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Johann Carl Friedrich Gauss Visits the Penguin Café ☕ : Calculate the sum of 1 +2+3 up to 100

The Penguin Café was buzzing again.

Gini was sketching little triangles on a napkin, Karl was stirring his cocoa and Flip, Pip, Flipflop, Waddle and even dolphin Walter had gathered around the big table.

Chef Pebble rang the tiny bell.

"Everyone, today's guest is very special."

A distinguished penguin-sized visitor stepped forward. "My name is Johann Carl Friedrich Gauss," he said politely. "And today, I have a math puzzle for you."

Calculate the sum of 1 + 2 + 3 + ... + 100

Karl blinked. "That's a lot of numbers. Do we really have to add them one by one?"

Penguin Gauss smiled. "That's exactly what my teacher once hoped I would do - just to keep me and my classmates busy. But there's a faster way."

He leaned over the table and started with something smaller.

"Let's first look at the numbers from 1 to 10," Penguin Gauss said.

He imagined them as a triangle of rocks:

o
oo
ooo
oooo
ooooo
oooooo
ooooooo
oooooooo
ooooooooo
oooooooooo

Flipflop tilted his head. "It looks kind of... half empty."

"Exactly!" said Penguin Gauss. "Now watch."

He copied the triangle, rotated it, and placed it next to the first one, filling in the missing space:

oxxxxxxxxxx
ooxxxxxxxxx
oooxxxxxxxx
ooooxxxxxxx
oooooxxxxxx
ooooooxxxxx
oooooooxxxx
ooooooooxxx
oooooooooxx
oooooooooox

Suddenly, everyone could see it.

"It's a rectangle!" Gini exclaimed.

"Yes," said Penguin Gauss. "It has 10 rows and 11 columns, so it contains 10 × 11 = 110 rocks."

Walter clicked happily. 🐬 "But we only want one triangle."

"Right," said Penguin Gauss. "So we divide by 2."

110 / 2 = 55

"And that," Penguin Gauss concluded, "is the sum of the numbers from 1 to 10."

"Now," said Karl, "what about 1 to 100?"

Penguin Gauss didn't even need to draw it.

"If we build the same kind of rectangle, we get

100 * 101 = 10100."

Pip whispered, "That's easy to calculate in your head!"

Penguin Gauss nodded. "And since that rectangle is made of two triangles, we divide by 2."

10100 / 2 = 5050

Chef Pebble raised a mug. "To clever thinking!"

Before leaving, Penguin Gauss posed one more challenge.

"What happens if we add only the uneven numbers, starting with 1?"

Gini started counting: 1, 3, 5, 7, 9, ...

As they added them, something curious appeared.

1 = 1

1 + 3 = 4

1 + 3 + 5 = 9

1 + 3 + 5 + 7 = 16

Waddle gasped. "These are squares!"

"Exactly," said Penguin Gauss. "The sum of the first n uneven numbers is n * n."

Karl grinned. "So the sum of those numbers equals the product n * n?"

Penguin Gauss bowed. "Mathematics is full of beautiful patterns - you just have to look for them."

As Penguin Gauss left the Penguin Café, Gini carefully folded the napkin with the triangles, Karl wrote 5050 on the chalkboard, and everyone agreed:

Sometimes, the smartest way to add numbers... is not to add them at all. 🐧✨

references

Rock groups by Steven Strogatz