Benjamin Cornish

A network diagram showing the orbits of numbers with starting values between 1-250, and path lengths of at least 10. Six cycles have been identified, and coloured.

My first attempt at a #DiscreteDynamicalSystem

Take a number, n
If (n-1) is prime then map n -> n^2
If (n-1) is composite then map n-> Aliquot sum of (n-1)
Numbers 1-19 all end in a cycle. I am not sure about 20.

eg. 6 -> 36 -> 13 -> 16 -> 9 -> 7 -> 6.

#mathsky
#recreationalmaths
#mathsart

Competition time:

What are the "Rules", and what happens to 20?

(Spoiler - my PC has counted 31 steps from 20 and hit a number too big to process, no cycle in sight)

I've got 5 numbers now! With a link to Fermat primes

A non-prime-power is a number not written as p^k where p is prime and k is a natural number.

Try this.

Take a number, n, split it into proper prime divisors.

Write those, ascending, as the place values for a new term base n.

Repeat this recursively.

I think only 3 non-prime-powers will terminate.

10-> 25_10
25-> 5_25
5->0

12-> 23_12
27-> 3_27
3-> 0.

What is the third number?

ACB? How does this scale with higher n-gons? What about a cube or higher dimensions?

Competition time:

A sequence stats 1, k, ...

Each new term is the product of the last two subtract the sum of the other previous terms.

Which sequence grows faster k=1 or k=2 ?

#oeis

oeis.org/A117030

I'm highly productive

I have an alarm set for whenever the UNIX time stamp is prime. That helps motivate at deterministic yet hard to predict intervals. isthisprime.com/now/

Just had a fun conversation about Primitive Pythagorean Triples with Carol Jacoby from: theartofmathematicspodcast.com

Can you find a coordinate M:(a,b) such that |OM|=|MA|=k, with O:(0,0) & A:(20,20) and k an integer?

You should be able to listen in at the end of March

#talkingmaths
#mathsky

Happy half term holidays!

I am celebrating by releasing another episode of #TheMathematicianspod. Find out about Systems A & B today with a double bill of 'Babylonian' Mathematicians.

www.podbean.com/eas/pb-vjqv3...

#talkingmaths
#mathsky
#mathspodcast

Again in mod 2, a companion sequence

oeis.org/A392906

[0]
[1, 0]
[1, 1, 0]
[0, 1, 0, 0]
[0, 0, 1, 0, 0]
[0, 0, 1, 0, 0, 0]
[0, 0, 0, 1, 1, 0, 0]
[0, 0, 0, 1, 1, 0, 0, 0]
[0, 0, 1, 0, 0, 1, 1, 0, 0]
[0, 0, 0, 0, 1, 0, 1, 0, 0, 0]
[0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0]

New oeis.org/A393283 just dropped.

Take a triangular array starting (0),(1,0) Now draw a ray from the tip down to the next term. Each new term is the sum of terms on or adjacent to this ray.

0
1,0
1,1,0
2,3,2,0

Try this in mod 2 for a #cellularautomata.

#mathsky
#sequences
#recreationalmaths

I just lost a game of rock paper scissors, four - nill, to a lobster

The whole image has a square grid marked in faint blue lines. At the top, a heading in blue reads "Iceberg diagrams". At the bottom, there is a horizontal blue line, with the space below it shaded in blue. Five identical grey shapes with black edges sit at different heights relative to the blue line. The edges are made by joining gridpoints horizontally, vertically or diagonally. Starting at the top left, the edge goes one step right, one step diagonally right and down, two steps down, one step left, one step diagonally left and down, one step diagonally left and up, one step up, one step diagonally right and up, and one step up, to come back to the start. The leftmost shape is fully above the blue line and has the number 8 written above it. The next shape is one gridline lower and has the number 6 written above it. The next shape is one gridline lower and has the number 0 written above it. The next shape is one gridline lower and has the number -5 written above it. The next shape is one gridline lower and is now fully below the blue line, and has the number -8 written above it.

Here is a full set of iceberg diagrams. The score is how many full squares there are, counting the ones above the waterline as positive and the ones below the waterline as negative.

Given a list of integers (eg 8, 6, 0, -5, -8) can you make an iceberg that will produce exactly that list?

#MathSky

Looking forward to watching. I managed to make my masters thesis rely on the HBT. Was it necessary? Probably not. Was it sufficient to delight me? Yes

My latest obsession with maths teacher bingo has now grown to a 5*6 grid... I might have to stop here.

#alevelmaths

New #mathspodcast episode just dropped of the #themathematicianspod, if you want to listen to #Baudhayana again, but with slightly updated mathematics, you will love the episode on #Apastamba

A tower is built with one brick on top of another. Come meet India's second brick!

#talkingmaths
#mathsky

Nine 'rules' of ratio tables laid out in a 3 by 3 grid format.

Some 'aspects/rules' of ratio tables ... collected together as an aid to proportional reasoning.
'El. V–16' = Euclid's Elements Book 5 Proposition 16

#UKMathsChat
#iTeachMath

a(5) = 7 since the iteration gives [2, 2, 4, 3, 2, 1, 2, 2], with 7 terms before the pair 2,2 repeats.

New sequence just dropped to the #oeis

Start with the numbers [2,2,...] Then each new number is the product of the last two, written in mod n. The sequence stops when there is a repeating pair of consecutive numbers.

a(n) is the length of the above sequence.

oeis.org/A389543

#mathsky

Who wants to play: "Guess what I've been teaching"

#alevelmaths

Day 2 is ready

Oh I wrote that on the board just to check you were paying attention. *Ahem*

(In my class we are all allowed to make mistakes... "But these were just the mistakes I made that you noticed, I wonder how many you missed")

A cross-section of the celestial sphere annotated with transliterated Sanskrit terminology.

A diagram for a diagram-less tradition; gradually making some sense of Brahmagupta's Khandakhadyaka. With a couple of shadow measurements and the angle of the rising sun he could calculate the time of day. Astonishing really.

3*3 bingo card of phrases I say a lot each day, eg "have you checked the formula book?"

I've started a "Maths lesson Bingo" with my students. Can anyone suggest some phrases for my next card to be 4*4?

#alevelmaths

That's not a position vector, it's just 3 components wearing a trench coat.

Vector-bet Vector-bet Game

New game just dropped: #Vector-bet

Can you create a letter of the alphabet using a polynomial with vector coefficients?

Have a play for yourself with my #GeoGebra widget:
www.geogebra.org/m/hv99xfkj

Featuring drag and drop vector tools and a squiggly pen!

#sillymaths
#mathsky

Geogebra plot of P_5(t) with coefficients as 2D vectors, the drawing is almost like a signature.

Follow up from my #alevelmaths student:

If you use 2D vectors for a high order polynomial, the resulting plot is, once again, 2D - however it is highly disorganised: Here we see a Quintic Equation.

#mathsky

Geogebra plot of a quartic equation in t, with coefficients as 3D vectors

A great observation from my #Alevelmaths student: If you create a polynomial with vector coefficients $r=a+bt+ct^2+dt^3...$ you can get higher dimensional equivalents of the polynomial. Here is a 3D quartic equation.

#mathsky