The Impossible Puzzle

Dutch mathematician Hans Freudenthal proposed this puzzle in 1969 — at first it appears impossible because so little information is given.

X and Y are two different whole numbers greater than 1. Y is greater than X, and their sum is no greater than 100. S and P are two logicians; S knows the sum X + Y, and P knows the product X × Y. S and P both reason perfectly, and both know everything I’ve just told you.

S says, “P does not know X and Y.”
P says, “Now I know X and Y.”
S says, “Now I also know X and Y.”

What are X and Y?

	Select Click for Answer
X = 4 and Y = 13. S knows the sum is 17, and P knows the product is 52. S makes a list of all the addends that could produce 17 and fulfill the other conditions above: 2 + 15 (product: 30) 3 + 14 (product: 42) 4 + 13 (product: 52) 5 + 12 (product: 60) 6 + 11 (product: 66) 7 + 10 (product: 70) 8 + 9 (product: 72) S knows that P is thinking of one of the products listed above; she can say that P doesn’t know X and Y because each of these products has multiple solutions (e.g. 30 = 2 × 15 = 3 × 10). P has been building his own list — in his case, he’s listing pairs of factors that produce 52. It’s a short list — there are only two possibilities that meet the conditions: 2 × 26 4 × 13 When S makes her first announcement, P understands that her candidate list is like the one above — it’s one in which all of the products are ambiguous. And this eliminates the first candidate on P’s list. If X = 2 and Y = 26, then their product would be 52, as P expects, but S would know that their sum was 28, and she couldn’t then claim that P didn’t know the numbers — her list of suspects would look like this: 2 + 26 (product: 52) 3 + 25 (product: 75) 4 + 24 (product: 96) 5 + 23 (product: 115) … That fourth item, X = 5 and Y = 23, gives a product of 115, which has no other set of factors that meet the requirements. So if 28 were the sum, S couldn’t have declared so confidently that P was still in the dark. Realizing this, P can eliminate 2 × 26 from his list, leaving only one possibility: X is 4 and Y is 13. He says, “Now I know X and Y.” And by reconstructing his thinking, S can determine these values herself. (Thanks, Jon.)

Click for Answer

September 24, 2017 | Puzzles · Science & Math

Memorial

Balloonist Andy Collett was floating over South Gloucestershire in July 2012 when he spotted something striking: a heart-shaped meadow in the center of a stand of oaks. “This was the most amazing sight I have ever seen from the sky,” he told the Telegraph. “It was a perfect heart hidden away from view — you would not know it was there.”

It turned out that 70-year-old Winston Howes had planted the wood to commemorate his 33-year marriage to his wife, Janet, who had died 17 years earlier of heart disease. In the months after her death he filled a six-acre field with thousands of oak saplings but left a heart-shaped clearing in the center, its point aimed at her childhood home.

“I came up with the idea of creating a heart in the clearing of the field after Janet died,” Howes said. “Once it was completed we put a seat in the field, overlooking the hill near where she used to live. I sometimes go down there, just to sit and think about things. It is a lovely and lasting tribute to her which will be here for years.”

The clearing was not visible from the road and remained a family secret until Collett spotted it. “You can just imagine the love story,” he said.

September 23, 2017September 22, 2017 | Oddities

Pedal Pushers

Cycling is popular in Trondheim, Norway, but the 130-meter hill Brubakken is more than some riders can manage. So the city installed the world’s first bicycle lift — press the start button and a plate will appear under your right foot and push you up the hill at 3-4 mph, rather like a ski lift.

With a maximum capacity of 6 cyclists per minute, the system has pushed more than 200,000 cyclists to the top of the hill in its 15 years of operation.

September 22, 2017 | Technology

Winkie

On Feb. 24, 1942, a bedraggled carrier pigeon arrived at an RAF bomber base on Scotland’s Fifeshire coast. She was covered with oil and appeared exhausted.

The sergeant who examined her, George Ross, alerted his superior officer. The bird had been carried aboard a Bristol Beaufort bomber that had crashed in the North Sea after taking enemy fire over Norway the previous day. The pilot had been unable to radio his position as they went down, and rescue planes had been searching the freezing waters in vain all night for some sign of the four-man crew.

The bird’s arrival told Ross that they’d been searching in the wrong place. She had flown for 16 hours, but with oil-smeared wings couldn’t have covered more than 140 miles in that time. The rescue operation had been searching beyond that range. When they moved closer to shore they discovered the crewmen, freezing but safe, in a rubber dinghy within 15 minutes.

When the fuselage had broken up, the pigeon had somehow escaped into the oily water, struggled free, and then flown across 120 miles of ocean to the base, despite a natural fear of the dark and a dislike of wide expanses of water. When she arrived she was so exhausted that she was closing one eye intermittently.

“Winkie” was awarded the Dickin Medal at a dinner that December. She was cited “for delivering a message under exceptional difficulties and so contributing to the rescue of an Air Crew while serving with the RAF in February 1942.”

September 22, 2017September 21, 2017 | History

Black and White

lincoln chess problem

By Robert A. Lincoln. White to mate in two moves.

	Select Click for Answer
1. Ka5! Kxa2 2. Kb4#

September 21, 2017 | Puzzles

The RNA Tie Club

In 1954, James Watson and George Gamow formed a “gentleman’s club” to “solve the riddle of the RNA structure and to understand how it built proteins.” There were 20 members, each of whom was designated by an amino acid:

Member	Training	Tie Designation
George Gamow	Physicist	ALA
Alexander Rich	Biochemist	ARG
Paul Doty	Physical Chemist	ASP
Robert Ledley	Mathematical Biophysicist	ASN
Martynas Ycas	Biochemist	CYS
Robley Williams	Electron Microscopist	GLU
Alexander Dounce	Biochemist	GLN
Richard Feynman	Theoretical Physicist	GLY
Melvin Calvin	Chemist	HIS
Norman Simons	Biochemist	ISO
Edward Teller	Physicist	LEU
Erwin Chargaff	Biochemist	LYS
Nicholas Metropolis	Physicist, Mathematician	MET
Gunther Stent	Physical Chemist	PHE
James Watson	Biologist	PRO
Harold Gordon	Biologist	SER
Leslie Orgel	Theoretical Chemist	THR
Max Delbrück	Theoretical Physicist	TRY
Francis Crick	Biologist	TYR
Sydney Brenner	Biologist	VAL

“We were just drinking California wine and we got the idea,” Gamow recalled. Each member was given a black woolen necktie with an RNA helix embroidered in green and yellow (above are Crick, Rich, Orgel, and Watson).

Each also received a gold tiepin with the three-letter abbreviation of his amino acid (which led several people to ask Gamow why his pin bore the wrong monogram).

Adopting the motto “Do or die, or don’t try,” they met twice a year to share ideas, cigars, and alcohol. Several went on to become Nobel laureates — but it fell to Marshall Nirenberg, a non-member, to finally decipher the code link between nucleic and amino acids.

September 21, 2017September 20, 2017 | Science & Math

Initial Velocity

In 2008, University of Michigan psychologist Jesse Chandler and his colleagues examined donations to disaster relief after seven major hurricanes and found that a disproportionately large number of donations came from people who shared an initial with the hurricane (e.g., people named Kate and Kevin after Hurricane Katrina).

It’s not clear why this is. It’s known that generally people attend to information with unusual care if it’s somehow relevant to themselves; in the case of a hurricane this may mean that they’re more likely to remember concrete information about victims and thus be more likely to donate.

Possibly they also feel more intense negative feelings (or a greater sense of responsibility) when the storm shares their initial. In that case, “Exposure to a same-initial hurricane makes people feel worse, and the most salient way to repair this feeling is the opportunity to donate money to Katrina.”

(Jesse Chandler, Tiffany M. Griffin, and Nicholas Sorensen, “In the ‘I’ of the Storm: Shared Initials Increase Disaster Donations,” Judgment and Decision Making 3:5 [June 2008], 404–410.)

September 20, 2017September 20, 2017 | Language · Science & Math

A Pi Fractal

Jack Hodkinson, who created the Corpus-Christi-College-shaped prime number that I mentioned last week, has sent along a remarkable followup produced by string rewriting. Here are the rules:

$pi fractal - rewriting rules$

The top row lists pixels of various colors; on each pass we’ll be replacing each of these with the 3 × 3 square beneath it. Start with a single color:

$pi fractal - genesis$

Following the associated rule produces a 3 × 3 array:

$pi fractal - iteration 1$

And so on:

$pi fractal - iteration 2$

$pi fractal - iteration 3$

After four iterations the rules produce a portrait of π:

$pi fractal - iteration 4$

And, charmingly, if we keep going, smaller πs begin to appear, due to the presence of dark pixels in the third image:

$pi fractal - iteration 5$

$pi fractal - iteration 6$

$pi fractal - iteration 7$

$pi fractal - iterationl 8$

And so on forever. “After iteration 8, it’s turtles all the way down.”

For those who want more information, Jack explains the rewriting rules in detail here.

(Note: To conserve bandwidth I’ve had to reduce the last two images above — you can find the full-resolution PNGs in this imgur gallery.)

September 20, 2017September 19, 2017 | Science & Math

Balance

https://commons.wikimedia.org/wiki/File:Neil_Young_2008_Firenze_02.jpg — Image: Wikimedia Commons

In a 2013 radio interview, Graham Nash recalled visiting Neil Young in 1972:

The man is totally committed to the muse of music. And he’ll do anything for good music. And sometimes it’s very strange. I was at Neil’s ranch one day just south of San Francisco, and he has a beautiful lake with red-wing blackbirds. And he asked me if I wanted to hear his new album, Harvest. And I said sure, let’s go into the studio and listen.

Oh, no. That’s not what Neil had in mind. He said get into the rowboat.

I said get into the rowboat? He said, yeah, we’re going to go out into the middle of the lake. Now, I think he’s got a little cassette player with him or a little, you know, early digital format player. So I’m thinking I’m going to wear headphones and listen in the relative peace in the middle of Neil’s lake.

Oh, no. He has his entire house as the left speaker and his entire barn as the right speaker. And I heard Harvest coming out of these two incredibly large loud speakers louder than hell. It was unbelievable. Elliot Mazer, who produced Neil, produced Harvest, came down to the shore of the lake and he shouted out to Neil: How was that, Neil?

And I swear to God, Neil Young shouted back: More barn!

Asked in 2016 whether this story was true, Young said, “Yeah, I think it was a little house-heavy.”

September 19, 2017September 19, 2017 | Entertainment

Four-Sided Story

University of Ljubljana electrical engineer Izidor Hafner devised this maze on a tetrahedron, presented as an unfolded plane plan. Can you find a path from one dot to the other? To do so you’ll have to fold up the figure in your head:

	Select Click for Answer
(From Crux Mathematicorum, December 2000. See Hafner’s Wolfram Demonstrations page for mazes on cylinders, polyhedra, tori, Moebius strips, Riemann surfaces, and more.)

September 19, 2017September 18, 2017 | Puzzles