Awaiting the dawn sat three prisoners wary,
A trio of brigands named Tom, Dick and Mary.
Sunrise would signal the death knell of two;
Just one would survive, the question was who.

Young Mary sat thinking and finally spoke.
To the jailer she said, “You may think this a joke,
But it seems that my odds of surviving till tea
Are clearly enough just one out of three.

But one of my cohorts must certainly go,
Without question, that’s something I already know.
Telling the name of one who is lost
Can’t possibly help me. What could it cost?”

The shriveled old jailer himself was no dummy.
He thought, “But why not?” and pointed to Tommy.
“Now it’s just Dick and me!” Mary chortled with glee,
“One in two are my chances, and not one in three!”

Imagine the jailer’s chagrin, that old elf.
She’d tricked him. Or had she? Decide for yourself.

— Richard E. Bedient, “The Prisoner’s Paradox Revisited,” American Mathematical Monthly, March 1994

1234567891, 12345678901234567891, and 1234567891234567891234567891 are prime.

So are

19
197
1979
19793
197933
1979339
19793393 and
197933933.

And so are

742950290870000078092059247
742950290871010178092059247
742950290872020278092059247
742950290873030378092059247
742950290874040478092059247
742950290875050578092059247
742950290876060678092059247
742950290877070778092059247
742950290878080878092059247 and
742950290879090978092059247.

If the nth term of the Fibonacci series is prime, then n also is prime (where n > 4). For example, the 17th term, 1597, is prime, and 17 is prime.

05/27/2017 UPDATE: Further to the first series, 1979339333 is also prime! (Thanks, Alon.)

Given a book of matches, a box of thumbtacks, and a candle, how can you fix the candle to the wall so that its wax won’t drip onto the table below?

By 1941 Russian botanist Nikolai Vavilov had created the largest seed bank in the world, a collection of 400,000 seeds, roots, and fruits whose genetic material held the future of Soviet agriculture. Unfortunately it was located in Leningrad, which Hitler encircled that summer and began to starve.

The siege of Leningrad lasted two years and cost more than a million lives, and Vavilov’s scientists endured it surrounded by edible plants. “As they slowly starved, they refused to eat from any of their collection containers of rice, peas, corn and wheat,” two survivors remembered in 1993. “They chose torment and death in order to preserve Vavilov’s gene bank.”

The collection filled 16 rooms, in which no one was allowed to remain alone. Workers stored potatoes in the basement and guarded them in shifts, “numb with cold and emaciated from hunger.” Botanist Dmitri Ivanov died preserving thousands of packets of rice; peanut specialist Alexander Stchukin died at his writing table. In all, nine scientists and workers chose to die of starvation rather than eat the plants. Vavilov himself died in a labor camp in 1943, but today his bank is the largest collection of fruits and berries in the world.

(Thanks, Mike.)

This equation:

122 + 542 + 695 = 211 + 364 + 784

… remains valid when the digits of each term are permuted in the same way:

122 + 542 + 695 = 211 + 364 + 784
122 + 524 + 659 = 211 + 346 + 748
212 + 452 + 965 = 121 + 634 + 874
221 + 425 + 956 = 112 + 643 + 847
212 + 254 + 569 = 121 + 436 + 478
221 + 245 + 596 = 112 + 463 + 487

And everything above remains valid if you square each term.

(Discovered by Albert Gloden.)

This sentence is true.

Is that sentence true or false?

Draw a chord AB through a point P inside a circle, and the product PA × PB is constant — it has the same value for every chord through P.

“Again we have perfect democracy,” write Eli Maor in The Pythagorean Theorem. “Every chord has the same status in relation to P as any other.”

A stationmaster waves his flag, and a train begins to move. There is a last moment of rest, and a first moment of motion.

But this is a problem. If time is infinitely divisible, then there is a moment between these two moments. Is it a moment of rest or of motion?

(From Robin Le Poidevin, Travels in Four Dimensions, 2003.)