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:

Monday, October 17, 2016

Pan Balance 1: Cats and fractions

Usually in elementary mathematics, teaching pan scale balances are used only for display them at the time of equilibrium, to verify that two quantities are equivalent.

In this case, apart of this use, it is also useful imbalance in the balance. 

Many years ago (2003), Wendy Petti of MathCats.com and me, we developed the "MathCats Balance":

" choose from a wide range of objects to place on this scale - from electrons to galaxies!
" So how can we balance thin cats with fat cats? You might try multiplying each side by the number shown on the opposite side of the balance. Will 2 x 6 thin cats balance with 5 fat cats? Yes, 12 thin cats do balance with 5 fat cats".

MathCats Balance App (Google Play)
MathCats Balance (Amazon) 

Many years after this, I developed under the same idea a pan balance of fractions.
Inspired in a old photograph of Maria Montessori and his son Mario:

 ( from Getty Images: www.gettyimages.es/fotos/maria-montessori )

Pan balance to weigh fractions:

This imbalance, when the imbalance ratio under certain conditions is proportional to the ratio of content of the dishes, is also the result of the division. The slope of a straight line. In this case, the result of the division of fractions.

To view it, you can multiply the contents of each dish, until the balance is obtained, the numbers for which has multiplied each dish are in turn the result  fraction of the division.
The program only multiply by prime numbers, because any number can be built with them.

 This is the "Fractions Balance" Android App:

1/2 + 1/3 + 1/6 
with: 1/3 + 1/5 + 1/6 + 1/8 + 1/10 + 1/12
And with: 1/3 + 1/4 + 1/5 + 1/8 + 1/12

I hope it helps on teaching division of fractions.

Wednesday, October 5, 2016

Place Value Time

After the experience of my old "Time Calculator" (for mathcats.com 2004) I tried to develop an app that explains how this webpage subtract time.

Over the years I have received all kinds of comments on how to make this calculation.
This time, I tried to explain graphically how the original application does.

And I think I've managed to build what should have been the original application.
What was missing was to be able to modify the difference between the two dates.

(and save last configuration) 

I must thank all those who have been making comments over many years

Here you can find the new: "Place Value Time"  (for Android devices) 
( https://play.google.com/store/apps/details?id=com.nummolt.time.place.value  ) 

 Here is the video of the 'Place Value Time' Android app:

 In the video:
  • - From countdown to elapsed time.
  • - my age.
  • - when I should have celebrated my 500,000 hours of life.

The app is available on Google Play, Samsung Galaxy Apps and Amazon.
Place Value Time - Google Play
Place Value Time - Amazon
Is a paid app.(the paid version of the previous 'Time Calculator' (Android)

Friday, September 16, 2016

Explore Ahmes 2/n table RMP and Akhmin n/17 table with 'Proper Fractions'

This summer, by accident I developed the 'Proper Fractions' app. (Pr.Fr. Amazon)

I was developing a new version of the 'Equivalent Fractions' But the development has been stopped for unknown reasons.
This left on my hands a circle, a square and a number line of fractions, all of them interactive.

Surprisingly for me, it proved to be very useful for exploring in unit fractions.
One issue that concerned me since I developed "Old Egyptian Fractions" for Mathcats.
I decided to do some videos to show it. It is a collection in a YouTube Playlist:

The main videos in the list:
"Explore The Ahmes 2/p table of the Rhind Mathematical Papyrus": 

Explore the Akhmim n/17 Table:

Proper Fractions It is a tool that can be useful in solving problems of Egyptian fractions:
Like the MathCats game:

More on Egyptian Fractions:
Milo Gardner:
Egyptian Unit Fractions:

I have only to add as a thank you: 
All this would not have been possible without the help and support of: +Milo Gardner (cryptanalyst)

Registered in MathTools (MathForum) http://mathforum.org/mathtools/tool/272979/ 

Saturday, June 11, 2016

Exploring Mersenne primes 2^p -1

Just a note:
My app "Touch Integers ℤ (+ - × ÷)" is useful to explore the Mersenne primes:
Here there's a tiny exploration from 2^2 -1 to 2^23 -1.
We found all numbers prime, except when p==11 and p==23
Like everybody knows:

The search is made with:  "Touch Integers ℤ (+ - × ÷)" Android app:

///////     UPDATE:  (August 2016)    ////////
Same exploration on Wagstaff Prime Numbers:  

 Woodall Primes:

Euler's Lucky numbers:


And Goldbach's Conjecture:

The exploration has been made with:  "Touch Integers ℤ (+ - × ÷)" Android app: