trominoes

   the straight trominoes can be packed in a 3 by n rectangle in various ways   

Golomb Vase

CSR math art can be relaxing and enjoyable as meditative math a new way of math thinking that is completely non competitive and focuses on careful craft like thinking and comparisons
Golomb vase requires careful cuisennaire rod counting to discover the numbers used and how they relate to a sequence to do with Golomb Rulers

 

Yates ordering in an artsy fashion

Yates ordering of P(P{1,2}) and a comparison with standard lexicographic ordering

GCD boats puzzle

   there has been some turbulent weather and so it may be that one or more of these GCD boats has a mast damaged by wind, what is the idea behind these GCD boats, and do any of them need some repair work on the masts ?   

life is a mystery

tribotetra morph

tribotetra morph concept  { within and between transpositions }

comparisons from sequences related to the " Orloj " clock

Consider these sequences 121212121, 123212321, 1234321234321, 12345432123454321, Can you get all the counting numbers 1, 2, 3, 4, ... from contiguous segments of these sequences ? One of the sequences from the OEIS about the <1234321> example is sequence A028355 It is a kind of problem where intuition seems valid but proving is more difficult It seems <121> and <1234321> do have the surprising property that one can always find connected segments that generate each of the successive counting numbers in turn. This is not true for <12321> and <123454321>

the 210 puzzle

two hundred and ten, 210, may seem like a non eventful little number, but it is the 4th primorial and therefore quite important. the primorials are products of consecutive primes starting from 2,
so they start with 2 then 2*3 = 6 , then 2*3*5 = 30 and 2*3*5*7 = 210
what's more amazing about 210, is there are 4 = 2*2 prime factors and the
number of distinct factors is (1+1)*(1+1)*(1+1)*(1+1) = 16 = 4*4 and the
sum of all the distinct factors is 576 = 24*24
furthermore, all the 16 factors can be represented by rectangles and packed into a 24 by 24 square

the 120 puzzle

find rectangles for all the divisors ( or factors ) of 120 without repeats so the sum of the areas of these rectangles is 360, and they can be packed into 3 copies of a rectangle with area 120 unit squares

 

 

Sheen of Chess

  orange deluxe fairy chess of sheen   

Hamiltonian paths

   The 6 Hamiltonian paths on a 16 square, 4 by 4 square lattice grid   
   tilt the laptop screen at various angles for interesting colour & light effects  

 

 

the splendour and intrigue of colour ensembles

  Tatami mats in a 2 by n-1 room { from the Narayana's cow sequence }  

 
  Snazzy   

patterns for change { diptych }

squaring the square

no 3 in a line problem

tesselation plus padding them out with Mrs Perkins Quilts

semi Fibonacci numbers

flora and fauna resting in peace

ps and qs and queues

 R< 3F4F5F  <B 

an easy folding pattern puzzle

  This is a representation of a Woodall number.
A Woodall number is any natural number of the form n*(2^n) - 1
for some natural number n. The first few Woodall numbers are:  1, 7, 23, 63, 159, 383, 895, … (sequence A003261 in OEIS).  What is the Woodall number represented ?   

omega3

   three layers of information omega, bigomega, and the differences between these two values   

fractal sequence variations

The Jaded Dynamic Neutrosophic Stratosphere

  SOR~ ; \??/ ; ~10 >> BIG   

 

  SOR~ ; /?\ ; ~10   

turning the light on divisor chains

two similar art works based on the concept of divisor chains, a simple example to illustrate
start with a number, say 3.  clearly 3 | 3  ( 3 divides 3 )
3, 1  is OK as  1 | (3+1) ;
  3, 1, 2  is OK as  2 | (3+1+2)  ;
    3, 1, 2, 6  is OK as  6 | (3+1+2+6)
      3, 1, 2, 6, 4  is OK as  4 | (3+1+2+6+4)
        3, 1, 2, 6, 4, 8  is OK as  8 | (3+1+2+6+4+8)
the idea is to consider earliest ( smallest ) whole numbers not yet used
until we find a value is OK, satisfying the required property.
the sum is up to 24.  at this stage, adding 5 isn't going to help, neither is adding 7, nor 9,
nor 10, nor 11, what about 12 ?
12 does the job nicely as 24 + 12 = 36 and 12 | (3+1+2+6+4+8+12)
the puzzle is to discover from the grouping and ordering of the coloured rectangles
what are the divisor chains represented from permutations of { 1, ..., n }

Sieves

Even Sieve

Gradual Replacement Sieve

thinking with concepts

Lucas numbers and 33

abstract iceberg puzzle

 puzzle: what is the common idea used for the icebergs ?

subsequences

Fairy Board

From the Fairy Board identify:
[1] The 2 ways to arrange 4 non-attacking kings on a 4 X 4 board with 1 in each row and column.
See Hertzsprung's problem: ways to arrange n non-attacking kings on an n X n board,
with 1 in each row and column  (A002464)
[2] The 4 ways to place a non-attacking white and black rook on 2 X 2 chessboard.
See Number of ways to place a non-attacking white and black rook on n X n chessboard  (A035287)
[3] One of the 4 ways to place 10 nonattacking superqueens on a 10 X 10 board.
See Number of ways of placing n nonattacking superqueens on an n X n board  (A051223)

the blues

Blue23

Sunny day in the Ocean Depths of Ultraworld

Sheen of blues

tracing rook paths

Polyknights and F3Layer2

Directional Contiguous Polyknights with 5, 6 and 7 moves, where a 1-move is a stationary knight

F3Layer2 is the pattern arising from applying 3 factorial
to 3 colours and then again to the 3 blocks of dimensions 2 by 3