Science & Math
Multiplication Palindromes
12 × 42 = 24 × 21
12 × 63 = 36 × 21
12 × 84 = 48 × 21
13 × 62 = 26 × 31
23 × 96 = 69 × 32
24 × 63 = 36 × 42
24 × 84 = 48 × 42
26 × 93 = 39 × 62
36 × 84 = 48 × 63
46 × 96 = 69 × 64
14 × 82 = 28 × 41
23 × 64 = 46 × 32
34 × 86 = 68 × 43
13 × 93 = 39 × 31
Immortal Truth
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.)
Math Notes
Nature, Nurture
Identical twins Jack Yufe and Oskar Stohr were born in 1932 to a Jewish father and a Catholic mother. Their parents divorced when the boys were six months old; Oskar was raised by his grandmother in Czechoslovakia, where he learned to love Hitler and hate Jews, and Jack was raised in Trinidad by his father, who taught him loyalty to the Jews and hatred of Hitler.
At 47 they were reunited by scientists at the University of Minnesota. Oskar was a conservative who enjoyed leisure, Jack a liberal workaholic. But both read magazines from back to front, both wore tight bathing suits, both wrapped rubber bands around their wrists, both liked sweet liqueur and spicy foods, both had difficulty with math, both flushed the toilet before and after using it — and both enjoyed sneezing suddenly in elevators to startle other passengers.
See Doppelgangers.
Hendecadivisibility
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.
Fugitive Truth
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.
Warm Work
(Please don’t try this.)
[T]ar … boils at a temperature of 220°, even higher than that of water. Mr. Davenport informs us, that he saw one of the workmen in the King’s Dockyard at Chatham immerse his naked hand in tar of that temperature. He drew up his coat sleeves, dipped in his hand and wrist, bringing out fluid tar, and pouring it off from his hand as from a ladle. The tar remained in complete contact with his skin, and he wiped it off with tow. Convinced that there was no deception in this experiment, Mr. Davenport immersed the entire length of his forefinger in the boiling cauldron, and moved it about a short time before the heat became inconvenient. Mr. Davenport ascribes this singular effect to the slowness with which the tar communicates its heat, which he conceives to arise from the abundant volatile vapour which is evolved ‘carrying off rapidly the caloric in a latent state, and intervening between the tar and the skin, so as to prevent the more rapid communication of heat.’ He conceives also, that when the hand is withdrawn, and the hot tar adhering to it, the rapidity with which this vapour is evolved from the surface exposed to the air cools it immediately. The workmen informed Mr. Davenport, that, if a person put his hand into the cauldron with his glove on, he would be dreadfully burnt, but this extraordinary result was not put to the test of observation.
– David Brewster, Letters on Natural Magic, 1868
“A Very Small Dinner Party”
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.
Math Notes
