angle is $\huge \frac{180^0-x^0}{2}$ , which would be less than $\huge 90^0$ so long as is any value greater than $\huge 0$ .

Therefore, the base angles of any isosceles triangle must be acute angles.

Color The Rest!

When coloring a map, each region must be filled with a single, solid color and no two regions with touching edges can be the same color.

Given that these two regions have already been colored red, what is the minimum number of colors needed (including the red already used) to color the entire map?

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Correct Answer

First note that every remaining region touches at least one of the red regions, so red cannot be used any more.

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Then, notice that each of the numbered regions in the diagram above, touches all of the other $\huge \fn_phv 3$ numbered regions. This means that $\huge 4$ unique coloursmust be used for these regions, as shown. The remaining $\huge 2$ regions can be coloured any of blue, green or purple (in either order).

Therefore, the total number required is $\huge \boxed{5}$ . This isn't a relevant case of the four colour map theorem as some of the map is already completed.

Two Animals Are Skydiving On The Moon

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A whale and a chihuahua dog are in free fall while skydiving on the moon (there is no air resistance).

Which animal has a greater acceleration?

Correct Answer

Objects are said to be in free fall when theonly force acting on them is gravity. Therefore the only force that we need to consider in this situation is the gravity acting on each animal. It's tempting to assume that because the whale has a greater mass than the chihuahua, that it will have a greater acceleration. But it turns out that their acceleration is the same. We can examine the motion of objects in free fall using Newton's Second Law. According to Newton's Second Law, an object's acceleration is:

$\huge \fn_phv a=\frac{F}{m}$

Now we can plug in for the only force acting, the gravitational force, which is expressed as theweight of an object, or the mass of an object multiplied by gravitational acceleration (of the moon), $\huge g_{moon}$

$\huge a=\frac{F}{m}=\frac{W}{m}=\frac{mg_{moon}}{m}$

$\huge a=g_{moon}$

The Solutions To This Problem Are More Beautiful Than You'd Expect

There's A Clever Solution...

In this array puzzle, each shape has a specific value. The number next to each row or column represents the sum of the values in that row or column.

What number should replace the question mark

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Correct Answer

We begin by comparing the rightmost column and the bottom row.

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From this comparison, we see that when we replace a yellow triangle with a red star, the total value decreases by $\huge 5$ , since

$\huge \fn_phv 19-14=5$

Now we can look at the middle row.

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This is almost what we want, except we need to replace the yellow triangle with a red star. As we've already seen, when we do that the value decreases by $\huge 5$ , so the value of the top row will be

$\huge 19-5=14$

Formula One Race Cars

Formula One racecars have an ingenious feature: wings attached to the body similar to an inverted airplane wing.

How can strategically placed wings enable F1 cars to go faster around turns

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Correct Answer

At high speeds, air flowing over and around the car can produce significant forces. Sometimes these forces can cause the driver to lose control.

For example, differences between the flow rate of air over and under the vehicle can causelift(the same principle that an airplane exploits to fly) and affect the grip between the tires and the road. This is particularly dangerous as the cars navigate certain turns of theCircuit de Monacomuch faster than commercially available vehicles would.

But Formula One cars have an ingenious feature: wings attached to the body similar to aninverted airplane wing. How can strategically placed wings enable F1 cars to go faster around turns?

It's Will Always Bounce Back...

Why does this toy always bounce back no matter how hard it's punched

puzzle

Correct Answer

Center of mass for this bodies lies exactly at the body. In what ever direction it is pushed centre of mass position is not changed. Which means centre of mass displacement is 0. This implies all the bodies will come to their initial position.

$\huge \fn_phv x_{cm}=\frac{m_1+m_2x_2+..}{m_1+m_2+...}$

Where $\huge x_{cm}$ is the change in position of centre of mass which is 0.

$\huge \Rightarrow\;\;\;m_1x_1+m_2x_2+...=0$

This says that the sum of product of masses and change in position of corresponding bodies is equal to . This implies that all the bodies' have reached their initial positions.

This Isn't Your Average 99 Cent Store¦

Marie shops at a store where all prices end in 99 cents ($0.99, $1.99, $2.99, etc.). She ends up spending $33.89.

How many items did she purchase?

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Correct Answer

$\huge 1$ item means that the cent value of the total will $\huge \boxed{99}$

$\huge 2$ items means that the cent value of the total will be $\huge 99+99 \equiv \boxed{98}\text{ (mod 100)}$

$\huge n$ times means that the cent value of the total will be $\huge \boxed{100-n}$

$\huge \Rightarrow\;\;100-n=89 \Rightarrow\;\;\boxed{n=11}$

(The next possibility would be $\huge 111$ items to get the correct pence, but $\huge 111 \times 0.99$ is the minimum

which is way more than $\huge 33.89$ )

The Burning Rope

Socks In The Dark

Ten red socks and ten blue socks are all mixed up in a dresser drawer. The 20 socks are exactly alike except for their colour. The room is in pitch darkness and you want two matching socks. What is the smallest number of socks you must take out of the drawer in order to be certain that you have a pair that match

Correct Answer

With two socks it is possible to have one red and one blue. But with three there is always a matching pair since either you will have chosen three of the same colour, or a matching pair and an odd one out. Answer: Three socks.

Lily Pad

You start with a single lily pad sitting on an otherwise empty pond. You are told that the surface area of the single lily pad doubles everyday and that it takes 24 days for the single lily pad to cover the surface of the pond.

If instead of one lily pad you start with eight lily pads (each identical to the single lily pad), how many days will it take for the surface of the pond to become covered?

Correct Answer

If you can figure out the relationship between 8 lily pads and one lily pad you will get the answer! Since one Lily pad doubles every day, after 3 days it will be equivalent to starting with 8 lily pads! Then since one Lily pad covers the whole pond in 24 days and we are starting after three days for 8 lily pads, the answer is 24-3 = 21 days! Answer: 21 days

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Source: https://www.makeageek.com/puzzle/assuming-the-pattern-continues/e3698522

Assuming the Pattern Continues What Are the Next Three Terms in the Pattern C E G I K

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