Geometry

Geometry is the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogues. Expected questions involve calculating lengths, angles, areas, and volumes of various shapes, understanding geometric theorems, and solving problems related to spatial reasoning.

Solid Geometry / Geometry in Space Trigonometry Spherical Geometry Plane Geometry Vectors

• Question

  Divide the given shape into four congruent parts:

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  Topics:

  Geometry -> Plane Geometry -> Symmetry Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries)

• Question

  Dissect the given shape into four congruent parts:

  ​​​​​​​

  Topics:

  Geometry -> Plane Geometry -> Symmetry Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries)

• Question

  Dissect the given shape into four congruent parts:

  

  Topics:

  Geometry -> Plane Geometry -> Symmetry Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries)

• Question

  The numbers `1,2,3,4,5` are written at the vertices of a regular pentagon, with each number at exactly one vertex. A trio of vertices is called successful if it forms an isosceles triangle, where the number at its apex is greater than the numbers at the other two vertices, or where the number at its apex is smaller than the numbers at the other two vertices.

  Find the maximum number of successful trios that can exist.

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  Topics:

  Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Geometry -> Plane Geometry -> Symmetry Minimum and Maximum Problems / Optimization Problems

• Dissect into four parts

  Geometric shapes are called congruent if they coincide when superimposed. Cut the following shape into four congruent parts:

  

  Topics:

  Geometry -> Plane Geometry -> Symmetry Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries)
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Final, Grades 3-4 Question 3

• Cut a Boat in Half

  Geometric shapes are called congruent if they coincide when placed on top of each other. Cut the shape into two congruent parts

  

  Topics:

  Geometry -> Plane Geometry -> Symmetry Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Final, Grades 5-6 Question 6

• Where is the point?

  In a convex hexagon ABCDEF, triangles ACE and BDF are congruent and regular. Show that the three segments connecting the midpoints of opposite sides of the hexagon intersect at one point.

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  Topics:

  Geometry -> Plane Geometry -> Symmetry Geometry -> Plane Geometry -> Triangles -> Triangle Congruence Geometry -> Vectors
  Sources:
  • Gillis Mathematical Olympiad, 2019-2020 Question 3

• THE CHRISTMAS PUDDING

  "Speaking of Christmas puddings," said the host, as he glanced at the imposing delicacy at the other end of the table. "I am reminded of the fact that a friend gave me a new puzzle the other day respecting one. Here it is," he added, diving into his breast pocket.

  "'Problem: To find the contents,' I suppose," said the Eton boy.

  "No; the proof of that is in the eating. I will read you the conditions."

   

  "'Cut the pudding into two parts, each of exactly the same size and shape, without touching any of the plums. The pudding is to be regarded as a flat disc, not as a sphere.'"

  "Why should you regard a Christmas pudding as a disc? And why should any reasonable person ever wish to make such an accurate division?" asked the cynic.

  "It is just a puzzle—a problem in dissection." All in turn had a look at the puzzle, but nobody succeeded in solving it. It is a little difficult unless you are acquainted with the principle involved in the making of such puddings, but easy enough when you know how it is done.

  Topics:

  Geometry -> Plane Geometry -> Symmetry Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 168

• BOARDS WITH AN ODD NUMBER OF SQUARES

  We will here consider the question of those boards that contain an odd number of squares. We will suppose that the central square is first cut out, so as to leave an even number of squares for division. Now, it is obvious that a square three by three can only be divided in one way, as shown in Fig. `1`. It will be seen that the pieces A and B are of the same size and shape, and that any other way of cutting would only produce the same shaped pieces, so remember that these variations are not counted as different ways. The puzzle I propose is to cut the board five by five (Fig. `2`) into two pieces of the same size and shape in as many different ways as possible. I have shown in the illustration one way of doing it. How many different ways are there altogether? A piece which when turned over resembles another piece is not considered to be of a different shape.
  Topics:

  Geometry -> Plane Geometry -> Symmetry Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 290

• THE CIGAR PUZZLE

  I once propounded the following puzzle in a London club, and for a considerable period it absorbed the attention of the members. They could make nothing of it, and considered it quite impossible of solution. And yet, as I shall show, the answer is remarkably simple.

  Two men are seated at a square-topped table. One places an ordinary cigar (flat at one end, pointed at the other) on the table, then the other does the same, and so on alternately, a condition being that no cigar shall touch another. Which player should succeed in placing the last cigar, assuming that they each will play in the best possible manner? The size of the table top and the size of the cigar are not given, but in order to exclude the ridiculous answer that the table might be so diminutive as only to take one cigar, we will say that the table must not be less than `2` feet square and the cigar not more than `4`½ inches long. With those restrictions you may take any dimensions you like. Of course we assume that all the cigars are exactly alike in every respect. Should the first player, or the second player, win?

  Topics:

  Combinatorics -> Game Theory Logic -> Reasoning / Logic Geometry -> Plane Geometry -> Symmetry
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 398