The Mirror Problem

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In 1868, 8-year-old Alice Raikes was playing with friends in her London garden when a visitor at a neighbor’s house overheard her name and called to her.

“So you are another Alice,” he said. “I’m very fond of Alices. Would you like to come and see something which is rather puzzling?” He led them into a room with a tall mirror in one corner.

‘Now,’ he said, giving me an orange, ‘first tell me which hand you have got that in.’ ‘The right,’ I said. ‘Now,’ he said, ‘go and stand before that glass, and tell me which hand the little girl you see there has got it in.’ After some perplexed contemplation, I said, ‘The left hand.’ ‘Exactly,’ he said, ‘and how do you explain that?’ I couldn’t explain it, but seeing that some solution was expected, I ventured, ‘If I was on the other side of the glass, wouldn’t the orange still be in my right hand?’ I can remember his laugh. ‘Well done, little Alice,’ he said. ‘The best answer I’ve had yet.’

“I heard no more then, but in after years was told that he said that had given him his first idea for Through the Looking-Glass, a copy of which, together with each of his other books, he regularly sent me.”

Bootstraps Everlasting

I recently visited an Eastern sage and asked him, ‘Is it possible to live for ever?’ ‘Certainly,’ he replied, ‘You must undertake to do two things.’ ‘What are they?’ ‘Firstly, you must never again make any false statements.’ ‘That’s simple enough. What is the second thing I must do?’ ‘Every day you must utter the statement “I will repeat this statement tomorrow.” If you follow these instructions faithfully you are certain to live forever.’

— Jacqueline Harman, letter to the Daily Telegraph, Oct. 8, 1985

Misc

  • Q is the only letter that does not appear in any U.S. state name.
  • 6455 = (64 – 5) × 5
  • North Dakota’s record high temperature (121°F) is higher than Florida’s (109°F).
  • UNNOTICEABLY contains the vowels A, E, I, O, and U in reverse order.
  • “An odd thought strikes me: We shall receive no letters in the grave.” — Samuel Johnson

Pandigital Squares

Square numbers containing all 10 digits unrepeated:

320432 = 1026753849
322862 = 1042385796
331442 = 1098524736
351722 = 1237069584
391472 = 1532487609
456242 = 2081549376
554462 = 3074258916
687632 = 4728350169
839192 = 7042398561
990662 = 9814072356

The Mirror

From Albert Beiler, Recreations in the Theory of Numbers (1964):

1 + 4 + 5 + 5 + 6 + 9 = 3 + 2 + 3 + 7 + 8 + 7

Pair each digit on the left with one on the right (for example, 13, 42, 53, 57, 68, 97). The sum of these six numbers will always equal its mirror image:

13 + 42 + 53 + 57 + 68 + 97 = 79 + 86 + 75 + 35 + 24 + 31

This works for all 720 possible combinations.

Most remarkably, you can square every term in these equations and they still hold:

132 + 422 + 532 + 572 + 682 + 972 = 792 + 862 + 752 + 352 + 242 + 312

Perpetual Notion

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The balls on the right exert greater torque than those on the left, so the wheel ought to turn forever, right?

Sadly, the balls on the left are more numerous.

“If at first you don’t succeed,” wrote Quentin Crisp, “failure may be your style.”

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:

beiler equation math

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.)