“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.

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.)

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.

Discovered by A.S. Bang in 1897.

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.

He, then, who says that something true exists either only asserts that something true exists or proves it. And if he merely asserts it, he will be told the opposite of his mere assertion, namely, that nothing is true. But if he proves that something is true, he proves it either by a true proof or by one that is not true. But he will not say that it is by one not true, for such a proof is not to be trusted. And if it is by a true proof, whence comes it that the proof which proves that something is true is itself true? If it is true of itself, it will be possible also to state as true of itself that truth does not exist; while if it is derived from proof, the question will again be asked ‘How is it that this proof is true?’ and so on ad infinitum. Since, then, in order to learn that there is something true, an infinite series must first be grasped, and it is not possible for an infinite series to be grasped, it is not possible to know for a surety that something true exists.

— Sextus Empiricus, Against the Logicians