In Scripta Mathematica, March 1955, Pedro A. Pisa offers an unkillably valid equation:

123789 + 561945 + 642864 = 242868 + 323787 + 761943

Hack away at its terms, from either end, and it remains true:

Stab it in the heart, removing the two center digits from each term, and it still balances:

1289 + 5645 + 6464 = 2468 + 3287 + 7643

Do this again and it still balances:

19 + 55 + 64 = 28 + 37 + 73

Most amazing: You can square every term above, in every equation, and they’ll all remain true.

12/03/2016 UPDATE: Reader Jean-Claude Georges discovered that the equalities remain valid when any combination of digits is removed consistently across terms. For example, starting from

123789 + 561945 + 642864 == 242868 + 323787 + 761943,

removing the 1st, 3rd and 5th digit from each number:

x2x7x9 + x6x9x5 + x4x8x4 == x4x8x8 + x2x7x7 + x6x9x3

gives

279 + 695 + 484 = 488 + 277 + 693 (= 1458)

and squaring each term gives

279² + 695² + 484² = 488² + 277² + 693² (=795122).

Amazingly, the same is true for any combination — for example, the equations remain valid when the 1st, 2nd, 4th, and 6th digits of each term are removed. (Thanks, Jean-Claude.)

To discover whether a number is divisible by 11, add the digits that appear in odd positions (first, third, and so on), and separately add the digits in even positions. If the difference between these two sums is evenly divisible by 11, then so is the original number. Otherwise it’s not.

For example:

11 × 198249381729 = 2180743199019

Sum of digits in odd positions = 2 + 8 + 7 + 3 + 9 + 0 + 9 = 38

Sum of digits in even positions = 1 + 0 + 4 + 1 + 9 + 1 = 16

38 – 16 = 22

22 is a multiple of 11, so 2180743199019 is as well.

When I conduct a psychological experiment, my expectations might influence the outcome.

That’s called the experimenter expectancy effect. Does it exist? Well, we could do an experiment to detect it …

… but if it exists then it would bias the experiment, and if it doesn’t then we’d detect nothing. Either way, it seems, we can’t reliably assess what’s happening.

From Lewis Carroll’s A Tangled Tale: The governor of Kgovjni gives a dinner party for his father’s brother-in-law, his brother’s father-in-law, his father-in-law’s brother, and his brother-in-law’s father — and invites a single person:

Males are denoted by capitals, females by small letters. The governor is E, and his guest is C.

See “Proof That a Man Can Be His Own Grandfather,” No Reunion, and The Half-Bastard.

Relativism either applies to itself or it doesn’t.

If it does, then it’s only relatively true.

If it doesn’t, then there’s an absolute truth.

If the moon orbits the earth, always presenting the same face to us, does it rotate on its own axis?

It seems a simple question, but its appearance in the London Times in April 1856 set off a war among the English intelligentsia:

• “A ship sailing round the world presents to the fishes always the same face as the Moon does to us. Coming home again, it will surely not be said that the ship has performed a [rotation].”
• “Let him perforate a small ivory ball to represent the Moon, pass a wire through it, and bend this wire into a circle of a foot in diameter, and then push the ball round the circumference. Will there then remain any doubt of her not rotating on her axis?”

The answer, as William James would note in his parable of the squirrel, is that “which party is right depends on what you practically mean” by the term in question. Today we’d say that the moon rotates about its axis in the same time it takes to orbit the earth.

Incidentally, Lewis Carroll submitted two letters, but the Times didn’t print them. Perhaps it’s just as well — he was far ahead of everyone else: “I noticed for the first time the fact that though [the moon] only goes 13 times round the earth in the course of the year, it makes 14 revolutions round its own axis, the extra one being due to its motion round the sun.”