One problem on the 1982 Scholastic Aptitude Test depicted a circle A adjoining a larger circle B. The problem read:

In the figure above, the radius of circle A is one-third the radius of circle B. Starting from position shown in figure, circle A rolls around circle B. At the end of how many revolutions of circle A will the center of circle A reach its starting point?

(A) 3/2
(B) 3
(C) 6
(D) 9/2
(E) 9

The College Board had expected the answer 3, but three students pointed out that the correct answer is 4. The Board explained, “The circumference of the large circle is three times the circumference of the small circle. If the small circle were to rotate along a straight line segment equal in length to the circumference of the large circle it would make three revolutions.”

“However, the motion of the small circle is not in a straight line, but rather around the large circle. This revolving action around the large circle contributes an extra revolution as circle A rolls around Circle B.” Here’s the same idea using coins of equal size:

Math scores for some 300,000 students had to be recalculated. See The Pup Tent Problem.

In Abraham Verghese’s 2009 novel Cutting for Stone, a character declares that 10213223 “is the only number that describes itself when you read it. ‘One zero, two ones, three twos, and two threes!'”

Refining that a bit, we can imagine a number N in which the first digit counts the number of zeroes in N, the second counts the number of ones, and so on. Such numbers are called autobiographical, and they’re remarkably few: 1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000.

But numbers needn’t look only inward — MIT mathematician Tanya Khovanova identified these “mutually praising” pairs, which inventory one another’s contents:

6300000100 and 7101001000
130 and 1101
2210 and 11200

“These numbers are too shy to advertise themselves,” she writes. “But Alice praises Bob, because Bob praises Alice. It’s a very advantageous flattery pattern!”

(Tanya Khovanova, “Autobiographical Numbers,” arXiv preprint arXiv:0803.0270 [2008]. Somewhat similar: John Conway’s “look and say” sequence.)

The prime factorization of 255987 is 3³ × 19 × 499.

That’s not very interesting. But when this is expressed in binary, it reads:

111110011111110011 = 11¹¹ × 10011 × 111110011

Each side of that equation comprises the same string of digits.

In 1917, W.S. Andrews inscribed a sphere with five circles of latitude and six of longitude.

Then he arranged the numbers 1 to 62 on the intersections such that each of the 11 circles bears 12 numbers totaling 378.

We think of light speed as dizzyingly fast, but it’s still disastrously slow given the roomy universe we find ourselves in. NASA astronomer James O’Donoghue designed these videos to illustrate. If we could bend it around Earth, light would orbit the planet in 0.13 seconds:

And it takes only 1.255 seconds to reach the moon:

But it takes fully 3 minutes to reach our neighboring planet Mars at its closest approach:

Light from the sun takes more than 8 minutes to reach Earth. To reach the closest star takes more than 4 years.

The programming language efghij is literally object-oriented — its logic is expressed by arranging ordinary items in meaningful tableaus:

In this example, the tray and the cutting board constitute code blocks. The oblong plate, however, has a clothes peg on it, which expresses a for loop. The fork is a local variable. The fork variable is initialized to 1 within the for loop’s initial-value expression. The sauce with the rubber band indicates that the number of iterations is equal to the function’s argument. The stuff on the cutting board is the loop body. Using a stapler, the fork is multiplied by the sauce and the result assigned to the fork using a drinking glass. A hammer decrements the sauce, which demonstrates that the function’s parameter can be used as a local variable too. Finally, the toilet roll indicates that the return value is the final value of the fork.

That produces a factorial. The program below outputs the phrase “Hello, world!” — it’s just a pillow bearing a piece of paper on which those words are written.