In a 2005 experiment, psychologist Petter Johansson and his colleagues presented each subject with two photographs of women’s faces and asked which they found more attractive. In each case the experimenter then presented the “chosen” photograph and asked the subject to explain their choice. But in fact, using sleight of hand, the experimenter had exchanged the photos and was presenting the one that the subject hadn’t picked.

Only 13 percent of the subjects noticed the change; the rest went on to confabulate an explanation justifying a choice they hadn’t made. And yet, in post-test interviews, 84 percent of the subjects said that they would have detected such a switch if one had been made.

A subsequent experiment involving supermarket taste tests of jam showed the same effect: Subjects indicated an initial preference and then (after the samples had been surreptitiously swapped) failed to recognize that they were now tasting the rejected variety and went on to justify that choice.

Johansson and his colleagues called this choice blindness. “We do not doubt that humans can form very specific and detailed prior intentions,” they wrote, “but as the phenomenon of choice blindness demonstrates, this is not something that should be taken for granted in everyday decision tasks.”

(Petter Johansson et al., “Failure to Detect Mismatches Between Intention and Outcome in a Simple Decision Task,” Science 310:5745 [Oct. 7, 2005], 116-119.) (Thanks, Colin.)

The more similar two options are, the more difficult it is to decide between them, and the less consequential the decision becomes. A rational decider might find herself spending the most time on the least important decisions.

Philosopher Edward Fredkin writes, “The more equally attractive two alternatives seem, the harder it can be to choose between them — no matter that, to the same degree, the choice can only matter less.”

To avoid this, the decider might resolve to apportion her decision-making time by the importance of the decision. But this requires assessing the importance of every decision, which requires evaluating the means she’s using to make those assessments, and so on.

Psychologist Gary Klein writes, “[[I]f I want to optimize, I must also determine the effort it will take me to optimize; however, the subtask of determining this effort will itself take effort, and so forth into the tangle that self-referential activities create.”

A gentleman had ordered a roasted stork for dinner, and as the legs were deemed the most savory part, he was greatly exasperated when the bird came upon the table with only one leg. The cook, it seems, had a sweetheart, and she had cut off one for him. However, when her master called her to account, she boldly asserted that storks had but one leg. To prove this, she proposed that they should repair to the bank of the river on the following morning, and settle the question by ocular demonstration. They went accordingly, and behold, there were a dozen storks, showing but one leg. ‘Hoo!’ said the master, upon which each stork showed his other leg. ‘There!’ said the gentleman, ‘you see those storks have two legs.’ ‘Yes,’ said the cook, ‘but you cried “hoo!” at them: I pray you to remember that you did not cry “hoo!” to the one I cooked yesterday.’

— Boccaccio

The following pair of sentences employ 2 ‘0’s, 2 ‘1’s, 9 ‘2’s, 5 ‘3’s, 5 ‘4’s, 4 ‘5’s, 5 ‘6’s, 2 ‘7’s, 3 ‘8’s and 3 ‘9’s.

The sentences above and below employ 2 ‘0’s, 2 ‘1’s, 8 ‘2’s, 6 ‘3’s, 5 ‘4’s, 6 ‘5’s, 3 ‘6’s, 2 ‘7’s, 2 ‘8’s and 4 ‘9’s.

The previous pair of sentences employ 2 ‘0’s, 2 ‘1’s, 9 ‘2’s, 5 ‘3’s, 4 ‘4’s, 6 ‘5’s, 4 ‘6’s, 2 ‘7’s, 3 ‘8’s and 3 ‘9’s.

(From Lee Sallows and Victor L. Eijkhout, “Co-Descriptive Strings,” Mathematical Gazette 70:451 [March 1986], 1-10.)

Draw a rectangle and pick a point inside it. Now the sum of the squares of the distances from that point to two opposite corners of the rectangle equals the sum to the other two opposite corners.

Above, the red squares have the same total area as the blue ones.

The upper edge of the setting sun is sometimes seen to take on a green tinge, an effect of atmospheric refraction. Normally this is apparent only briefly, but for Richard Byrd’s Antarctic expedition of 1928-1930 it lasted more than half an hour:

Here the sun descends so slowly that it seems to roll along the horizon and as it will be only two days until it is above the horizon all the time for the rest of the summer it clings interminably before, with seeming reluctance, dropping from sight. As its downward movement is so prolonged the last rays shimmer above the barrier edge as it moves eastward, appearing and reappearing from behind the irregularities of the barrier surface. It trembles and pulsates, producing a vibration light of great beauty.

The night the green flash was seen some one ran into the administration building and called, ‘Come out and see the green sun.’

There was a rush for the surface and as eyes turned southward, they saw a tiny but brilliant green spot where the last ray of the upper limb of the sun hung on the skyline. It lasted an appreciable length of time, several seconds at least, and no sooner disappeared than it flashed forth again. Altogether it remained on the horizon with short interruptions for thirty-five minutes.

When it disappeared momentarily it seemed to have been shut off by a tiny spurt, an inequality in the skyline caused by the barrier surface.

“Even by moving the head up a few inches it would disappear and reappear again and after it had finally disappeared from view it could be recaptured by climbing up the first few steps of the [antenna] post.”

(From an account by witness Russell Owen, San Francisco Chronicle, Oct. 23, 1929.)

Draw a triangle, pick a point on one side, and draw a path as shown, with each segment parallel to a side of the triangle.

The path with always be closed — you’ll always return to your starting point.

Discovered by German mathematician Gerhard Thomsen.