## Seked

If a pyramid is 250

cubitshigh and the side of its base 360cubitslong, what is itsseked?

Take half of 360; it makes 180. Multiply 250 so as to get 180; it makes of a cubit. A cubit is 7 palms. Multiply 7 byThe seked is palms [that is, ].

A’h-mose.The Rhind Papyrus. 33 AD

The image above presents one of the problems included in the Rhind Papyrus, written by the scribe Ahmes (A’h-mose) circa 33 AD. This is a description of all the hieroglyphs, as translated by August Eisenlohr:

The question for the reader, after going carefully over the English translation is: What does *seked* mean?

## Trigonometry

I have just realized that I haven’t posted a good puzzle here in a long time, so here it goes one on Trigonometry, that the average student of Calculus should be able to tackle: you can use **anything** you think it could help: derivatives, symmetry, periodicity, integration, summation, go to several variables, differential equations, etc

Prove that, for all real values it is

or, in a more compact notation,

## Geolocation

Recall the **First Spherical Law of Cosines**:

Given a unit sphere, a

spherical triangleon the surface of the sphere is defined by the great circles connecting three points , , and on the sphere. If the lengths of these three sides are (from to (from to and (from to and the angle of the corner opposite is then

In any decent device and for most computer languages, this formula should give well-conditioned results down to distances as small as around three feet, and thus can be used to compute an accurate geodetic distance between two given points in the surface of the Earth (well, ok, assuming the Earth is a *perfect sphere*). The geodetic form of the law of cosines is rearranged from the canonical one so that the latitude can be used directly, rather than the colatitude, and reads as follows: Given points and with positions and respectively, the distance between the two points is given by the following formula.

where is the radius of the Earth in miles (well, ok, the average radius of the Earth…)

A nice application of this formula can be used for **geolocation** purposes, and I recently had the pleasure to assist a software company (thumb-mobile.com) to write such functionality for one of their clients.

Go to www.lizardsthicket.com in your mobile device, and click on “Find a Location.” This fires up the location services of your browser. When you accept, your latitude and longitude are tracked. After a fast, reliable and resource-efficient algorithm, the page offers the location of the restaurant from the Lizard’s chain that is closest to you. Simple, right?