Science & Math

Math Notes

math notes

Where’s the Error?

A man deposited $50 in a savings account, then withdrew it in various sums. When he’d recovered his $50 he was surprised to find $1 left in the account, though it had drawn no interest. When he inquired, the bank produced this ledger:

where's the error

Math Notes

2 + 5 + 6 = 13; 132 = 169
1 + 6 + 9 = 16; 162 = 256

Math Notes

83 = 512; 5 + 1 + 2 = 8
273 = 19683; 1 + 9 + 6 + 8 + 3 = 27


Passing through the quadrangle of Christ Church, Oxford, one day, the classical scholar Gilbert Murray encountered Albert Einstein sitting dreamily in thought.

Murray asked him what he was thinking about.

“I am thinking,” Einstein answered, “that, after all, this is a very small star.”


He who has not lost a thing has it.

You have not lost horns.

Therefore you have horns.


Is this a bad sum?


Not in a mirror:

turnabout reversal

Adapted by Martin Gardner from Henry Dudeney.

Math Notes

9 × 9 + 7 = 88
98 × 9 + 6 = 888
987 × 9 + 5 = 8888
9876 × 9 + 4 = 88888
98765 × 9 + 3 = 888888
987654 × 9 + 2 = 8888888
9876543 × 9 + 1 = 88888888
98765432 × 9 + 0 = 888888888

Relative Logic

You say that you have a dog.

Yes, and a villain of a one, said Ctesippus.

And he has puppies?

Yes, and they are very like himself.

And the dog is the father of them?

Yes, he said, I certainly saw him and the mother of the puppies come together.

And is he not yours?

To be sure he is.

Then he is a father, and he is yours; ergo he is your father, and the puppies are your brothers.

Let me ask you one little question more, said Dionysodorus, quickly interposing, in order that Ctesippus might not get in his word: You beat this dog?

Ctesippus said, laughing: Indeed I do; and I only wish that I could beat you instead of him.

Then you beat your father, he said.

— Plato, Euthydemus

“Small Problem”

Higgamus hoggamus
“Anton Von Leewenhoek
Has a small problem,” con-
Fided his wife.

Doesn’t disturb me; his
Blighting my life!”

— Theodore L. Drachman

Math Notes

2025 = (20 + 25)2
3025 = (30 + 25)2
9801 = (98 + 01)2

A Confederacy of Squares

When asked his age, mathematician Augustus De Morgan used to offer a clue: “I was x years of age in the year x2.” (He was 43 in 1849.)

That quirk puts De Morgan in a pretty exclusive club. Other members include Charles Atlas (who was 44 in 1936) and Jake Gyllenhaal (who will be 45 in 2025). Next up: Babies born in 2070 will be 46 in 2116.

Math Notes


Two business partners asked their lawyer to hold $20,000, making him promise to get both of their signatures before disbursing any of it.

As soon as one partner left town, the other pressed the lawyer for $15,000, citing an emergency. The lawyer reluctantly gave it to him, and he disappeared.

On his return, the other partner was irate, so the lawyer explained that he had donated the $15,000 out of his own pocket.

“Then give me the $20,000 you’re holding,” said the partner.

“All right,” said the lawyer. “Give me the two signatures.”

Math Notes

24 + 14 + 74 + 84 = 6514
64 + 54 + 14 + 44 = 2178

Infinite Composition

In Tristram Shandy, the title character laments that he’ll never be able to finish his autobiography, as he seems to need a year to record each day’s events. “It must follow, an’ please your worships, that the more I write, the more I shall have to write.”

But Bertrand Russell noted that if Shandy’s eventful life had lasted forever, no part of his biography would have remained unwritten — for the hundredth day would be recorded in the hundredth year, the thousandth in the thousandth, and so on. “This paradoxical but perfectly true proposition depends upon the fact that the number of days in all time is no greater than the number of years.”


Rest the ends of a yardstick on your index fingers. Now slowly draw your fingers together, trying to make them meet at some spot other than the center of the stick.

It’s impossible. When either finger leads, it bears more weight, which creates more friction, and the other catches up.

Math Notes

651 × 156 = 372 × 273

The Kruskal Count

Here’s a card trick devised by Rutgers physicist Martin Kruskal. Give a friend a deck of cards and ask her to follow these instructions:

  1. Think of a “secret number” from 1 to 10. (Example: 6)
  2. Shuffle the deck and deal the cards face up one at a time, counting silently as you go.
  3. When you reach the secret number, note the value of that card and adopt it as your new secret number. Aces count as 1; face cards count as 5. (Example: If the 6th card is a 4, then 4 becomes your new secret number.)
  4. Continue dealing, counting silently anew from 1 each time you adopt a new number. Remember the last secret card you reach.

That’s it. You just stand there and watch her deal. When she’s finished, you can identify her final secret card in any way you please, preferably through a grotesquely extortionate wager.

You can do this because you’ve simply played along. When she’s dealing, note the value of an early card and then silently follow the same steps that she is. Five times out of six, your “paths” through the deck will intersect and your final secret card will match hers. That’s far from obvious, though; the trick can be baffling if you refuse to explain it.

Kaprekar’s Constant

Choose four distinct digits and arrange them into the largest and smallest numbers possible (e.g., 9751 and 1579). Subtract the smaller from the larger to produce a new number (9751 – 1579 = 8172) and repeat the operation.

Within seven iterations you’ll always arrive at 6174.

With three-digit numbers you’ll aways arrive at 495.

Math Notes

410 + 610 + 710 + 910 + 310 + 010 + 710 + 710 + 710 + 410 = 4679307774


Abraham de Moivre correctly predicted the date of his own death.

He noted that he was sleeping 15 minutes longer each day and surmised that he would die on the day he slept for 24 hours. That date, he calculated, would be Nov. 27, 1754.

He was right.

Food for Thought

A force of 1 newton is about the weight of an apple.

Burglars Beware

Who says math is too abstract?

The Chvátal Art Gallery Theorem states that if you run an art gallery with n corners, you’ll need n/3 guards (at most) to watch the entire gallery—regardless of its shape.