22 June 2018

Pythagorean theorem

Move the orange and/or the magenta points in order to change the lenght of the sides of the right triangle and observe the changes in the areas of the respective squares. The Pythagorean theorem states that 'the square of the hypotenuse is equal to the sum of the squares of the other two sides of the right triangle'.

21 June 2018

Square root of five

\(\sqrt 5\) is an irrational number which is present in the golden ratio.

It is also the measure of the hypotenuse of a right triangle whose other two sides measuring \(1\) and  \(2\) units.

We have also:

\(\sqrt 5 = e^{i\pi}+2\phi\);

\(\sqrt 5 \approx \frac{85}{38}\).

20 June 2018

Dattel

\(3x^2+3y^2+z^2=1\)

19 June 2018

Result of five operations (version 2)

Think of a number. Take a paper and a pen to do the following operations:
  • Think of a number;
  • Calculate its double;
  • Add \(10\) units to the result;
  • Calculate its half;
  • Subtract the number that you thought.

The number that you get after these five operations was \(5\), was not it? Can you explain the trick?

18 June 2018

Oware


The Oware is a game belonging to the family of Mancala games, also known as sowing games or count and capture games, having these games an important role in many African and Asian societies.

The Oware is a game for two people, played on a board with \(12\) houses, \(2\) deposits and with \(48\) seeds. The goal is to collect as many seeds, winning the player who gets \(25\) or more seeds on his deposit.

Rules

Each player chooses his side on the board. To know who will start the game, one player collects a seed in one hand and, if the other player can guess, then this player starts the game. Otherwise, it will be the other player to start the game. The deposit of each player is to his right.

Early in the game each house has \(4\) seeds. The player who starts the game collects all the seeds of one of his houses and distributes one by one in the other houses in the anti-clockwise, and such seeds are placed in the houses of the opponent. If a house has more than \(12\) seeds, the player should not count on the starter house in the distribution of the seeds. The player must not move in houses with one seed while other houses have more seeds.

If the player at the end of putting the seeds found that there are opponents houses containing \(2\) or \(3\) seeds, including those that he just placed, then he can capture them and put them in his deposit. In the event that one of the players runs out of seeds on his side after making his move, the opponent must make a move to enter seeds in the house of the another player. If a player take a capture and let the opponent without seeds, he will be required to make a move in order to introduce seeds in the houses of the opponent. 

When a player runs out of seeds and the opponent can not play in order to introduce seeds in the houses of this player, the game ends and the opponent collects the seeds that are in his houses and places them in his deposit. Wins who has the most seeds. When the game is almost over and the number of seeds means that there is a situation that is repeated indefinitely, then each player takes the seeds that lie in his houses and win the one with more seeds in his deposit.

17 June 2018

Challenge #10

The following figure is composed of \(16\) matches that form \(5\) congruent squares.


Modify the position of only \(3\) matches in order to obtain \(4\) congruent squares.