The least knotted of all knots is a simple closed loop, the “unknot.” Certainly this is easy to spot on its own, but adding even a few twists can make it hard to recognize:

An elaborately draped loop can be quite difficult to distinguish from a knottier knot. Is this an unknot?

(Yes, it is.)

Surprisingly, while research is ongoing, it remains unknown whether the challenge of recognizing unknots is efficiently solvable — whether an algorithm can accomplish the task in polynomial time. It’s an open question.

In the 18th century Leonhard Euler famously addressed the question whether it was possible to walk through the city of Königsberg and return to one’s home having crossed each of seven bridges exactly once.

The answer, briefly, was no, but in 2013 network scientist Thilo Gross noticed that the city of Bristol has a similar layout, and here the task is possible: If you’re willing to walk 30 miles, you can cross each of the 45 bridges on this map and return to your starting point.

Details here.

A paradox by Columbia University logician Haim Gaifman:

line 1: The sentence on line 1 is not true.
line 2: The sentence on line 1 is not true.
line 3: The sentence on line 2 is not true.
line 4: The sentence on line 3 is not true.
line n + 1: The sentence on line n is not true.

All these sentence are equivalent, because each essentially restates its predecessor. Since the sentence on line 1 isn’t true, the sentence on line 3 isn’t true either. “[B]ut what I have just stated is not true, because it is the sentence on line 4 (or an obviously equivalent reformulation of it), and also this last statement of mine is not true, because it is the sentence on line 5, etc. None of these sentences can be successfully asserted, because none of them is true; but again I find myself slipping into nontruth: what I have just said is not true for it obviously includes the conjunction of these very same sentences; and also this last assertion is not true, and so on ad infinitum.”

(Haim Gaifman, “Pointers to Truth,” Journal of Philosophy 89:5 [May 1992], 223-261. See Yablo’s Paradox.)

Ghost leg is a method of establishing random pairings between any two sets of equal size. For example, it might be used to assign chores randomly to a group of people. The names of the participants are listed across the top of the diagram and the chores across the bottom, and a vertical line is drawn connecting each name to the chore below it. Then the names are concealed and each participant adds a “leg” to the diagram. A leg is a segment that connects two adjacent vertical lines (it must not touch any other horizontal line).

When the legs have been drawn, the names are revealed and a path drawn from each name to the bottom of the diagram. Each path must follow each leg that it encounters, jumping to the adjacent vertical line and continuing downward. When it reaches a chore at the bottom, it establishes a link between a name and a chore.

The benefit of this method is that it will work for groups of any size, reliably establishing a 1:1 correspondence between their elements. And it will work no matter how many horizontal lines are added. In Japanese it’s known as amidakuji.

From Lee Sallows:

A feature common to many geomagic squares is that the set of shapes they employ reveal an atomic structure. That is, they are built up from repeated copies of a single unit shape. Examples of this are piece sets composed of polyominoes, the unit shape then being a (relatively small) square.

For the would-be geomagic square constructor, a key advantage of the atomic property is that the shapes concerned are each describable in terms of the positions of their constituent atoms. Or, to put it another way, they can be represented by a set of numbers. Hence, unlike non-atomic shapes, they are readily amenable to analysis and manipulation by computer.

Take, for example, an algorithm able to identify and list each of the different ways in which a given planar shape can be tiled by some specified set of smaller shapes. Such a program might be challenging to write, but provided the pieces concerned are composed of repeated units, implementation ought to be straightforward. But could the same be said in the case of non-atomic pieces? Without a set of numbers to describe piece shapes, how are they to be represented in a digital computer?

This is worth noting since, as inspection will show, the shapes employed in the square above are plainly non-atomic. In line with this I can confirm that the only computer program involved in deriving this solution was a vector graphics editor used to create the drawing seen above.

(Thanks, Lee.)

Peter Nicholson’s Carpenter’s New Guide of 1803 contains an interesting technique:

To find a right line equal to any given Arch of a Circle. Divide the chord ab into four equal parts, set one part bc on the arch from a to d, and draw dc which will be nearly equal to half the arch.

Apparently this was an item of carpentry lore in 1803. In the figure above, if arc ad = bc, then cd is approximately half of arc length ab.

Nicholson warns that this works best for relatively short arcs: “This method should not be used above a quarter of a circle, so that if you would find the circumference of a whole circle by this method, the fourth part must only be used, which will give one eighth part of the whole exceedingly near.”

But with that proviso it works pretty well — in 1981 University of Essex mathematician Ian Cook found that for arcs up to a quadrant of a circle, the results show a maximum percentage error of 0.6 percent, “which I suppose can be said to be ‘exceedingly near.'” He adds, “[I]t would be of interest to know who discovered this construction.”

(Ian Cook, “Geometry for a Carpenter in 1800,” Mathematical Gazette 65:433 [October 1981], 193-195.)

In the 1970’s, a student of Maoist inclination asked [Columbia philosopher Sidney Morgenbesser] if he disagreed with Chairman Mao’s saying that a proposition can be true or false at the same time. Dr. Morgenbesser replied, ‘I do and I don’t.’

From Morgenbesser’s 2004 New York Times obituary.