Jumping Kangaroos

A puzzle by National Security Agency mathematician David B., from the agency’s October 2017 Puzzle Periodical:

Joey, the baby kangaroo has been kidnapped and placed at 2¹⁰⁰ on a number line.

His mother, Kandice the Kangaroo, is at 0 on the number line, and will try to save him. Kandice normally jumps forward 6 units at a time. Guards have been placed at n³ on the number line, for every integer n≥1. If Kandice lands on a number with a guard on it, she will be caught and her mission will fail. Otherwise, she will safely sneak past the guard. Whenever she successfully sneaks past a guard, she gets an adrenaline rush that causes her next jump (the first jump after passing the guard) to take her 1 unit farther than it normally would (7 units instead of 6). (After a single 7-unit jump, she resumes jumping 6 units at a time, until the next time she sneaks past a guard.)

Will Kandice the Kangaroo reach (or pass) her son Joey safely?

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Yes. First, note that two consecutive cubes are always separated by one more than a multiple of 6. To see this, note that (n+1)³ = n³ + 3n² + 3n + 1, so (n+1)³ – n³ = 3n(n+1) + 1. n must be either even or odd; if it is even, then n(n + 1) is even, and if it is odd, then n+1 is even, so again, n(n+1) is even. Therefore, 3n(n+1) is divisible by both 2 and 3, so it must be a multiple of 6, so the difference (n+1)³ – n³ is one more than a multiple of 6. Now, it’s easy to check that Kandice will safely jump over the first two guards (at 1 and 8). Suppose Kandice later gets caught and Gary the Guard is the first guard who catches her. Since passing the guard previous to Gary, Kandice would have finished one jump of size 7 and several jumps of size 6. Because the distance between the guards is one more than a multiple of 6 (as shown in this illustration), counting backwards we see this pattern would have forced Kandice to land on the previous guard also. But this is impossible, since Gary was the first guard to catch Kandice. So Gary does not exist and Kandice will never be caught! Key: Cube numbers are the positions of the guards. Evens and odds alternate, so if some number n isn’t even, then the next number (n+1) must be even.

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July 18, 2025July 17, 2025 | Puzzles

“A Matter of Method”

A Philosopher seeing a Fool beating his Donkey, said:

‘Abstain, my son, abstain, I implore. Those who resort to violence shall suffer from violence.’

‘That,’ said the Fool, diligently belaboring the animal, ‘is what I’m trying to teach this beast — which has kicked me.’

‘Doubtless,’ said the Philosopher to himself, as he walked away, ‘the wisdom of fools is no deeper nor truer than ours, but they really do seem to have a more impressive way of imparting it.’

— Ambrose Bierce, Fantastic Fables, 1899

July 18, 2025July 17, 2025 | Literature · Society

An American in Paris

A note sent by Mark Twain’s American traveling companion to his French landlord, 1867:

PARIS, le 7 Juillet. Monsieur le Landlord — Sir: Pourquoi don’t you mettez some savon in your bed-chambers? Est-ce que vous pensez I will steal it? La nuit passee you charged me pour deux chandelles when I only had one; hier vous avez charged me avec glace when I had none at all; tout les jours you are coming some fresh game or other on me, mais vous ne pouvez pas play this savon dodge on me twice. Savon is a necessary de la vie to any body but a Frenchman, et je l’aurai hors de cet hotel or make trouble. You hear me. Allons. BLUCHER.

“I remonstrated against the sending of this note, because it was so mixed up that the landlord would never be able to make head or tail of it; but Blucher said he guessed the old man could read the French of it and average the rest.”

(From The Innocents Abroad.)

July 17, 2025July 17, 2025 | Language

Current Affairs

A problem by G. Galperin, from the May-June 1995 issue of Quantum:

A raft and a motorboat depart simultaneously from Point A on a riverbank and begin drifting and speeding downstream, respectively, toward Point B. At the same moment, a second motorboat, of the same type as the first, sets out from Point B heading upstream. When the first motorboat reaches B, will the floating raft be closer to Point A or to the second motorboat?

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This intuitive solution is by V. Dubrovsky. Imagine a third point, C, that’s as far upstream from A as A is from B (so CA = AB). And imagine a third motorboat that parallels the movements of the first, setting out from point C and heading downstream at the same time. The third and second motorboats are equidistant from the raft when they begin their journeys, and both move at the same speed relative to the raft (and to the surface of the water). So they’ll remain equidistant from the raft as they progress toward it. When the first motorboat reaches B, the third will reach A. At that moment the raft will be the same distance from the second motorboat and from Point A (now occupied by the third motorboat). That’s the answer.

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July 17, 2025July 16, 2025 | Puzzles

First Things First

George Orwell’s six rules of writing, from “Politics and the English Language,” 1946:

Never use a metaphor, simile, or other figure of speech which you are used to seeing in print.
Never use a long word where a short one will do.
If it is possible to cut a word out, always cut it out.
Never use the passive where you can use the active.
Never use a foreign phrase, a scientific word, or a jargon word if you can think of an everyday English equivalent.
Break any of these rules sooner than say anything outright barbarous.

But “one could keep all of them and still write bad English.”

July 16, 2025July 16, 2025 | Language · Literature

Winifred’s Bloomers

English novelist Winifred Ashton had a disastrous gift for inadvertent double entendre. From Cole Lesley’s biography of Noël Coward:

The first I can remember was when poor Gladys was made by Noël to explain to Winifred that she simply could not say in her latest novel, ‘He stretched out and grasped the other’s gnarled, stumpy tool.’ The Bloomers poured innocently from her like an ever-rolling stream: ‘Olwen’s got crabs!’ she cried as you arrived for dinner, or ‘We’re having roast cock tonight!’ At the Old Vic, in the crowded foyer, she argued in ringing tones, ‘But Joyce, it’s well known that Shakespeare sucked Bacon dry.’ It was Joyce too who anxiously inquired after some goldfish last seen in a pool in the blazing sun and was reassured, ‘Oh, they’re all right now! They’ve got a vast erection covered with everlasting pea!’ ‘Oh the pleasure of waking up to see a row of tits outside your window,’ she said to Binkie during a weekend at Knott’s Fosse. Schoolgirl slang sometimes came into it, for she was in fact the original from whom Noël created Madame Arcati: ‘Do you remember the night we all had Dick on toast?’ she inquired in front of the Governor of Jamaica and Lady Foot. Then there was her ghost story : ‘Night after night for weeks she tried to make him come …’

“Why could she not have used the word ‘materialise’?” wrote Lesley, who was Coward’s secretary. “But then if she had we should never have had the fun.” See Shocking!

July 16, 2025July 15, 2025 | Language

Portent

One other oddity concerning π: If you add up the first three sextads in the decimal expansion, you get 1588419:

141592 + 653589 + 793238 = 1588419

That’s a little prophecy: If you now skip ahead 15 places you arrive at the string 88419:

3.1415926535897932384626433832795028841971693 …

(Communicated by P. Olivera.)

July 15, 2025July 14, 2025 | Science & Math

Unquote

“We have got to learn to think scientifically, not only about inanimate things, but about ourselves and one another. It is possible to do this.” — J.B.S. Haldane

“Nothing has such power to broaden the mind as the ability to investigate systematically and truly all that comes under thy observation in life.” — Marcus Aurelius

July 15, 2025July 14, 2025 | Quotations

The Graceful Pi-Way

This graph has 5 edges, and we’ve managed to label its vertices in a remarkable way: Each vertex bears some integer from 0 to 5, no two receive the same integer, and each edge is now uniquely identified by the absolute difference between its endpoints, such that this magnitude lies between 1 and 5 inclusive. Such a labeling is called graceful.

In 2008 Donald E. Knuth made a graph representing the contiguous 48 states and the District of Columbia in which each pair of states are connected if they’re joined by at least one drivable road. It turns out that this graph can be labeled gracefully.

And, amazingly, in 2020 T. Rokicki discovered that if you undertake an imaginary journey on Knuth’s map, starting in California and going up the Pacific coast and then along the Canadian border, you’ll visit successive vertices labeled 31, 41, 59, 26, 53, 58, 97, 93, 23, 84, 62, 64, 33, 83, and 27. These are the first 30 decimal digits of π!

Knuth called this a “graceful miracle.”

July 14, 2025July 14, 2025 | Science & Math

“A Wooden Alphabet”

‘The letters in this curious alphabet are all wood, chiefly twisted roots of the blue gum, and have not been altered in any way from their original growth; three girls collected them in their daily walks or rides for a period of six months, and the specimens were found in various places; frequently one was carried home on horseback for many miles. All are about two feet high. The “B” was the last found, and when the young ladies had almost despaired of ever getting one it was found in a heap of driftwood caught against a tree in the river.’ — Miss Cave, Vergemont, Clontreagh, Co. Dublin

— Strand, December 1902

July 14, 2025July 13, 2025 | Oddities