Math Avenue : Learn Math By Examples

Concentric Circles Challenge

Video solution on our YT channel
https://www.youtube.com/channel/UC9V7YZifrdxPnQYm13YIiVw?sub_confirmation=1

Concentric Circles Challenge | #geometry MathicsZahlenInternational Mat...

Let Socrates show us the math insight from that picture.

Socrates: Boboy, look at the triangle on the left. Do not look at the formula yet. What do you notice?

Boboy: Every row begins and ends with 1.

Socrates: Good. Now look at the 6 in Row 4. Where did it come from?

Boboy: From the two 3s directly above it.

Socrates: And 3 + 3?

Boboy: 6.

Socrates: Look at the 4 beside it. What numbers are directly above it?

Boboy: 1 and 3.

Socrates: And their sum?

Boboy: 4.

Socrates: So if I covered Row 5, could you reconstruct it from Row 4?

Boboy: I think so.

Socrates: Try.

Boboy: It begins with 1.

Then 1 + 4 = 5.

4 + 6 = 10.

6 + 4 = 10.

4 + 1 = 5.

And it ends with 1.

So Row 5 is:

1, 5, 10, 10, 5, 1.

Socrates: Excellent. You have just built another row of Pascal’s Triangle.

Now look at the table on the right.

What is the sum of Row 0?

Boboy: 1.

Socrates: Row 1?

Boboy: 2.

Socrates: Row 2?

Boboy: 4.

Socrates: Row 3?

Boboy: 8.

Socrates: Continue.

Boboy: 16, 32, 64.

Socrates: Forget Pascal’s Triangle for a moment.

Just look at these numbers:

1, 2, 4, 8, 16, 32, 64.

What is happening?

Boboy: Each one is twice the one before it.

Socrates: Can you write them as powers of 2?

Boboy:

1 = 2⁰

2 = 2¹

4 = 2²

8 = 2³

16 = 2⁴

32 = 2⁵

64 = 2⁶.

Socrates: Now compare the exponent with the row number.

Boboy: They are the same!

Socrates: Then finish this sentence:

The sum of all the numbers in Row n appears to be…

Boboy: 2ⁿ.

Socrates: Did I give you that rule?

Boboy: No.

Socrates: Where did you get it?

Boboy: From the pattern in the picture.

Socrates: Good.

But seeing a pattern gives us a conjecture.

Mathematics asks another question:

Why should it continue?

Boboy: Hmm.

Socrates: Let us look again.

Suppose one number in a row is 6.

When we build the next row, where does that 6 contribute?

Boboy: To the number below it on the left and the number below it on the right.

Socrates: So how many times does that 6 contribute to the next row?

Boboy: Twice.

Socrates: And what about every other number in the row?

Boboy: The same thing!

Socrates: Then if every number contributes twice to the next row, what happens to the total?

Boboy: It doubles!

Socrates: Exactly.

So if Row 5 has a sum of 32…

Boboy: Row 6 must have a sum of 64.

Socrates: And the next?

Boboy: 128.

Socrates: And the next?

Boboy: 256.

Socrates: Now we are no longer merely noticing that the totals double.

We can see why they double.

Now look at the challenge at the bottom.

The picture stops at Row 6.

Suppose I ask you for the sum of all the numbers in Row 100.

Would you construct all one hundred rows?

Boboy: No.

Socrates: Would you calculate every number in Row 100 and add them?

Boboy: Definitely not.

Socrates: Then what does what we have discovered tell you?

Boboy: The sum of Row 100 is 2¹⁰⁰.

Socrates: Which answer?

Boboy: B.

Socrates: Very good.

Boboy: But Socrates, why do mathematicians care so much about this triangle?

Socrates: A better question.

Suppose there are five students and you must choose two of them for a team.

How many different pairs can you make?

Boboy: Ten.

Socrates: Look at Row 5.

Boboy: 1, 5, 10, 10, 5, 1.

There is 10.

Socrates: Coincidence?

Boboy: I am beginning to suspect that you will say no.

Socrates: I would rather you discover that yourself.

Those numbers count combinations: the number of ways we can choose things.

Now, have you studied this?

(a + b)²

Boboy: Yes.

a² + 2ab + b².

Socrates: Ignore the letters for a moment. What are the coefficients?

Boboy: 1, 2, 1.

Socrates: Find them in the picture.

Boboy: Row 2!

Socrates: And what about

(a + b)³?

Boboy: That gives coefficients 1, 3, 3, 1.

Wait.

That is Row 3.

Socrates: So where have we now found the same triangle?

Boboy: In number patterns, combinations, and algebra.

Socrates: And combinations are one of the foundations of probability.

Boboy: So Pascal’s Triangle is not just a triangle of interesting numbers.

Socrates: Exactly.

Its importance is not merely in the numbers you see.

It is in the mathematical ideas those numbers connect.

Boboy: Is that why it is called Pascal’s Triangle? Did Pascal invent it?

Socrates: Another good question.

No.

The pattern was studied by mathematicians in different civilizations centuries before Blaise Pascal. It appeared in mathematical work in places such as China, Persia, and India.

Pascal later studied its properties systematically and developed important connections involving combinations and probability.

Boboy: So even the history of the triangle has a pattern of connections.

Socrates: Mathematics often does.

And we have barely opened this triangle.

Look at its diagonals.

Look at its symmetry.

There are other familiar number patterns hiding there.

Boboy: Which ones?

Socrates: I would rather not tell you.

Boboy: Why?

Socrates: Because then you would know what I see.

I want to know what you see.

The powers of 2 were not the end of the lesson, Boboy.

They were the door.

Now look again at the picture.

What pattern can you find in Pascal’s Triangle that Socrates and Boboy did not discuss?

Write what you see in the comments.

MATH AVENUE
Visual Math. Deeper Thinking.

MATH IN PICTURES
See it. Understand it. Remember it.

Once you see it, you cannot unsee it.

MATH IN PICTURES #3
Consecutive Squares Tell a Simple Story

Let Socrates show us the math insight from the picture.

Socrates: Boboy, keep looking at the picture above. Don’t calculate yet. What do you see in the first pair of squares?

Boboy: A 2 × 2 square and a 1 × 1 square.

Socrates: Good. Now use the picture. What remains when the smaller square is removed from the larger one?

Boboy: Three little squares.

Socrates: Can you see another way to count those three?

Boboy: Yes. Two across the top and one down the side.

Socrates: So what does the picture tell you?

Boboy: 2 + 1 = 3.

Socrates: Now move to the next diagram. What do you see?

Boboy: Three across the top and two more down the side.

Socrates: So?

Boboy: 3 + 2 = 5.

Socrates: Keep following the pictures. What comes next?

Boboy: 4 + 3 = 7.

Socrates: And after that?

Boboy: 5 + 4 = 9.

Socrates: And the next?

Boboy: 6 + 5 = 11.

Socrates: Why is that 6 + 5 and not 6 + 6?

Boboy: Because the corner square has already been counted in the row of six. So there are only five more squares going down the side.

Socrates: Excellent. Now look at all the diagrams together. What pattern do you see?

Boboy: Each new border is the sum of two consecutive numbers.

Socrates: Which two numbers?

Boboy: The side lengths of the two consecutive squares.

Socrates: Then, without calculating the squares, what can you say about

6² − 5²?

Boboy: The picture shows me that

6² − 5² = 6 + 5 = 11.

Socrates: Why?

Boboy: Because 6² − 5² is exactly the L-shaped border added when a 5 × 5 square becomes a 6 × 6 square.

Socrates: Now let’s see whether what you discovered works for any two consecutive squares.

Suppose the smaller square has side n.

What is the side of the next square?

Boboy: n + 1.

Socrates: Look back at the picture. If the same pattern continues, what should

(n + 1)² − n²

equal?

Boboy: The two consecutive side lengths added together:

(n + 1) + n.

Socrates: Which simplifies to?

Boboy: 2n + 1.

Socrates: Did you memorize that formula today?

Boboy: No.

I saw why it works.

Socrates: Then keep the picture in front of you. One last question:

100² − 99² = ?

Boboy: Oh.

I don’t need to calculate 100².

I don’t need to calculate 99².

I only need to see the border.

YOUR TURN:

Look back at the picture.

100² − 99² = ?

Can you explain WHY without multiplying?

MATH AVENUE
Visual Math. Deeper Thinking.

MATH IN PICTURES
See it. Understand it. Remember it.

Once you see it, you cannot unsee it.

Let Socrates show us the math insight from that image.

Socrates: Boboy, look at the yellow staircase in the picture. What do you see?

Boboy: Five rows.

Socrates: Count the squares in each row.

Boboy: 1, 2, 3, 4, and 5.

Socrates: So the yellow staircase shows us:

1 + 2 + 3 + 4 + 5

Do you know how many squares there are altogether?

Boboy: I could add them.

Socrates: You could. But today, let us try not to add.

Boboy: Then how will we know?

Socrates: Look at the picture. Make another staircase exactly like the first one. Turn it around and fit the two staircases together.

What shape do they make?

Boboy: A rectangle.

Socrates: Good. Now look at the completed rows. How many squares are across each row?

Boboy: Six.

Socrates: Show me why.

Boboy:

1 + 5 = 6
2 + 4 = 6
3 + 3 = 6
4 + 2 = 6
5 + 1 = 6

Every row has 6 squares!

Socrates: And how many rows does the rectangle have?

Boboy: 5.

Socrates: So what does the picture tell you about the whole rectangle?

Boboy:

5 rows × 6 squares = 30 squares.

Socrates: But look carefully. How many identical staircases make that rectangle?

Boboy: Two.

Socrates: Then how much of the rectangle belongs to one staircase?

Boboy: Half.

So:

30 ÷ 2 = 15.

Socrates: And what did one staircase represent?

Boboy:

1 + 2 + 3 + 4 + 5

So now I can see that:

1 + 2 + 3 + 4 + 5 = 15.

Socrates: Excellent.

But perhaps the picture is showing us more than just the answer to one problem.

Suppose the staircase did not stop at 5.

Suppose it had n rows.

In our picture, 5 was the number of rows.

So what replaces 5?

Boboy: n.

Socrates: Now look again at how the two staircases completed each other.

With 5 rows, the completed width was:

5 + 1 = 6.

If there were n rows, what would the completed width become?

Boboy:

n + 1.

Socrates: Exactly.

So our 5-row picture says:

5 rows × (5 + 1) squares

And the same picture with n rows would say:

n rows × (n + 1) squares.

Therefore, the two staircases together contain:

n × (n + 1)

squares.

Socrates: But do we want both staircases?

Boboy: No. We want only one.

Socrates: And one staircase is what fraction of the rectangle?

Boboy: One-half.

So we divide by 2!

n × (n + 1) ÷ 2

Socrates: And what does that one staircase represent?

Boboy:

1 + 2 + 3 + … + n

Socrates: Then the picture itself gives us:

1 + 2 + 3 + … + n = n(n + 1) ÷ 2

Socrates: Excellent, Boboy.

We did not begin with a formula and try to prove it.

We looked at the picture until the formula appeared.

Now one last challenge.

Look at the picture again.

What if the staircase had 100 rows?

1 + 2 + 3 + … + 100 = ?

Boboy: Wait… don’t tell me.

I can see it now.

MATH AVENUE
Visual Math. Deeper Thinking.

MATH IN PICTURES
See it. Understand it. Remember it.

Once you see it, you cannot unsee it.

After a year away, Math Avenue is back.

But we’re not coming back just to give you more problems to solve. We’re coming back to help you see mathematics differently.

Welcome to MATH IN PICTURES #001.

Sometimes, a picture can reveal what a page of equations cannot.

Look closely at the squares. Each new odd number adds another layer, and a beautiful pattern begins to appear:

1 + 3 + 5 + ··· + (2n − 1) = n²

Now take a moment and look at the pattern.

Can you see the answer to

1 + 3 + 5 + ··· + 99 = ?

Try to find it without adding all the numbers.

There’s a wonderfully simple way to see it.

Share what you notice in the comments. And if you found a different way, we’d love to see that too.

MATH AVENUE
Visual Math. Deeper Thinking.

MATH IN PICTURES
See it. Understand it. Remember it.

THE TRAGEDY OF MANY GRADUATE SCHOOL THESES

When the Thesis Is Finished Before the Question Has Truly Begun

Socrates entered a graduate school classroom.

Around him sat students with laptops open, surrounded by journal articles, citation managers, and documents with filenames such as:

Thesis_Final.docx

Thesis_Final_Revised.docx

Thesis_Final_Revised_FINAL.docx

and, after several meetings with the adviser,

Thesis_Final_Revised_FINAL_USE_THIS_ONE.docx.

At the front of the room stood the professor.

“Today,” she announced, “we will discuss your thesis proposals.”

Socrates raised his hand.

“Yes?”

“What is a thesis?”

The professor smiled.

“A substantial piece of scholarly research required for the completion of a graduate degree.”

Socrates frowned.

“So the purpose of a thesis is to complete a degree?”

“No, of course not.”

“Excellent. Then what is its purpose?”

“To contribute to knowledge.”

Socrates turned toward the students.

“And what knowledge are you hoping to contribute?”

Silence.

Finally, one student raised his hand.

“I’m still looking for a topic.”

Another student spoke.

“I already have mine.”

Socrates brightened.

“What question troubles you?”

“Oh, I don’t know yet.”

“But you already have a topic?”

“Yes.”

“What is it?”

“The Effects of Technology on Student Academic Performance.”

“And what do you wish to discover?”

“I’m waiting for my adviser to approve my research questions.”

Socrates nodded slowly.

“So you have a topic without a question.”

“Yes.”

“And soon you may have research questions without ever having actually wondered about them.”

The student looked at the professor.

The professor looked at the clock.

Another student raised her hand.

“My problem is Chapter Two.”

“What happens in Chapter Two?”

“The review of related literature.”

“And what is the problem?”

“I only have forty-three sources.”

Socrates looked concerned.

“How many truths are required?”

“Not truths. Sources.”

“Ah.”

“My adviser says I should have at least sixty.”

“And when you reach sixty, will the problem become clearer?”

“Not necessarily.”

“Will your argument become stronger?”

“Hopefully.”

“Will you know something you did not know before?”

The student hesitated.

“I’ll have sixty sources.”

Socrates nodded.

“A formidable achievement.”

The professor decided to intervene.

“Perhaps a historical example would help.”

She projected a photograph onto the screen.

A young man appeared.

“Claude Shannon,” she said.

Socrates looked at the photograph.

“What did he study?”

“Electrical engineering and mathematics.”

“And what was his thesis about?”

“In 1937, while completing his master’s degree at MIT, Shannon wrote a thesis called A Symbolic Analysis of Relay and Switching Circuits.”

One student whispered:

“That sounds terrible.”

The professor laughed.

“It helped change the world.”

The room became quiet.

Socrates turned toward her.

“How?”

“Shannon noticed something.”

Socrates smiled.

“Ah. We have reached a dangerous word.”

“What word?”

“Noticed.”

The professor continued.

“Electrical switching circuits used relays that could be in one state or another.”

“On or off?”

“Yes.”

“Two possibilities.”

“Yes.”

“And?”

“Shannon realized that the behavior of these circuits could be represented using Boolean algebra.”

“What is Boolean algebra?”

“A system of logic developed by George Boole in the nineteenth century. It deals with logical relationships such as true and false, AND, OR, and NOT.”

“So Boole already existed?”

“Yes.”

“And electrical switching circuits already existed?”

“Yes.”

“And Shannon invented neither?”

“No.”

Socrates became interested.

“Then what did he discover?”

The professor thought for a moment.

“The connection.”

Silence.

Socrates walked toward the board.

On one side he wrote:

BOOLEAN LOGIC

On the other:

ELECTRICAL SWITCHES

He drew a line between them.

“That was the great insight?”

“In essence, yes.”

Socrates stared at the line.

“That seems rather short.”

The professor smiled.

“The implications were not.”

Shannon showed that Boolean algebra could be used systematically to analyze and design relay and switching circuits.

True and false.

One and zero.

On and off.

Abstract logic could become electrical circuitry.

And a mathematical foundation for modern digital circuit design was laid.

One student slowly closed his laptop.

“So his master’s thesis helped make digital computing possible?”

“It became one of the foundational works behind digital circuit design.”

Socrates turned toward the professor.

“And how old was Shannon?”

“Twenty-one when he completed the thesis.”

Several graduate students suddenly became uncomfortable.

Socrates noticed.

He decided not to help.

After a moment, he asked:

“How many citations did Shannon have?”

The professor laughed.

“That is not the point.”

“How many pages?”

“Also not the point.”

“What research design?”

“Socrates.”

“What font?”

The students laughed.

“What similarity index?”

More laughter.

“Did he have five research questions?”

“Socrates.”

“Did he distribute questionnaires to three hundred electrical relays?”

Now even the professor laughed.

“No.”

“Did the relays answer using a five-point Likert scale?”

“No.”

“Strongly Agree, Agree, Neutral, Disagree, Strongly Disagree?”

“No.”

Socrates shook his head.

“Then I fear this thesis might encounter difficulties in some universities.”

The professor erased part of the board.

“To be fair, Socrates, graduate research needs structure.”

“I agree.”

“Students need methodology.”

“Certainly.”

“They must understand previous scholarship.”

“Of course.”

“They need evidence.”

“Absolutely.”

“And standards protect academic rigor.”

Socrates nodded.

“Then perhaps none of these things is the problem.”

“What is?”

Socrates pointed again at the line connecting Boolean logic and electrical switches.

“The problem begins when the structure meant to support inquiry becomes a substitute for inquiry.”

The room became quiet.

“When methodology becomes more important than curiosity.”

He looked at the students.

“When students learn how to format a question before learning how to be troubled by one.”

Then he looked toward the shelves filled with bound theses from previous years.

“When the purpose of research quietly becomes proving that research was performed.”

A student at the back raised her hand.

“But not everyone is Claude Shannon.”

“Of course not.”

“Most theses will never change the world.”

“Certainly.”

“Then isn’t the comparison unfair?”

Socrates smiled.

“My dear student, I am not asking every master’s thesis to help create the digital age.”

He pointed toward her laptop.

“I am asking whether the person writing it is genuinely trying to discover something.”

She became quiet.

“That something may be very small,” Socrates continued.

“A better way to teach fractions.

A pattern explaining why patients miss appointments.

A question about a poem that previous readers overlooked.

A small improvement in an algorithm.

A contradiction in an accepted explanation.

A problem in one community that nobody has carefully studied.”

He paused.

“The size of the discovery is not what makes inquiry real.”

“What does?”

“The desire to know.”

The first student looked again at his proposal.

The Effects of Technology on Student Academic Performance.

Socrates sat beside him.

“Why did you choose this?”

“Because there are many studies about it.”

“That explains why others studied it.”

Socrates leaned closer.

“Why do you want to study it?”

The student did not answer immediately.

Finally he said:

“I teach.”

Socrates waited.

“And?”

“I’ve noticed something strange.”

The professor looked up.

Socrates smiled.

There was that dangerous word again.

“Go on.”

“My students have access to more information than any students I’ve ever taught.”

“And?”

“They can find answers almost instantly.”

“And?”

The student hesitated.

“But some of them seem less willing to stay with a difficult problem.”

The room became quiet.

Socrates pushed the proposal back toward him.

“There.”

“There what?”

“Your thesis.”

The student stared at the document.

“But that’s not my title.”

“I know.”

“That’s not one of my approved research questions.”

“I know.”

“I haven’t even determined the methodology.”

“Wonderful.”

“Wonderful?”

“You have finally reached the beginning.”

The professor looked again at Shannon’s photograph.

“So what exactly are you suggesting?”

Socrates stood.

“That perhaps we have reversed the order.”

“What order?”

“We sometimes begin with the requirements.”

He counted them on his fingers.

“Choose a topic.

Write the title.

Prepare Chapter One.

Collect the literature.

Select the methodology.

Gather the data.

Defend the thesis.

Graduate.”

He stopped.

“And somewhere in that excellent sequence, we hope curiosity appears.”

The professor smiled.

“And Shannon?”

“Shannon appears to have begun somewhere else.”

Socrates walked back to the board.

He wrote:

NOTICE.

Then:

WONDER.

Then:

CONNECT.

Then:

TEST.

And finally:

EXPLAIN.

He stepped away.

“Perhaps research begins here.”

The professor looked at the board.

“But eventually Shannon still had to write the thesis.”

“Of course.”

“He still had to demonstrate his reasoning.”

“Yes.”

“He still needed mathematics.”

“Certainly.”

“He still needed rigor.”

“More than ever.”

Socrates smiled.

“I have never argued against rigor.”

He pointed toward the board.

“I merely prefer rigor in the service of a question rather than a question invented in the service of rigor.”

The class ended.

The students began closing their laptops.

Socrates remained beside the photograph of the twenty-one-year-old Claude Shannon.

“One final question,” he said.

Everyone stopped.

“So Shannon did not begin by asking how to finish a master’s degree?”

“Apparently not,” said the professor.

“He began with something he noticed?”

“Yes.”

“He saw a connection between ideas others already knew?”

“Yes.”

“And then followed that connection rigorously?”

“Yes.”

“And his master’s thesis helped establish a foundation upon which modern digital computing would grow?”

“Yes.”

Socrates looked around the room.

“Then perhaps the tragedy of graduate research is not that some students fail to finish their theses.”

He looked toward the shelves filled with beautifully bound volumes.

“Perhaps the greater tragedy is that a thesis can be finished before a question has truly begun.”

The room was silent.

One student slowly reopened his laptop.

He stared at the title he had chosen because there were plenty of related studies.

Then he deleted it.

And for the first time in graduate school, he did not ask:

“What topic can I finish?”

He asked:

“What have I noticed that I genuinely want to understand?”

Socrates smiled.

Now the thesis had begun.