A centered hexagonal number is a number that can be represented by a hexagonal lattice with a dot in the center, like so:

Starting at the center, successive hexagons contain 1, 7, 19, and 37 dots. The sequence goes on forever.

The sum of the first n centered hexagonal numbers is n³, and there’s a pretty “proof without words” to show that this is so:

Instead of regarding each figure as a hexagon, think of it as a perspective view of a cube, looking down along a space diagonal. The first cube here contains a single dot. How many dots must we add to produce the next larger cube? Seven, and from our bird’s-eye perspective this pattern of 7 added dots matches the 7-dot hexagon shown above. The same thing happens when we advance to a 3×3×3 cube: This requires surrounding the 2×2×2 cube with 19 additional dots, and from our imagined vantage point these again take the form of a hexagonal lattice. In the last image our 3³ cube must accrete another 37 dots to become a 4³ cube … and the pattern continues.

Reader Derek Christie sent in this surprising curiosity after Wednesday’s post about Borromean rings:

Both the ring and the karabiner clip are attached to the cord and can’t be removed.

Now complicate matters by clipping the karabiner onto the ring:

Quite unexpectedly, the cord can now be just pulled away:

(Thanks, Derek.) (A related perplexity: The Prisoners’ Release Puzzle.)

“Rules for the direction of the mind,” from an unfinished treatise by René Descartes:

1. The aim of our studies must be the direction of our mind so that it may form solid and true judgments on whatever matters arise.
2. We must occupy ourselves only with those objects that our intellectual powers appear competent to know certainly and indubitably.
3. As regards any subject we propose to investigate, we must inquire not what other people have thought, or what we ourselves conjecture, but what we can clearly and manifestly perceive by intuition or deduce with certainty. For there is no other way of acquiring knowledge.
4. There is need of a method for finding out the truth.
5. Method consists entirely in the order and disposition of the objects towards which our mental vision must be directed if we would find out any truth. We shall comply with it exactly if we reduce involved and obscure propositions step by step to those that are simpler, and then starting with the intuitive apprehension of all those that are absolutely simple, attempt to ascend to the knowledge of all others by precisely similar steps.
6. In order to separate out what is quite simple from what is complex, and to arrange these matters methodically, we ought, in the case of every series in which we have deduced certain facts the one from the other, to notice which fact is simple, and to mark the interval, greater, less, or equal, which separates all the others from this.
7. If we wish our science to be complete, those matters which promote the end we have in view must one and all be scrutinized by a movement of thought which is continuous and nowhere interrupted; they must also be included in an enumeration which is both adequate and methodical.
8. If in the matters to be examined we come to a step in the series of which our understanding is not sufficiently well able to have an intuitive cognition, we must stop short there. We must make no attempt to examine what follows; thus we shall spare ourselves superfluous labour.
9. We ought to give the whole of our attention to the most insignificant and most easily mastered facts, and remain a long time in contemplation of them until we are accustomed to behold the truth clearly and distinctly.
10. In order that it may acquire sagacity the mind should be exercised in pursuing just those inquiries of which the solution has already been found by others; and it ought to traverse in a systematic way even the most trifling of men’s inventions though those ought to be preferred in which order is explained or implied.
11. If, after we have recognized intuitively a number of simple truths, we wish to draw any inference from them, it is useful to run them over in a continuous and uninterrupted act of thought, to reflect upon their relations to one another, and to grasp together distinctly a number of these propositions so far as is possible at the same time. For this is a way of making our knowledge much more certain, and of greatly increasing the power of the mind.
12. Finally we ought to employ all the help of understanding, imagination, sense and memory, first for the purpose of having a distinct intuition of simple propositions; partly also in order to compare the propositions.
13. If we perfectly understand a problem we must abstract it from every superfluous conception, reduce it to its simplest terms and, by means of an enumeration, divide it up into the smallest possible parts.
14. The problem should be re-expressed in terms of the real extension of bodies and should be pictured in our imagination entirely by means of bare figures. Thus it will be perceived much more distinctly by our intellect.
15. It is generally helpful if we draw these figures and display them before our external senses. In this way it will be easier for us to keep our mind alert.
16. As for things which do not require the immediate attention of the mind, however necessary they may be for the conclusion, it is better to represent them by very concise symbols rather than by complete figures. It will thus be impossible for our memory to go wrong, and our mind will not be distracted by having to retain these while it is taken up with deducing other matters.

He’d planned a further 15 but did not finish the work. These 21 were published posthumously in 1701.

It is time to bury the nonsense of the ‘incomplete animal.’ As Julian Huxley, the eminent British biologist, once observed concerning human toughness, man is the only creature that can walk twenty miles, run two miles, swim a river, and then climb a tree. Physiologically, he has one of the toughest bodies known; no other species could survive weeks of exposure on the open sea, or in deserts, or the Arctic. Man’s superior exploits are not evidence of cultural inventions: clothing on a giraffe will not allow it to survive in Antarctica, and neither shade nor shoes will help a salamander in the Sahara. I am not speaking of living in those places permanently, but simply as a measure of the durability of men under stress.

— Paul Shepard, The Tender Carnivore and the Sacred Game, 1973

These circles display an odd property — the three are linked, but no two are linked.

A.G. Smith exhibited this curious variant in Eureka in 1967:

“I leave it to the reader the problem of finding whether Knotung is knotted, and if so, whether it is equivalent to the Borromean Rings, with which it shares the property that cutting any one loop releases the other two completely.”

Srinivasa Ramanujan devised this magic square to mark his own birthday. He began with a Latin square (upper right) in which the numbers 1, 2, 3, and 4 appear in each row, column, and long diagonal as well as in the four corners, the four central squares, the middle squares in the top and bottom rows, and the middle squares in the outermost columns. Note the adjustments that would be necessary to reduce the four top cells to zero, and arrange these adjustments in the diagonally reflected pattern shown in the upper left. Now adding these two squares together produces the square in the lower left, which gives us a formula for creating a magic square based on any date (in the format 1 January 2001). The example at lower right is based on Ramanujan’s own birthday, 22 December 1887 (so D = day = 22, M = month = 12, C = century = 18, and Y = year = 87). In this example all 16 numbers are distinct, but that won’t be the case with every date.

You and a friend agree to meet on New Year’s Day at the Mozart Café in Vienna. You fly separately to the city but are dismayed to learn that it contains multiple cafés by that name.

What now? On the first day each of you picks a café at random, but unfortunately you choose different locations. On the second day you could both go out searching cafés, but you might succeed only in “chasing each other’s tails.” On the other hand, if you both stay where you are, you’ll certainly never meet. What is your best course, assuming that you can’t communicate and that you must adopt the same strategy (with independent randomization)?

This distressingly familiar problem remains largely unsolved. If there are 2 cafés then the best course is to choose randomly between them each day. If there are 3 cafés, then it’s best to alternate between searching and staying put (guided by certain specified probabilities). But in cases of 4 or more cafés, the best strategy is unknown.

In 2007 a reader wrote to the Guardian, “I lost my wife in the crowd at Glastonbury. What is the best strategy for finding her?” Another replied, “Start talking to an attractive woman. Your wife will reappear almost immediately.”

Staggering fact: Science historian Clifford Truesdell estimates that “[a]pproximately one-third of the entire corpus of research on mathematics and mathematical physics and engineering mechanics published in the last three-quarters of the eighteenth century” was written by a single person, Leonhard Euler.

The work of compiling Euler’s scientific writings has been going on since 1908 and will fill 81 volumes when complete. Mathematician William Dunham writes, “A typical volume of the Opera Omnia is large, running from 400 to 500 pages — although some contain over 700. In size and weight, such a volume resembles its counterpart from (say) the Encyclopedia Britannica. No one short of an athlete could carry more than five or six at once, and to cart off the entire collection — over 25,000 pages in all – would require a forklift.”

Laplace wrote, “Read Euler, read Euler, he is the master of us all.”