In the December 2024 issue of Recreational Mathematics Magazine, Illinois State University mathematician Sunil Chebolu demonstrates that the digits of π can be generated using the positions of stars.

The probability P(x, y) that two randomly chosen positive integers, x and y, are relatively prime is 6/π². The stars in the night sky appear to be distributed randomly on the celestial sphere, and this suggests a novel experiment: Use the pairwise angular distances between stars to simulate a large random sample of numbers, then pick random pairs and determine the proportion of relatively prime pairs. Equating that with the theoretical probability above should generate an approximation of π.

In Nature in 1995, University of Aston mathematician Robert Matthews used the positions of 100 stars to compute π to within 0.5% of its correct value (3.12772). By expanding the sample to 1,000 stars, Chebolu got an average estimate of 3.141540567, an error of less than 0.002%.

“The above method can also be viewed as a statistical hypothesis test,” Chebolu notes. “Since the error in the estimate for π obtained using this method is pretty low, one may also argue that it supports the hypothesis that the bright stars are randomly distributed on the celestial sphere.”

(Sunil K. Chebolu, “Baking Star π,” Recreational Mathematics Magazine 11:19 [December 2024], 67-70.)

From Lee Sallows:

Thanks, Lee!

In 1940, George Gamow published Mr Tompkins in Wonderland, in which a bank clerk attends a lecture on relativity and then finds that Einstein’s principles have become apparent in the ordinary street scenes before him:

A single cyclist was coming slowly down the street and, as he approached, Mr Tompkins’s eyes opened wide with astonishment. For the bicycle and the young man on it were unbelievably shortened in the direction of the motion, as if seen through a cylindrical lens. The clock on the tower struck five, and the cyclist, evidently in a hurry, stepped harder on the pedals. Mr Tompkins did not notice that he gained much in speed, but, as the result of his effort, he shortened still more and went down the street looking exactly like a picture cut out of cardboard.

In later adventures Tompkins explores cosmology and quantum physics, again in an exaggerated world in which extreme effects become observable. Gamow called this a “fantastic but scientifically correct dream.”

“A red rose absorbs all colours but red; red is therefore the one colour that it is not.” — Aleister Crowley, of all people

Draw an arbitrary quadrilateral and divide each of its sides into three equal parts. Draw a line through adjacent points of trisection on either side of each vertex and you’ll have a parallelogram.

Discovered by Austrian engineer Ferdinand Wittenbauer.

07/03/2025 UPDATE: Reader Ross Ogilvie writes, “I realized that there is nothing special about dividing the sides into thirds. The construction still works so long as the two points adjacent to a vertex divide their respective sides in the same ratio. This ensures that the line connecting the adjacent points are parallel to a diagonal of the quadrilateral. In Wittenbauer’s construction, they divide the sides 1:2. The case of 1:1, so all the points are midpoints, is Varignon’s theorem. It’s even possible that different vertices have different ratios. I drew up a little demo if you would like to play around and see for yourself. Move the sliders a,b,c,d to change the ratios.”

In 1897, German mathematician August Leopold Crelle published a book listing all the products of pairs of 3-digit numbers. He offered it as a useful aid in the multiplication of large numbers. Suppose we want to multiply 26457081 by 247183. Divide each factor into 3-digit “chunks”: 026 457 081 and 247 183. Now think of each full chunk as a digit — instead of falling in the range 0-9, here each falls in the range 000-999; in effect we’re expressing each factor in base 10³. With that understanding we can proceed with the arithmetic in the usual fashion. The first few partial products are 183 × 081 = 14823, 183 × 457 = 83631, and 183 × 026 = 4758; indent these successively by three places, as shown below, and continue with the rest, following the same pattern:

       26457081
       x 247183

       20007
   112879 14823
  6422 83631
     4758

  6539740652823

The user can look up all the intermediate products in Crelle’s book, so all that remains is to do the final sums. Crelle gives some further examples in the book.

In 1989, Northwestern University mathematician R.P. Boas pointed out that the same method can be used to work through ungainly problems such as 2849365028828173 × 4183920538293052 = 11921516865208167208145227753996 even on a simple 8-digit calculator, the prevailing tool at the time. Cutting these factors into 4-digit chunks reduces all the intermediate products and sums to manageable size, and the 32-digit result is reached reliably even though it would normally be beyond the calculator’s capacity.

(R.P. Boas, “Multiplying Long Numbers,” Mathematics Magazine 62:3 [June 1989], 173-174.)

In the 1950s, mathematician C. Dudley Langford was watching his son play with blocks, two of each color, when he noticed that they formed a curious pattern: one block lay between the red blocks, two between the blue blocks, and three between the yellow blocks. Langford found that by rearranging the blocks he could add a green pair with four blocks between them.

This presented a clear challenge. He found solutions for as many as 15 pairs of blocks but came to believe that some smaller groupings (14 pairs, for example) could not produce a solution. He asked for a general investigation.

Today we know that a solution exists if and only if the number of blocks is 4k or 4k + 3, so Langford was right that no solution can be arranged with 14 pairs of blocks. The number of solutions for each quantity of pairs is listed here, and a few proofs are given here.

(C. Dudley Langford, “Problem,” Mathematical Gazette 42:341 [October 1958], 228.)