Beatty Sequences

Here’s another interesting source of complementary sequences. Take any positive irrational number, say \sqrt{2}, and call it X. Call its reciprocal Y; in this case Y = 1/\sqrt{2} = \sqrt{2}/2, 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:

beatty sequences

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).


Biologist F.W. Went points out that the physical size of human beings was a critical factor in their mastery of fire. Any flame must maintain a certain size in order to sustain the ignition temperature of its fuel, and a wood or coal fire in particular radiates so much heat that it must maintain a fairly large critical mass in order to keep burning; a small fire will go out.

“Interestingly enough,” Went writes, “a wood or coal fire above the critical size produces just the right amount of heat to warm man in a cave, or a room, or a camping site. But ants or small rodents would have to keep too far away to make a fire economical, or rather, they would be unable to bring up enough wood to keep the fire going. Therefore in an ant society fire is not an economical possibility, and they have developed without its benefits, by operating only while outside temperatures are within the physiological range. Man on the other hand has been able to move into very cold areas by using fire.”

“Man, with his remarkable brain, developed the use of fire, but … only a creature of man’s size could effectively control that fire,” writes Peter S. Stevens in Patterns in Nature (1974). “It happens that a small campfire is the smallest fire that is reliable and controllable. A still smaller flame is too easily snuffed out and a larger one too easily gets out of control. Prometheus was just large enough to feed the flames and to keep from getting burnt.”

(F.W. Went, “The Size of Man,” American Scientist, 56:4 [Winter 1968], 400-413.)

Sundaram’s Sieve

sundaram's sieve

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.)

Celestial Mechanics

Being an angel is hard work. In his 1926 essay “On Being the Right Size,” J.B.S. Haldane writes, “An angel whose muscles developed no more power weight for weight than those of an eagle or a pigeon would require a breast projecting for about four feet to house the muscles engaged in working its wings, while to economize in weight, its legs would have to be reduced to mere stilts.”

And this takes no account of the weight of the harp. In The Book of the Harp, John Marson notes that gold is about 10 times heavier than willow, once the favorite wood of Celtic harp makers. He calculates that a harp of gold would weigh 120 pounds, far more than the 70-80 pounds of the largest pedal harp.

Should we worry about this? Let us not forget that it was angels who destroyed Babylon for its people’s wrongdoings. In the Book of Revelation, chapter 18, verse 21 tells us: “And a mighty angel took up a stone like a great millstone, and cast it into the sea, saying, ‘Thus with violence shall that great city of Babylon be thrown down.'”

This becomes a public health matter. Even if harps aren’t thrown at us deliberately by vengeful angels, Marson writes, “there is always the danger of one being dropped accidentally from a great height, resulting in the kind of damage caused on occasion by meteorites — unless, of course, the Bible is indeed correct after all, and angels do not play harps.”

See Hesiod’s Anvil.

Straight and Narrow

wells triangle

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.

The Plate Trick

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.)

Complementary Sequences

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 pn. 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 pn is pn. 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, pn 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:

Pn: 3, 5, 8, 11, 16, 19 …

Qn: 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.

De Gua’s Theorem

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.


 A_{ABC}^{2} = A_{ABO}^{2} + A_{ACO}^{2} + A_{BCO}^{2}

Pascal’s Primes

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:

pascal prime table

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.)

The Keyhole


Draw circles C1 and C2 with the common chord PQ. Now choose a point A on the arc of C1 that’s outside of C2 and project it through P to B and through Q to C.

Surprisingly, the length of BC remains the same no matter where A is chosen on its arc of C1.