## Sympy should suffice

I have just received a copy of Instant SymPy Starter, by Ronan Lamy—a no-nonsense guide to the main properties of `SymPy`, the Python library for symbolic mathematics. This short monograph packs everything you should need, with neat examples included, in about 50 pages. Well-worth its money.

To celebrate, I would like to pose a few coding challenges on the use of this library, based on a fun geometric puzzle from cut-the-knot: **Rhombus in Circles**

Segments and are equal. Lines and intersect at Form four circumcircles: Prove that the circumcenters form a rhombus, with

Note that if this construction works, it must do so independently of translations, rotations and dilations. We may then assume that is the origin, that the segments have length one, and that for some parameters it is We let `SymPy` take care of the computation of circumcenters:

import sympy from sympy import * # Point definitions M=Point(0,0) A=Point(2,0) B=Point(1,0) a,theta=symbols('a,theta',real=True,positive=True) C=Point((a+1)*cos(theta),(a+1)*sin(theta)) D=Point(a*cos(theta),a*sin(theta)) #Circumcenters E=Triangle(A,C,M).circumcenter F=Triangle(A,D,M).circumcenter G=Triangle(B,D,M).circumcenter H=Triangle(B,C,M).circumcenter

Finding that the alternate angles are equal in the quadrilateral is pretty straightforward:

In [11]: P=Polygon(E,F,G,H) In [12]: P.angles[E]==P.angles[G] Out[12]: True In [13]: P.angles[F]==P.angles[H] Out[13]: True

To prove it a rhombus, the two sides that coincide on each angle must be equal. This presents us with the first challenge: Note for example that if we naively ask SymPy whether the triangle is equilateral, we get a `False` statement:

In [14]: Triangle(E,F,G).is_equilateral() Out[14]: False In [15]: F.distance(E) Out[15]: Abs((a/2 - cos(theta))/sin(theta) - (a - 2*cos(theta) + 1)/(2*sin(theta))) In [16]: F.distance(G) Out[16]: sqrt(((a/2 - cos(theta))/sin(theta) - (a - cos(theta))/(2*sin(theta)))**2 + 1/4)

Part of the reason is that we have not indicated anywhere that the parameter `theta` is to be strictly bounded above by (we did indicate that it must be strictly positive). The other reason is that `SymPy` does not handle identities well, unless the expressions to be evaluated are perfectly simplified. For example, if we *trust* the routines of simplification of trigonometric expressions alone, we will not be able to resolve this problem with this technique:

In [17]: trigsimp(F.distance(E)-F.distance(G),deep=True)==0 Out[17]: False

Finding that with `SymPy` is not that easy either. This is the second challenge.

How would the reader resolve this situation?

## Project Euler with Julia

Disclaimer: Project Euler discourages the posting of solutions to their problems, to avoid spoilers. The three solutions I have presented in my blog are to the three first problems (the easiest and most popular), as a means to advertise and encourage my readers to “push the envelope” and go beyond brute-force solutions.

Just for fun, and as a means to learn Julia, I will be attempting problems from the Project Euler coding exclusively in that promising computer language.

The first problem is very simple:

Multiples of 3 and 5

If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multip les is 23.Find the sum of all the multiples of 3 or 5 below 1000.

One easy-to-code way is by dealing with arrays:

julia> sum( filter( x->((x%3==0) | (x%5==0)), [1:999] ) ) 233168

A much better way is to obtain it via summation formulas: Note that the sum of all multiples of three between one and 999 is given by

Similarly, the sum of all multiples of five between one and 995 is given by

Finally, we need to subtract the sum of the multiples of both three and five, since they have been counted twice. This sum is

The final sum is then,

## 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?

## Nezumi San

On January first, a pair of mice appeared in a house and bore six male mice and six female mice. At the end of January there are fourteen mice: seven male and seven female.

On the first of February, each of the seven pairs bore six male and six female mice, so that at the end of February there are ninety-eight mice in forty-nine pairs. From then on, each pair of mice bore six more pairs every month.

- Find the number of mice at the end of December.
- Assume that the length of each mouse is 4
sun. If all the mice line up, each biting the tail of the one in front, find the total length of miceJinkō-ki, 1715

## Ruthless Thieves Stealing a Roll of Cloth

Some thieves stole a long roll of silk cloth from a warehouse. In a bush far from the warehouse, they counted the length of the cloth. If each thief gets 6

hiki, then 6hikiis left over, but if each thief takes 7hikithen the last thief gets no cloth at all. Find the number of thieves and the length of the cloth.Jinkō-ki, 1643

## Stones, balances, matrices

Let’s examine an easy puzzle on finding the different stone by using a balance:

You have four stones identical in size and appearance, but one of them is heavier than the rest. You have a set of scales (a balance): how many weights do you need to determine which stone is the heaviest?

This is a trivial problem, but I will use it to illustrate different ideas, definitions, and the connection to linear algebra needed to answer the harder puzzles below. Let us start by solving it in the most natural way:

- Enumerate each stone from 1 to 4.
- Set stones 1 and 2 on the left plate; set stones 3 and 4 on the right plate. Since one of the stones is heavier, it will be in the plate that tips the balance. Let us assume this is the left plate.
- Discard stones 3 and 4. Put stone 1 on the left plate; and stone 2 on the right plate. The plate that tips the balance holds the heaviest stone.

This solution finds the stone in two weights. It is what we call *adaptive measures*: each measure is determined by the result of the previous. This is a good point to introduce an algebraic scheme to code the solution.

**The weights matrix**: This is a matrix with four columns (one for each stone) and two rows (one for each weight). The entries of this matrix can only be or depending whether a given stone is placed on the left plate , on the right plate or in neither plate For example, for the solution given above, the corresponding matrix would be**The stones matrix**: This is a square matrix with four rows and columns (one for each stone). Each column represents a different combination of stones, in such a way that the*n*-th column assumes that the heaviest stone is in the*n*-th position. The entries on this matrix indicate the weight of each stone. For example, if we assume that the heaviest stone weights*b*units, and each other stone weights*a*units, then the corresponding**stones matrix**is

Multiplying these two matrices, and looking at the sign of the entries of the resulting matrix, offers great insight on the result of the measures:

Note the columns of this matrix code the behavior of the measures:

- The column indicates that the balance tipped to the left in both measures (and therefore, the heaviest stone is the first one)
- The column indicates that the heaviest stone is the second one.
- Note that the other two measures can’t find the heaviest stone, since this matrix was designed to find adaptively a stone supposed to be either the first or the second.

Is it possible to design a solution to this puzzle that is not adaptive? Note the solution with two measures given (in algebraic form) below:

Since each column is different, it is trivial to decide after the experiment is done, which stone will be the heaviest. For instance, if the balance tips first to the right (-) and then to the left (+), the heaviest stone can only be the third one.

Let us make it a big harder: Same situation, but now we don’t know whether the stone that is different is heavier or lighter.

The solution above is no good: Since we are not sure whether *b* is greater or smaller than *a*, we would obtain two sign matrices which are virtually mirror images of each other.

In this case, in the event of obtaining that the balance tips twice to the left: which would be the different stone? The first, which is heaviest, or the fourth, which is lightest? We cannot decide.

One possible solution to this situation involves taking one more measure. Look at the algebraic expression of the following example, to realize why:

In this case there is no room for confusion: if the balance tips three times to the same side, then the different stone is the first one (whether heavier or lighter). The other possibilities are also easily solvable: if the balance tips first to one side, then to the other, and then to the first side, then the different stone is the third one.

The reader will not be very surprised at this point to realize that three (non adaptive) measures are also enough to decide which stone is different (be it heavier or lighter) in a set of twelve similar stones. To design the solution, a good weight matrix with twelve columns and three rows need to be constructed. The trick here is to allow measures that balance both plates, which gives us more combinations with which to play. How would the reader design this matrix?