The Cognitive Reflection Test

Answer these questions:

A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost? _____ cents
If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets? _____ minutes
In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake? _____ days

The correct answers are 5 cents, 5 minutes, and 47 days, but each question also invites a quick, intuitive response that’s wrong. In order to succeed, you have to suppress your “gut” response and reflect on your own cognition deeply enough to see the error. Psychologist Shane Frederick devised the three-question test in 2005 to illustrate these two modes of thought, unreflective and reflective, which he called System 1 and System 2.

Scores on the CRT correlate with various measures of intelligence, patience, and deliberation, but cognitive ability alone isn’t strongly correlated with CRT scores: If you’re not disposed to answer impulsively then the problems aren’t hard, and if you do answer impulsively then cognitive ability won’t help you. A sample of students at MIT averaged 2.18 correct answers, Princeton 1.63, Carnegie Mellon 1.51, Harvard 1.43; see the link below for more.

(Shane Frederick, “Cognitive Reflection and Decision Making,” Journal of Economic Perspectives 19:4 [2005], 25-42.) (Thanks, Drake.)

March 17, 2019March 18, 2019 | Science & Math

Slitherlink

https://commons.wikimedia.org/wiki/File:SurizaL%C3%B6sung.png — Image: Wikimedia Commons

In this original logic puzzle by the Japanese publisher Nikoli, the goal is to connect lattice points to draw a closed loop so that each number in the grid denotes the number of sides on which the finished loop bounds its cell, as above: Each cell bearing a “1” is bounded on 1 side, a “2” on 2 sides, and so on.

Here’s a moderately difficult puzzle. Can you solve it? (A loop that merely touches a cell’s corner point without passing along any side is not considered to bound it.)

https://commons.wikimedia.org/wiki/File:Slitherlink-example.png — Image: Wikimedia Commons

	Select Click for Answer
Image: Wikimedia Commons

March 16, 2019 | Puzzles

A for Effort

So many more men seem to say that they may soon try to stay at home so as to see or hear the same one man try to meet the team on the moon as he has at the other ten tests.

This ungainly but grammatical 41-word sentence was constructed by Anton Pavlis of Guelph, Ontario, in 1983. It’s an alphametic: If each letter is replaced with a digit (EOMSYHNART = 0123456789), then you get a valid equation:

   SO     31
 MANY   2764
 MORE   2180
  MEN    206
 SEEM   3002
   TO     91
  SAY	 374
 THAT   9579
 THEY   9504
  MAY    274
 SOON   3116
  TRY    984
   TO     91
 STAY   3974
   AT     79
 HOME   5120
   SO     31
   AS     73
   TO     91
  SEE    300
   OR     18
 HEAR   5078
  THE    950
 SAME   3720
  ONE    160
  MAN    276
  TRY    984
   TO     91
 MEET   2009
  THE    950
 TEAM   9072
   ON     16
  THE    950
 MOON   2116
   AS     73
   HE     50
  HAS    573
   AT     79
  THE    950
OTHER  19508
+ TEN    906

TESTS  90393

Apparently this appeared in the Journal of Recreational Mathematics in 1972; I found the reference in the April 1983 issue of Crux Mathematicorum, which confirmed (by computer) that the solution is unique.

March 15, 2019March 17, 2019 | Science & Math

Black and White

tuxen chess problem

By Harry Viggo Tuxen. White to mate in two moves.

	Select Click for Answer
1. Qh1! and the queen will mate. From Hvar Dag, 1917.

March 15, 2019March 14, 2019 | Puzzles

Making Do

When the Allies secured New Guinea’s Goodenough Island in October 1942, they left a small Australian occupation force to hold this important position against the Imperial Japanese. They succeeded through deception: The Australians built dummy structures (including a hospital), pointed logs at the sky to suggest anti-aircraft guns, wove jungle vines into barbed wire, lighted numerous “cooking fires” at night, and sent messages in easily broken code that suggested that a full brigade occupied the island.

It worked. The small force held the island until December 28, and a new garrison arrived the following year.

March 14, 2019 | History

The Roving Princess

A puzzle by University College London mathematician Matthew Scroggs: A princess lives in a row of 17 rooms. Each day she moves to a new room adjacent to the last one (e.g., if she sleeps in Room 5 on one night, then she’ll sleep in Room 4 or Room 6 the following night). You can open one door each night. If you find her you’ll become her prince. Can you find her in a finite number of moves?

	Select Click for Answer
Here’s one way to do it. Imagine a chessboard with 17 files and a large number of ranks. Each file represents one room, and each rank represents one night. On each night (rank), mark the room (square) where the princess is sleeping. Because the princess moves to an adjacent room each day, the marker will always move “diagonally” on the board, and hence will always occupy squares of the same color. Check the 17th room on the first night, the 16th on the second night, and so on. Suppose the 17th square in the first row is white. If the princess started on a white square, this will find her, as the diagonal you’re drawing must eventually intercept her path. If you reach Room 1 without finding her, this means she’s been traversing black squares. Go immediately back to Room 17 (which will be represented by a black square on row 18), and work your way down as before. This time your paths must cross.

Click for Answer

March 14, 2019March 17, 2019 | Puzzles

Distortion

https://www.youtube.com/watch?v=BaCzOuHYuB8

Austrian artist Peter Kogler uses twisting lines and geometric shapes to generate dramatic illusions in ordinary spaces.

“The black-and-white grid provides a maximum contrast which has a very strong visual presence,” he says. “The structure of the image is comprehensive and completely surrounds the beholder. In a sense, you are standing in the picture, and the work can be experienced physically.”

More at his website.

March 13, 2019 | Art

Accessory

In 1812 Percy Shelley and his wife Harriet had committed themselves to a vegetarian diet. During their residence in Ireland that March, Harriet sent a note to a friend in Dublin:

Sunday morng.
17 Grafton Street

Mrs. Shelley’s comps. to Mrs. Nugent, and expects the pleasure of her company to dinner, 5 o’clock, as a murdered chicken has been prepared for her repast.

Isaac Bashevis Singer once said, “I am a vegetarian for health reasons — the health of the chicken.”

March 13, 2019March 12, 2019 | Literature · Society

Bertrand’s Problem

French mathematician Joseph Bertrand offered this observation in his Calcul des probabilités (1889). Inscribe an equilateral triangle in a circle, and then choose a chord of the circle at random. What is the probability that this chord is longer than a side of the triangle? There seem to be three different answers:

https://commons.wikimedia.org/wiki/File:Bertrand1-figure.svg — Image: Wikimedia Commons

1. Choose two random points on the circle and join them, then rotate the triangle until one of its vertices coincides with one of these points. Now the chord is longer than a side of the triangle when its farther end falls on the arc between the other two vertices of the triangle. That arc is one third of the total circumference of the circle, so by this argument the probability is 1/3.

https://commons.wikimedia.org/wiki/File:Bertrand2-figure.svg — Image: Wikimedia Commons

2. Choose a radius of the circle, choose a point on that radius, and draw a chord through that point that’s perpendicular to the radius. Now imagine rotating the triangle so that one of its sides also intersects the radius perpendicularly. Our chord will be longer than a side of the triangle if the point we chose is closer to the circle’s center than the point where the triangle’s side intersects the radius. The triangle’s side bisects the radius, so by this argument the probability is 1/2.

https://commons.wikimedia.org/wiki/File:Bertrand3-figure.svg — Image: Wikimedia Commons

3. Choose a point anywhere in the circle and draw the chord for which this is the midpoint. This chord will be longer than a side of the triangle if the point we chose falls within a concentric circle whose radius is half the radius of the larger circle. That smaller circle has one-fourth the area of the larger circle, so by this argument the probability is 1/4.

Further methods yield still further solutions. After more than a century, the implications of Bertrand’s conundrum are still being discussed.

March 12, 2019 | Science & Math

Stagecraft

Through his innovative stage machines, architect Nicola Sabbatini summoned lightning, fire, hell, storms, gods, and clouds to the sets of 17th-century Venetian operas. The effect could be spectacular — characters braved moving waves, flew through the air, and descended into the underworld.

His illusions, which came to be known as scènes à l’italienne, were best viewed from “the prince’s seat,” the center of the seventh row, where “all the objects in the scene appear better … than from any other place.” The scene above, undertaken with stage designer Giacomo Torelli, depicts Apollo’s palace as a city among the clouds in Francesco Sacrati’s La Venere Gelosa (1643).

But they didn’t always work. Where one libretto read, “Here one sees descend an enormous machine, which arrives at the level of the gloria from the level of the floor of the stage, forming a majestic stairway of clouds, by which Jove descends, accompanied by a multitude of deities and celestial goddesses,” one critic wrote, “A stairway of clouds? For shame! / pardon me, architect: / it was a ladder to climb to the roof.”

March 12, 2019March 11, 2019 | Art