Here’s another interesting source of complementary sequences. Take any positive irrational number, say , and call it X. Call its reciprocal Y; in this case , or about 0.7. Add 1 to each of X and Y and we get

1 + X ≈ 2.4

1 + Y ≈ 1.7.

Now make a table of the approximate multiples of 1 + X and 1 + Y:

If we drop the fractional part of each number in the table, we’re left with two complementary sequences — every number 1, 2, 3, … appears in one sequence or the other, but never in both.

They’re called Beatty sequences, after Sam Beatty of the University of Toronto, who discovered them in 1926. A pretty proof by A. Ostrowski and J. Hyslop appears in the March 1927 issue of the American Mathematical Monthly and in Ross Honsberger’s Ingenuity in Mathematics (1970).

In 1934, Indian mathematician S.P. Sundaram proposed this “sieve” for finding prime numbers.

In the first row of a table, write the arithmetic progression 4, 7, 10, …, with the first term 4 and a common difference of 3.

Copy these values into the first column, and then complete each row with its own arithemetic progression, with common differences of 3, 5, 7, 9 …, in successive rows.

Now, remarkably, for any natural number N > 2, if N occurs in the table then 2N + 1 is not a prime number, and if N does not occur in the table, then 2N + 1 is a prime number. (For example, 17 appears in the table, so 35 is not prime; 23 does not appear in the table, so 47 is prime.)

(From Ross Honsberger, Ingenuity in Mathematics, 1970.)

A pleasing fact from David Wells’ Archimedes Mathematics Education Newsletter:

Draw two parallel lines. Fix a point on one line and move a second point along the other line. If an equilateral triangle is constructed with these two points as two of its vertices, then as the second point moves, the third vertex of the triangle will trace out a straight line.

Thanks to reader Matthew Scroggs for the tip and the GIF.

Theoretical physicist Paul Dirac offered this example to show that some objects return to their original state after two full rotations, but not after one.

Hold a cup water in one hand and rotate it through 360 degrees (in either direction). You’ll have to contort yourself to accomplish this without spilling any water, but if you continue rotating the cup another 360 degrees in the same direction, you’ll find that you return to your original state.

The same principle can be demonstrated using belts. In the video below, the square goes through two full rotations and we find that the belts have returned to their original state. This would not be the case after a single rotation. (Here two belts are attached to the square, but the trick works with any number of belts.)

Another interesting item from James Tanton’s Mathematics Galore! (2012):

Write down a sequence of positive integers that never decreases. The list can include duplicates. As an example, here’s a list of primes:

2, 3, 5, 7, 11, 13

Call the sequence p_n. Now, a “frequency sequence” records the number of members less than 1, less than 2, and so on. For the list of primes above, the frequency sequence is:

0, 0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6

Pleasingly, the frequency sequence of the frequency sequence of p_n is p_n. That is, if we take the frequency sequence of the list 0, 0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6 above, we get 2, 3, 5, 7, 11, 13 again.

Now add position numbers to each of the two lists, p_n and its frequency sequence — that is, add 1 to the first element of each, 2 to the second, and so on. With the primes that gives us:

P_n: 3, 5, 8, 11, 16, 19 …

Q_n: 1, 2, 4, 6, 7, 9, 10, 12, 13, 14, 15, 17, 18, 20 …

These two sequences will always be complementary — all the counting numbers appear, but they’re split between the two sequences, with no duplicates.

French mathematician Jean Paul de Gua de Malves discovered this three-dimensional analogue of the Pythagorean theorem in the 18th century.

If a tetrahedron has a right-angled corner (such as the corner of a cube), then the square of the area of the face opposite that corner is the sum of the squares of the areas of the other three faces.

Above,

In Pascal’s triangle, each number is the sum of the two immediately above it:

In 1972, Henry Mann and Daniel Shanks found a curious connection between the triangle and prime numbers. Stagger the triangle’s rows so that row n starts at column 2n:

Now a column number is prime precisely when the numbers in that column are each divisible by their row number. For instance, in the diagram above, column 13 has two entries — 10, which is divisible by 5, and 6, which is divisible by 6 — so 13 is prime. The numbers in column 12 are not all evenly divisible by their row numbers, so 12 is not prime.

“It’s a nifty and surprising result,” writes James Tanton in Mathematics Galore! (2012), “but it is not a formula that allows us to find prime numbers with ease.”

(Henry B. Mann and Daniel Shanks, “A Necessary and Sufficient Condition for Primality, and Its Source,” Journal of Combinatorial Theory, Series A 13:1 [1972], 131-134.)

If two unit circles are tangent externally as shown, and from a point P on one circle rays PQ and PR are drawn intersecting both circles, then arc lengths x + y = z.

From Claudi Alsina and Roger B. Nelsen, Icons of Mathematics, 2011.