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.)
“How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?” — Albert Einstein
“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.” — Eugene Wigner
filipendulous
adj. hanging by a thread
A cube suspended by a corner casts a hexagonal shadow.
From Royal V. Heath in Scripta Mathematica, March 1955:
0264 + 4125 + 5610 = 0165 + 5214 + 4620
… remains valid if you split each term with a multiplication sign:
02 × 64 + 41 × 25 + 56 × 10 = 01 × 65 + 52 × 14 + 46 × 20
… or an addition sign:
02 + 64 + 41 + 25 + 56 + 10 = 01 + 65 + 52 + 14 + 46 + 20
Remarkably, everything above holds true if you square each term.
“If anything is possible, then it is possible to prove that something is impossible. And if it is possible to prove that something is impossible, then necessarily, something is impossible.”
— Roy Sorensen, Vagueness and Contradiction, 2001
Retired Pittsburgh math teacher Walter W. Horner devised this doubly magic square in 1955:
Each row, column, and long diagonal produces both a sum of 840 and a product of 2058068231856000.
And Rodolfo Marcelo Kurchan of Buenos Aires discovered this remarkable square in 1991:
Each number contains all 10 digits — and so does the magic sum, 4129607358.