Tuesday, May 30, 2017

Collatz Conjecture

Mathematics and Nature: 

(from Wikipedia:  Collatz Conjecture )
The 3x+1 problem is a conjecture in mathematics which can be summarized as follows: 

Take any positive integer n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1.

Image from Wikipedia: 

Try yourself  the Collatz Conjecture with this tool:

Video Numberphile: (Collatz Conjecture)


And The briliant response to this video by Adrian Wadley:

Video Answer by: Video from: Adrian Wadley @wadleysf


Another way to show the Collatz Tree by: Matei Copot:  

Post made after a collection of messages on twitter: https://twitter.com/macajo/status/869506214891315201

Monday, February 20, 2017

Math Garden

Spring Math: Math Garden

In this app, the Natural Numbers are represented as Vegetables (plants): 

Fundamental Theorem of Arithmetic:
(From Wikipedia)
In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is prime itself or is the product of prime numbers, and that this product is unique, up to the order of the factors.

Therefore, if each decomposition of a number into its prime factors is unique, each number will have a unique form when it is shown as a plant.
But it is not exactly like that. Depends on the order in which the factors are multiplied.

This is shown in the following app videos:

There are two versions of Math Garden (Android) 

Kids Math Garden:

Math Garden:

The general operation is similar to that of plants:
Seeds fall from the sky
They should be planted. And then grow the plant corresponding to the seed (and the number chosen).
When the time comes to harvest dandelions, the plant is plucked from the ground.
Then the seeds that it has generated are released.
And the seeds return to heaven.

The particular operation is similar to that of the numbers:
If two unseeded seeds overlap, they add up.
If two seeds are planted, the plants multiply.

The plants are structured in branches, forking in function of the prime numbers that compose the factorization of the number.

Plants multiplied underground, often have a structure different from plants planted in a single blow.
(Multiplied plants do not have the usual order of a well-made factorization)
Each plant generates as many little dandelions as the number that indicates its seed, Having any of the structures that may have.

The program has 30 furrows one behind the other to be able to plant.

I hope it will be useful to teach maths. 

Math Garden:
Pythagorean Garden:
Pythagorean Garden

105 Plant In the Math Garden
Variants: The 6 subspecies:
(In invented mathematical plants, of course!!!)
Order of multiplication  vs. Factorization

UPDATE:  (August 2017)


Origin of Math Garden:
Jessica's Drawing of 96
(From Simon Gregg' work)

And Nummolt's:
"Touch Natural Numbers"  
"Touch Integers Z"

Saturday, January 7, 2017

Pan Balance 2: Fractions Scroll (Fulcrum And Lever)

The Well-sorted Irreducible Fractions

After the development of the MathCats balance and the Balance of Fractions, I realized that the new fulcrum of the MathCats Balance had not been sufficiently exploited.
The fulcrum of the MathCats Scale was passive. Was expressing only the result of the supposed slope of the balance. It was not really interactive.

In the new Android program, Fractions Scroll, the fulcrum of MathCats is already interactive. It responds to the touch, being able to move the fractions with the finger. From left to right (swipe)

This causes the slope indicator corresponding to the chosen fraction (on the right) to be tilted,

Scrolling up and down widens or reduces the range of fractions used in the program: Increases or decreases the maximum denominator and numerator used.

In the paid version: "Fractions Scroll Gravity Lever" this interaction can also be obtained by tilting the device during the fifteen seconds of use of the accelerometers after pressing the corresponding button.
In this version, when the accelerometers are running, the bar that indicates the slope corresponding to the chosen fraction always remains horizontal.

Fractions Scroll Gravity Lever Video: https://www.youtube.com/watch?v=F_fH5XMlIRI

Fractions Scroll Gravity Lever: Also available at Amazon:

Blog about MathCats Balance and Fractions Balance: