Create two columns, one starting with the numbers 12 and 18 and the other with 5 and 5. Continue each column, deriving each new number by adding the two that precede it:

In the Journal of Recreational Mathematics, James Davis writes, “Forming successive pairs with adjoining numbers from each column one finds the ratio of the two numbers in each pair converges to π!” How can this be?

“The alert reader will suspect there is a trick in this method, as I did when π first presented it to me. The labor of several hours of computation coupled with trial and error produced half of the secret of the method. It is obviously based somehow on the fact that φ (the golden mean, which equals , can be closely approximated by the nifty pseudo equation below:”

“Can the reader decipher π’s technique for making herself with φ?”

3139971973786634711391448651577269485891759419122938744591877656925789747974914319
422889611373939731 is prime, whether it’s spelled forward or backward. Further, if it’s cut into 10 pieces:

… each row, column, and diagonal is itself a reversible prime.

Discovered by Jens Kruse Andersen.

In 1922, after the death of his mother, Carl Jung felt “I had to achieve a kind of representation in stone of my innermost thoughts and of the knowledge I had acquired. Or, to put it another way, I had to make a confession of faith in stone.”

He began to build a structure on the shores of Lake Zurich in Switzerland. It began as a regular two-story house, “a maternal hearth,” but over the years he added a towerlike annex with a “retiring room” for withdrawal and contemplation, and a courtyard and loggia.

At 80, after his wife’s death, “I suddenly realized that the small central section which crouched so low, so hidden, was myself!” He added an upper story, an extension of his own personality no longer hidden behind the “maternal” and “spiritual” towers. “Now it signified an extension of consciousness achieved in old age. With that the building was complete.”

The final building, he saw, symbolized the structure of his own psyche, the full emergence of his personality in adulthood. “Unconsciously built at the time, only afterward did I see how all the parts fitted together and that a meaningful form had resulted: a symbol of psychic wholeness.” “At Bollingen,” he wrote, “I am in the midst of my true life, I am most deeply myself.”

I place $20 in a box.

So do you.

Now the box contains $40, and we both know it.

I sell the box to you for $30.

And we both walk away with a $10 profit.

(Thanks, Ben.)

Choose any number and write down its divisors:

14
1 2 7 14

Then write down the number of divisors of each of these divisors:

14
1 2 7 14
1 2 2 4

Now the square of the sum of this last group will always equal the sum of its members’ cubes:

(1 + 2 + 2 + 4)² = 1³ + 2³ + 2³ + 4³

Discovered by Joseph Liouville.

Do holes exist? That is, if a hole is merely a void or vacancy in a surrounding substance, then properly speaking is the hole a thing in itself? Philosopher Roberto Casati writes, “Ask any person to tell you what holes are — ‘real,’ everyday holes, not the abstract holes of geometry – and he will likely elaborate upon absences, nonentities, nothingnesses, things that are not there. Are there such things?”

A hole story from John Timbs’ 1873 A Century of Anecdote:

A gentleman, one Sunday morning, was attracted to watch a young country girl on the high road from the village to the church, by observing that she looked hither and thither, this way and that upon the road, as if she had lost her thimble. The bells were settling for prayers, and there was no one visible on the road except the girl and the gentleman, who recognised in her the errand-maid of a neighbouring farmer. ‘What are you looking for, my girl?’ asked the gentleman, as the damsel continued to pore along the dusty road. She answered, gravely: ‘Sir, I’m looking to see if my master be gone to church.’ Now, her master had a wooden leg.

In number theory, a Holey prime is a prime number made up exclusively of the digits 4, 6, 8, 9, and 0 — digits whose Arabic numerals contain “holes.” Ironically, these get pretty substantial: The largest known specimen is a 4 followed by 16,131 9s.

What’s remarkable about these numbers?

They form a perfect magic square. Each row, column, and diagonal adds to 81.

W.S. Andrews wrote, “Considering its constructive origin and interesting features, this square, notwithstanding its simplicity, may be fairly said to present one of the most remarkable illustrations of the intrinsic harmony of numbers.”