Welcome to tuxgraphics.org

no preservatives added blog | computer and graphics corner | herbal treasure chest | splash into math | periodic table

multiply by 9 with your fingers

Gini and Karl learn the magic of 9 with their flippers

It was a quiet afternoon at the Penguin Café. Gini was doodling numbers in the frost on the window, while Karl tried to memorize the multiplication table.

"I always mix up the 9s," Karl admitted. "There are so many of them!"

Gini grinned. "Then it's time for a penguin trick - no chalk, no calculator... just your flippers."

They both held out their two flippers in front of them, ten fingers in a row.

"Let's give every finger a number from 1 to 10," said Gini. "Left to right, nice and tidy."

"Now," Gini continued, "pick the number you want to multiply by 9."

Karl chose 3.

"Fold down the finger with the number 3," said Gini.

Karl looked at his flippers.

The fingers to the left showed the tens.

The fingers to the right showed the ones.

Karl counted. "Two fingers on the left... seven on the right."

"So that makes 27," Gini said. "And 3 * 9 = 27!"

Karl's eyes widened. "That was fast!"

They kept going:

Fold finger 1 → 0 on the left, 9 on the right → 9

Fold finger 2 → 1 and 8 → 18

Fold finger 4 → 3 and 6 → 36

Fold finger 5 → 4 and 5 → 45

Each time, the pattern stayed the same.

Flip and Pip joined in, folding and unfolding fingers. Even Waddle counted carefully, and dolphin Walter tried with his fins (with mixed results 🐬).

The Pattern Behind the Magic 🔍

Karl suddenly noticed something.

"Hey! Every time we move one finger over, the number on the right goes down by one..."

"...and the number on the left goes up by one," Gini finished.

"And look!" Karl added.

"The two digits always add up to 9!"

Gini nodded. "That's the secret of multiplying by 9. The hands show the pattern for us."

Chef Pebble watched from behind the counter.

"Very useful," she said. "Especially when counting café prices without a calculator."

Karl wiggled his fingers proudly.

"Who needs memorizing when you have flippers?"

And from that day on, whenever the number 9 appeared, Gini and Karl just held out their hands - and let their fingers do the thinking.

After everyone finished folding fingers and giggling, Flipflop adjusted his tiny glasses.

"Okay," he said, "the trick is cool - but why does it work?"

The café grew quiet.

Flipflop lined up his fingers again.

"When you multiply by 9, you're really doing this: 9 = 10 - 1."

Karl frowned. "So... almost ten?"

"Exactly!" said Flipflop.

"When we count our fingers, we secretly count in tens."

He pointed to the folded finger.

"If you fold down finger number n, then:

there are n - 1 fingers on the left → that's the tens

there are 10 - n fingers on the right → that's the ones"

Gini's eyes lit up. "So for 7 * 9..."

"Six fingers on the left and three on the right," Flipflop said. "Which makes 63."

"And the digits always add up to 9," Karl added, "because one side goes up by one while the other goes down by one!"

Flipflop nodded. "The hands aren't doing magic. They're showing the pattern."

Chef Pebble smiled. "That's the best kind of trick - the kind that teaches."

Here is the trick again with a few pictures:

To learn how to multiply a number from your multiplication table by 9 you can use your hands to help you.

Put your 2 hands with your 10 fingers in front of you and give every finger a numbers like this:

Now take the number you want to multiply with and take that finger away.

Count the fingers to the right of this fingers. These are the ones. Then count the fingers to the left. These are the tens. Put them together and you have the result.

1 * 9

If you take the finger with the number 1 away you can count 9 fingers to the right and 0 to the left so the correct answer is 9.

2 * 9

If you take the finger with the number 2 away you can count 8 fingers to the right and 1 to the left so the correct answer is 18.

3 * 9

If you take the finger with the number 3 away you can count 7 fingers to the right and 2 to the left so the correct answer is 27.

4 * 9

If you take the finger with the number 4 away you can count 6 fingers to the right and 3 to the left so the correct answer is 27.

5 * 9

If you take the finger with the number 5 away you can count 5 fingers to the right and 4 to the left so the correct answer is 45.

6 * 9

If you take the finger with the number 6 away you can count 4 fingers to the right and 5 to the left so the correct answer is 54.

7 * 9

If you take the finger with the number 7 away you can count 3 fingers to the right and 6 to the left so the correct answer is 63.

8 * 9

If you take the finger with the number 8 away you can count 2 fingers to the right and 7 to the left so the correct answer is 72.

9 * 9

If you take the finger with the number 9 away you can count 1 finger to the right and 8 to the left so the correct answer is 81.

10 * 9

If you take the finger with the number 10 away you can count 0 fingers to the right and 9 to the left so the correct answer is 90.

Did you notice that to the right you always counted one less and to the left you always counted one more? And if you took the sum of the digits they always added up to 9?