An odd detail from the autobiography of Bertrand Russell:
“The summers of 1903 and 1904 we spent at Churt and Tilford. I made a practice of wandering about the common every night from eleven till one, by which means I came to know the three different noises made by night-jars.”
He adds, “(Most people only know one.)” He says nothing more.
“A red rose absorbs all colours but red; red is therefore the one colour that it is not.” — Aleister Crowley, of all people
Paris newspapers once carried an ad offering a cheap and pleasant way of travelling for the price of 25 centimes. Several simpletons mailed this sum. Each received a letter of the following content:
‘Sir, rest at peace in bed and remember that the earth turns. At the 49th parallel — that of Paris — you travel more than 25,000 km a day. Should you want a nice view, draw your curtain aside and admire the starry sky.’
The man who sent these letters was found and tried for fraud. The story goes that after quietly listening to the verdict and paying the fine demanded, the culprit struck a theatrical pose and solemnly declared, repeating Galileo’s famous words: ‘It turns.’
— Yakov Perelman, Physics for Entertainment, 1913
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.)
From an appreciation of Ernest Rutherford by C.P. Snow in the November 1958 issue of The Atlantic:
Worldly success? He loved every minute of it: flattery, titles, the company of the high official world. He said in a speech: ‘As I was standing in the drawing room at Trinity, a clergyman came in. And I said to him: “I’m Lord Rutherford.” And he said to me: “I’m the Archbishop of York.” And I don’t suppose either of us believed the other.’
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.)
From Lee Sallows:
Thanks, Lee!
If a sphere is inscribed in a tetrahedron, and segments are drawn on each face connecting its vertices to the point of tangency, then the same three angles will appear on each face.
And the 12 triangles produced will form congruent pairs across each edge of the tetrahedron.
Into an empty vase drop balls numbered 1 to 10. Remove ball 1. Add balls numbered 11 to 20. Remove ball 2. Continue in this way, spending half an hour on the first transaction, 15 minutes on the next, and so on. After one hour all the transactions will be finished.
Obviously, in the end the vase will contain infinitely many balls, since with each step more balls have been added than removed.
But, equally obviously, after an hour the vase will be empty — since the time of each ball’s removal is known.
