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# Thinking with Words and Visuals

Dashed lines mean *add*, and solid lines mean *multiply*.

For example, two dashed lines point from ? and , so we add those numbers together to fill in the circle they point at.

What number must go in the yellow marked circle?

## Correct Answer

We can work backwards starting from the we know the circles that lead to it multiply to be , and one of the values is . Since the other value has to be .

Now we know the bottom portion is the question mark needs to be for the addition problem to work.

Incidentally, this process is equivalent to solving for in the equation . It's fairly natural in the charted form to work backwards from the , and this is exactly what needs to be done to solve when the equation is in algebraic form.

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When cubes are assembled, one of the cubes fills the center, leaving cubes visible on the outside. How many "corner" cubes are there (with 3 sides visible), how many are "edge" cubes (with 2 sides visible) and how many are "center" cubes (with 1 side visible)?

## Correct Answer

**8 corner, 12 edge, 6 center**

The "corner" cubes are marked in orange, "edge" cubes are marked in blue, and "center" cubes are marked in magenta in the visualization above.

Each of the smaller cubes in the by by configuration that is touching a vertex of the larger cube has sides visible; there are vertices of the larger cube, so there are corner cubes.

Each of the sides of the larger cube has "edge" cubes as marked where sides are visible.

Each of the sides of the larger cube has "center" cube in the middle with only side visible.

The radius of the circle is . 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.

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 .

By Pythagoras’ theorem,

Answer is 20

Consider the sequence beginning with the five terms below:

Assuming the pattern continues, what is the next term in the sequence?

## Correct Answer

OR

So.. Next will be

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 or .

Now we can see that Annie must be .

That makes Dante and Becca , leaving Carlos as .

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

## Correct Answer

Partition the diagram into triangles, which each have an area of .

The purple-shaded region is made up of triangles with an area of , so its area is

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

## Correct Answer

Correct Answer:

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

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

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

## Correct Answer

**Correct ****Answer:****15**

Define the ratio: OR is to is proportional to is to

Multiply both:

Solve further:

As shown in diagram,

Length of red line =

You are at a charity event and have purchased ticket for a raffle.

Prior to the drawing, you are told that there are people at the event (including yourself), and about of the people at the event purchased tickets to the raffle. Of those people, equal numbers of people purchased and 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?

## Correct Answer

Correct answer:

Of the people, about purchased tickets. purchased ticket, purchased tickets, and purchased tickets. This gives a total of tickets purchased. Thus, with one ticket, the probability to win is

Find the sum of all solutions to the equation

Hint: Don't forget about the case where the base is and the exponent is even!

## Correct Answer

Correct answer: **2**

We must consider three cases:

**The exponent is . **Solving gives and .

checking that this doesn't make the base equal ** **, which it does not.

**The base is . **Solving gives and .

**The base is **and the exponent is even**. **Solving gives and . Substituting these into the exponent shows that the exponent is only even when .

Thus, the sum of all solutions is

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, and the other two sides (lengths 5 and 6) would need to be the two legs, and 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

We have

Therefore, the triangle is not right.

Also, we can see from these calculations that

**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 **

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

## Correct Answer

Correct Answer:

There are six separate triangles in the diagram, each with a total of

Therefore the total number of degrees between the six triangles is .

The six unshaded angles in the center of the figure sum to .

All of the green angles must sum to .

If you **multiply** a blue number by an orange number, what color will the product be?

## Correct Answer

The blue numbers are the odd numbers, and the orange numbers are the even numbers, so the question can be rephrased as follows: If you multiply an odd number by an even number, will the product be odd or even?

The product of an odd number and an even number is *always* even, regardless of the specific values of the numbers. An odd number must be of the form and an even number must be of the form (where and are integers); the product of these numbers is

Since divides this product, it must be even. This reasoning does not depend on the specific values of and so this result is true for all and thus, the product of an odd number and an even number is always even, and therefore, if you multiply a blue number by an orange number, the product will be orange.

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

## Correct Answer

Here is a way to solve this puzzle in steps:

**Step 1**

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

- Because the center segment, is a radius of both circles, we know the circles must be the same size and therefore, that all of the radii in this picture are the same length.
- Both of the two circles, and would have a circumference of because the radius of both circles is given as . However, the answer is not because the red outline is composed of only part of the circumference of each circle. We need to know how much of each circumference has been removed because it lay within the overlap of the two circles.

**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 and meet. Extend these lines all the way across the circles:

Now we can see the following:

- Each of the line segments reaching out from the center of either of the two circles is a radius of either one circle or the other.
- That means that the two triangles with line segment as one side are both equilateral triangles and that the radii lines drawn in are each apart.
- Because there are in a circle, the part of the circumference of each circle cut off by being in the overlapping area is of the circumference of each circle.

**Step 3**

** **We can now say that the red perimeter is two thirds of the rim of circle and two thirds of the rim of circle .

Therefore, altogether, the perimeter will be the circumference of either circle or ,

What is

## Correct Answer

and are supplementary because the angles share a pair of parallel lines.

Therefore,

How many isosceles triangles are drawn in this regular pentagon?

## Correct Answer

There are isosceles triangles in the diagram given:

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

Using the diagram above, and are sides of a regular pentagon, so they are congruent, which means is an isosceles triangle. Likewise, and are sides of a regular pentagon, so they are also congruent, which means is an isosceles triangle.

Since is an isosceles triangle, , and since the angles of a triangle add up to and since the interior angle of a regular pentagon is ,. Likewise, from isosceles , . Since is an isoceles triangle.

Since the angles of add up to , , which means and . Since

Likewise,

Since , is an isosceles triangle, and since ,

is an isosceles triangle.

Rectangular pieces of square are removed to obtain shape . (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 yields the same total length as the two horizontal sides of figure .

Similarly, adding together all of the vertical line segments in figure yields the same total length as the two vertical sides of figure .

Therefore, the perimeters of and are the same.

Suppose that the equation has real solutions.

Let the greater solution of the equation be .

Now let and find the value of such that the greater solution of the equation

.

## Correct Answer

Instead of directly using , we will simply use as a generalization (assuming of course), and then obtain a formula we can just plug in.

Plugging in our value of , we get that

Is it possible to fill each square in with an arithmetic operation so that the right side of the equation is

## Correct Answer

It is possible. For example,

We can evaluate this expression using the order of operations. Division and multiplication are evaluated first, from left to right. That gives us which is clearly true once we perform the addition.

Alternatively, we could also get a total of from

## Correct Answer

can be expressed as . Thus, the expression becomes

I have **3 unit **circles (A unit circle has a radius of **1**) . Two of them are externally tangent to each other, and the third one passes through the tangent point, cutting two symmetrical areas from the two circles, as shown above. What is the shaded area?

## Correct Answer

The shaded area can be split up into regions with right angles and unit legs, and then rearranged into a by rectangle, as shown below.

Therefore, the area of the shaded region is the same as the area of the by rectangle, which is .

I started to write out this equation:

Is the equation true or false.

## Correct Answer

Hence, the answer is

Suppose you are visiting an island with **knights** who always tell the truth, **knaves** who always lie, and **jokers** who can do either.

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.

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?

## 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.)

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.

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

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

## 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.

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

Which position is most likely when it comes to rest?

## Correct Answer

The center of mass of a solid hemisphere of radius is from the center of the sphere normal to the flat surface of the hemisphere.

Since the density of iron is and that of wood is , the center of mass of the ball is approximately 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

*is the mass of the ball,*

**m***the acceleration due to gravity and*

**g****the distance from the flat surface. As**

*h**and*

**m***are constant, the lowest potential energy occurs when*

**g****is**

*h***smallest. of the four positions. has the smallest**

**and hence the most stable position.**

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

## Correct Answer

If the vertex (top) angle measures , then the base angle is , which would be less than so long as is any value greater than .

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

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?

## Correct Answer

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

Then, notice that each of the numbered regions in the diagram above, touches all of the other numbered regions. This means that unique colours **must** be used for these regions, as shown. The remaining regions can be coloured any of blue, green or purple (in either order).

Therefore, the total number required is . This isn't a relevant case of the four colour map theorem as some of the map is already completed.

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 the *only* 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:

Now we can plug in for the only force acting, the gravitational force, which is expressed as the **weight** of an object, or the mass of an object multiplied by gravitational acceleration (of the moon),

The big square is divided into nine congruent squares. Give your answer in degrees.

## Correct Answer

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

## Correct Answer

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

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

Now we can look at the middle row.

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 , so the value of the top row will be

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

## 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 cause **lift**(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 the *Circuit de Monaco*much faster than commercially available vehicles would.

But Formula One cars 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?

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

## 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.

Where is the change in position of centre of mass which is 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.

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?

## Correct Answer

item means that the cent value of the total will

items means that the cent value of the total will be

times means that the cent value of the total will be

(The next possibility would be items to get the correct pence, but is the minimum

which is way more than )

A rope burns non-uniformly for exactly one hour. How do you measure 45 minutes, given 2 such ropes

## Correct Answer

First start burning rope 1 at both ends, and rope 2 at one end only. When rope 1 finishes burning (after 30 minutes) light the other end of rope 2. 45 minutes will be up when rope 2 finishes burning.

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.

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