# Mathematics

The documents are copyrighted under the CC-BY-SA license, which permits freely downloading, copying, printing, etc. You may also modify the document provided that my name remains and that you distribute it under the same license.

## Bagrut Exams

פתרונות לכל השאלות בבחינות הבגרות במתמטיקה, שאלון 806, מהשנים תשע"ד עד תשע"ח.

.המסמך כולל גם פתרונות חלופיים, הנחיות לתהליך הפתרונות, מלכודות שיש להימנע מהם, ועוד

.נספח ב' ייצוג גרפי של משפים בגיאומטריה: קובץ נפרד להדפסה בצבע

."ניתן להוריד ולהעתיק את המסמך בחינם לפי תנאי רישיון "ייחוס-שימוש-לא-מסחרי-שיתוף זהה

## Enrichment

### The many guises of induction

Induction is often presented as a mechanical procedure for proving properties of sequences, but it widely used in other areas of mathematics (geometry, trigonometry, logic) and in computer science (data structures, automata, formal languages). This document tries to show that induction is a uniform concept although it appears in many guises. (Some of the examples and exercises require a background in computer science and discrete mathematics.)

Version 1.6.1, 14 April 2019:

### Surprising constructions with straightedge and compass

The following interesting constructions are presented in full detail at the level of secondary school mathematics:

1. A "collapsing" compass perform any construction that a "fixed" compass can. This is theorem 2 of Euclid's Elements, but many subsequent proofs are incorrect.
2. Trisection. Although angles cannot be trisected using a straightedge and  compass, the construct can be performed using other tools.
3. It is not possible to square a circle with straightedge and compass, because pi cannot be constructed. However, 355/133 is very close to the precise value of pi. This chapter presents Ramanujan's clever construction of this value.
4. Any construction possible with straightedge and compass can be performed with a compass only!
5. Any construction possible with straightedge and compass can be performed with a straightedge only, provided that a single circle exists somewhere in the plane.
6. Can there be non-congruent triangles with the same perimeter and area? The answer is yes! Here we show how to derive the values 156/35, 101/21, 41/15 of the edges of the triangle that have the same perimenter and area as the triangle with sides 3, 4, 5. (This results is not about construction with straightedge and compass but I included it anyway.

Version 1.0.0 (11 February 2019):

## Langford's problem

In the following arrangement of colored blocks: There is one block between the red blocks, two blocks between the blue blocks and three blocks between the green blocks. Expressed in numbers, the bag of numbers {1,1,2,2,3,3} can be arranged in a sequence 312132 such that between the two occurrences of i there are i blocks. Langford's problem asks if this is always possible for {1,1,...,n,n}. Donald Knuth showed that solutions to Langford's problems can be easily found using SAT solvers, such as LearnSAT that I developed.

### Minesweeper is NP-Complete

Richard Kaye showed that a puzzle based on the minesweeper game is NP-complete. This document presents Kaye's result with detailed explanations of the construction.

### Pythagorean triples

SAT solving is capable of solving mathematical problems beyond the reach of "normal" mathematical techniques. This document introduces SAT solving and gives an overview of the proof by Heule and Kullman that: in any division of the natural numbers into two disjoint subsets, at least one subset will contain a Pythagorean triple, that is, three numbers such that a2 = b2 + c2.

### Three-dimensional rotations

This is a tutorial on Euler angles and quaternions for describing rotations in three dimensions.

Version 2.0.1, 19 July 2018:

## Instructions for formatting the documents

The LaTeX source files for these documents can be found on GitHub at:

https://github.com/motib/mathematics

The formatting was done using MikTeX and TeXWorks:

https://miktex.org

The Cumulus Hebrew fonts were used:

http://www.ma.huji.ac.il/~sameti/tex/culmusmiktex.html

Notes:

• TeXWorks is included in the MikTeX distribution.
• When writing Hebrew LaTeX in TeXWorks, place mathematics (\$\$, \[\], etc.) on separate lines to avoid difficulties with the cursor movement.