I left school a "Cut the Knot" kind of guy, Occam's razor states "Entia non sunt multiplicanda praeter necessitatem". It means the simplest solution IE. fewest errors, shortest route, least steps taken etc. without letting it multiply beyond necessity (or a students attention) is the best way to solving any of life's problems. It'says so much more for our learning processes though. Why not Cut the knot, and why not drop the barometer of the roof. If someone can not present the problem with complexities as to eliminate the simple answer, then why bother with the complex answer. If the Rope was a durable non breakable material, it may be far fetched but why not add a little wonder. It presents us with a reason to solve it, if there is sense of urgency, or improbability. If I were playing a video game, imagine I'm stuck on a roof, i have a barometer and a parachute (a little James Bond like i agree, what are the chances of these two useful items being up here with Me?) My parachute only opens at X height in feet, Can i jump? and How far From X am I?. Just imagine an exam taken in this way, if you were to cut the programs in half, you could eliminate cheating, where 50% of the class would have a random(ish*~) wrong answer below X and would see several different outcomes around them,if they could steal the equation at least they're thinking and if they can cut the knot.. GOOD LUCK TO THEM ...

(*~I say random ish because to me Random=Surprise, My Mp3Player and most programs I've seen, be it two columns of sequence in numerate or one always start (column one) number one,track one or the last track i was playing. I'm yet to be surprised from the get go i implore you, find a way to leapfrog the start position without eliminating the possible "wrap around function" (Like in music's (Dz) base 12 / Base 8 C-Major Harmonic Scale C,D,E,F,G,A,B)in which he jumps past the last track in my mp3 universe and land back where he started(Last Known value+1(i+1=?)). Or reach the end and come back of course(maybe bounce of the wall). Anyway...

Occam's razor "Entities must not be multiplied beyond necessity."

This reminds me of the first times table i saw: 16

09 12

04 06 08

01 02 03 04

(n^2) Squared and all n's below (or up to n to n^2 and on toward the right +n to infinity), no need for repartition.

Leaving out the infinite nought's and the equal opposit's eg.

16 12 08 04 0 0 04 08 12 16

12 09 06 03 0 0 03 06 09 12

08 06 04 02 0 0 02 04 06 08

04 03 02 01 0 0 01 02 03 04

00 00 00 00 0 0 00 00 00 00

Y by X & X by Y

This simple truth is an empty space filled with an amount of two's eg. 0+2+2+2+2+2+2+2+2+2+2+2+2+2+2 is a waste of time to write and near impossible to count at times, and telling you that i wrote it 14times showed me the fundamentals of addition, subtraction, multiplication and division. In one table from 1^2 to 4^2. This way i could reverse the question to the one which I was more familiar. The axis are made up of n?+n? multiple times in multiple directions with negatives all over the axis or't third dimension*~ etc etc etc.... Better to start with nought to explain the first step, but emitting it's repetitive nature.

*~ 3D example (X)(Y)Z.

Number One must be a square example of a prime as it has the same function in the cube. We are going to need a 3D projection of our times table cube, and a stone that defies gravity and can bounce within the confides of said cube for the later experiment but I'll show you any way.

01

04 08 xx xx

02 xx xx xx

09 18 27

06 12

03

xx 32 48 64

xx 24 36

xx 16

xx

25 50 75 100 125

xx 40 60 80

15 30 45

10 20

05

My drawings with colours and giving it the 3D illusion by raising the immediate right column up one..

The xx represents a variable(similar in value but not form) of these 3D rectangles added together in the repeated column below.

Repeated thus:

x 3 3 3 1

3 3 1 x 6 6 3

3 1 6 3 x 6 3

1 3 3 x 3

1(1) +7(8) +19(27) +37(64)

Three's on the out side, six's in the middle and leading to the Cube Prime 1 at the top.

1x1

2x2 2x1

3x3 3x2 3x1

4x4 4x3 4x2 4x1

In primary school a teacher once gave us some different coloured blocks, we were basically asked to be creative and see how many different ways we could arrange them. This reminded me of my good night equations (There were Seven in my house and like the Walton's there were many good nights exchanged and many variables to take into account) and this gave me an opportunity to explore them, I've seen it called a handshake problem and, I'm sure you understand it but here is what we should all take from it:

A A+A 0+1(1)

A a+a A+B +3(4)

B B+A B+B

A a+a a+b A+C +5(9)

B b+a b+b B+C

C C+A C+B C+C

The amount of times goodnight was exchanged

a= 0

b=+1 (1) A+B

c=+2 (3) a+b, B+C, C+A

d=+3 (6) a+b, b+c, c+a, A+D, B+D, C+D

Tn= (n^2/2) + n/2

The amount of goodnight's said to each other.

a=0

b=+2(02) A+B b+a

c=+4(06) A+C B+C c+a c+b

d=+6(12) A+D B+D C+D d+a d+b d+c

Tn * 2 (repeated function ie. *2) The other Variable was, saying goodnight to there there teddy bear, my pet monster, or sadly to them self. But they could hug, have two (independent of thought) heads, or it could decrease with two kids in separate rooms and mum and dad visit each separately and only the mum gives hugs. The structure still stays at the most, Occam's Razor being the simple root and the most complex root being the point where most would say F*&k It CUT THE KNOT.

Though my probability fascination started with humble beginnings, I must admit I try to exploit it once a year at the grand national. But mathematics a gambler does not make, more likely a bookmaker. So i used my powers for good. To Find how many times the answer's appeared on the Times Table would require an odd shaped table at least for there odds value. eg. If i were to toss a stone onto this theoretical table, what are the chances of landing on any Number/Answer{n and all<n} placed in a better order perhaps or placed on a die of equivalent sides. Finding and documenting all these up to 600 was as fun as it was reading, yesterday that Dirichlet had found a way to list them and find the decaying parts that make it more seemingly random. With pen to paper like me, back in the 19th century. I was listing the division table once and asked how this question can also be the solution 2/3=2/3 (Two divided by three equals two thirds) the teacher told me to write it differently, i think that was besides the point, all the way round the point in fact, but great knot cutting to avoid a complex answer that was self explanatory studying a few of then. 23/3= 7 remainder 2 that still needs dividing by the 3 (7r2/3).

(Closest Known) Divisors Root's

How far from being a rectangle or square they are.

16 (16/16=1)

15 (16/15=1& 1 of 15 away)

14 (16/14=1& 2 of 14 away)

13 (16/13=1& 3 of 13 away)

12 (16/12=1& 4 of 12 away)

11 (16/11=1& 5 of 11 away)

10 (16/11=1& 6 of 10 away )

09 (16/9=1 & 7 of 9 away)

08 16 (16/8=2)

07 14 (16/7=2 & 2 of 7 away)

06 12 (16/6=2 & 4 of 6 away)

05 10 15 (16/5=3 & 1 of 5 away)

04 08 12 16 (16/4=4)

03 06 09 12 15 (16/3=5 & 1 of 3rd away)

02 04 06 08 10 12 14 16 (16/2=8)

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 (16/1=16)

The principle is in there for gathering the odds, though the English takes away the form (pattern) we can use this as the skeleton to get the equation's odds of Two dice (X)Y With x rolled first.

And so

00 + X/1 +X/2 ((STOP at 0 we don't need the other fractions to infinity they explain a deeper degree of accuracy)

01=1

02=1 1

03=1 1 1

04=2 1 1 1

05=2 1 1 1 1

06=2 2 1 1 1 1

...

A New divisor is added at 1, 4, 9... 2, 6, 12... 3, 8, 15... = 1 + 3 + 5... 2 + 4 + 6... 3 + 5 + 7... etc (The other fraction's or 0's are added later)

I had taken the teachers Square amount of(n^2) different calculations to commit to memory and made (n^2/2+n/2)a smaller none infinite, more manageable triangular number of calculations. None infinite because if it was less than a half, I'd add up from Nought and more, down from the Square. I see it like a rocket at 45deg to infinity as it climbs it gathers a part of each sum (n-1+n)0,(n-1+n)1,(n-1+n)3,(n-1+n)...?

similar to a knight's move in chess only one right an one up(as 0:1 is to 1:2 as 1:0 is to 2:1) without repeating repeating myself one can take a combination of the two, Coordinate VS Functions and find the value.

First stop on the integer rocket are all "Unique Factors and there odd relatives"

Or Divisors as I later found out they are called.

eg.

2= 2*1

6= 6*1 4= 4*1

3*2 2^2

24= 24*1 64= 64*1

12*2 32*2

8*3 16*4

6*4 8^2

So a times table without repeated function like narrowing it down to it's Closest Known Squareroot (irregular squares also OR Squares with Wrecked Angles as i like to call them) can be made thus..

(DE2 Lowest Possible amount of (X)Y to form a distinguishable characteristics of primes and all the other divisor amount's)

06 14 24 36 50 66 = 0+6+8+10.............

05 12 21 32 45 60 = 0+5+7+9+11.........

04 10 18 28 40 54 = 0+4+6+8+10.........

03 08 15 24 35 48 = 0+3+5+7+9+11.....

02 06 12 20 30 42 = 0+2+4+6+8+10.....

01 04 09 16 25 36 = 0+1+3+5+7+9+11.

Ignoring nought OR starting at 0 and accumulating each number along the X axis we get all the unique forms of rectangle, starting with the equal one's or Squares obviously 1:1 (0:1 is a line thus a perfect rectangle, but the times table allows this to disappear. If we share 1 by no people there is still one remainder to go over nothing to infinity, but it exists). Then the perfect rectangles up "2:1" 3:2... I tried the chess piece trick here also 1UP:1LEFT at squares and just 1UP like a pawn, to produce the weird table in it's simplest form. So if we know multiplying a number especially a prime or it's closest square bellow, by the same prime repeatedly it will increase at an equal(or relative in some way) rate of divisors.

*2= 0 0 0 0 0 1 1 1 1 1 1 1

*2= 0 0 0 0 1 1 1 1 1 1 2 2

*2= 0 0 0 1 1 1 1 1 2 2 2 2

*2= 0 0 1 1 1 1 2 2 2 2 2 2

*1= 0 1 1 1 2 2 2 2 2 3 3 3

0 1 2 3 4 5 6 7 8 9 10...

This alone would help my understanding without the squares being worth *1 and all others worth *2 but i wasn't about to declare 1 as prime, just similar in its adjusted properties to the odds structure.

1:0 3:1 5:2... thus adding at 0(0) + 1(1) + 3(4) + 5=(9)

Just in the same way we collect Tn (Where 0 MUST be true)

We can calculate it's increase

0=0

1=1/2 + 1/2 = 1

2=2 + 1 = 3

3=4&1/2 + 1&1/2 = 6

4=8 + 2 = 10

eg. If we just use the exponential value's for squares to eliminate repeated function (and to avoid the Prime SQ (1) which is harder to explain there relationship without) then we can sum up All the unique Factors ignoring the ghost skeleton which gives two or even four respectfully for 1, because if we have a distinguishing feature ie a red die and a blue one X&Y = BXRY, BYRX, RXBY, RYBX.

0=0 always

2 : 1=1, 2=2, 4=3 8=4, 16=5...

3 : 1=1, 3=2, 9=3, 27=4, 81=5...

3 : 2=2, 6=4, 18=6, 54=8, 162=10...

If we add variables they vary. So *2 : from the 3rd degree of (*3:2)

18=6, 36=10, 72=12 144=16, 288=18, 576=21

curiouser & curiouser...

Each Factorial Degree

Ignore the remainder's or take them from the original... or all sort(Occam's Razor, we'll ignore them for we know there importance)

r0 x/2 x/3 x/4 (00r01)...etc

01r00

02r00 01r00

03r00 01r01 01r00

04r00 02r00 01r01 01r00

05r00 02r01 01r02 01r01 01r00

06r00 03r00 02r00 01r02 01r01 01r00

07r00 03r01 02r01 01r03 01r02 01r01 01r00

08r00 04r00 02r02 02r00 01r03 01r02 01r01 01r00

09r00 4 was the last time it could divide and so is the sum of all previous *2divisors so if we take n-1 away(n+1, n*?) the difference is the nth value of divisors (...)

0=0

#Even's

1=1

2=3

3=5

#Odds (because of the weird square value)

4=8

5=10

6=14

7=16

8=20

#Even's

9=23

10=27

Because of our base 10 system each number ending in n and increasing by its self follow these patterns. I tried using an odd base system to eliminate evens appearing as clearly but its just replaced an still as apparent only fraction's stay true, which i used to keep on track. This didn't help, but the importance of place value and the unique patterns they create tickles Ones curiosity.

And so, each column respectfully separated at *10's (The Base 60 was good, 60/12 with the n&x i describe below, whoo! to much fun, i almost got lost with 12 unique divisors as apposed to our 4)...

00xTable {0}

01xTable {0,1,2,3,4,5,6,7,8,9}

02xTable {0,2,4,6,8}

03xTable {0,3,6,9} {2,5,8} {1,4,7}

04xTable {0,4,8} {2,6}

05xTable {0,5}

06xTable {0,6} {2,8} {4}

07xTable {0,7} {4} {1,8} {5} {2,9} {6} {3}

08xTable {0,8} {6} {4} {2}

09xTable {0,9} {8} {7} {6} {5} {4} {3} {2} {1}

Adding a new character for each prime would help spot them but would get complex fast. eg. 0<,a,b,c,bb,d,bc,e... ?

The Ancient Sumerians then Babylonians with the base 60 system had it best "12 insides of knuckle's on one hand pointed to with natures own appossable thumb, and times with place value on which other hand you choose out of the 5 fingers. And so with n12 fingers and x3(and it's varied degree's of divisibility ie 1 3rd of x) amount of knuckles, and no need to count on my toes as my grandfather always jokes we count to 60 with relatively no repetition where 12, 24, 36... can be expressed twice (Base 10= 0-9 10-99..)(Base 60= 0-59 60-359... (Base 12= per finger = 0-11...).

AND SO.

If we simplify these "Factorial Degree" further still and we start at the square root

If we say:

^2root of n = X

(Truncate n/(X-0) = 0Y > Nought) + (Truncate n/(X-1) = 1Y > Nought)...

All-(X-1)+(X-2)... = (X-1)^2+X/2 = (X-1)Tn (Triangular number: i came up with X*(X/2)+(X/2) i was very pleased to see it some years later simplified with the two x's obviously having the same goal thus the exponential.)

Then if n=15

X=3

0y=5

1Y=7

2y=15

3ySTOP=0

(-1X)Tn=3 ANSWER=24

And then take n-1 away

Then if n=14

X=3

0y=4

1Y=7

2y=14

3ySTOP=0

(-1X)Tn=3 ANSWER=22

No. of DE2 Divisors=2

If we continue in this way we find all the odds for The Lowest divisor table.

Thanks again for the great site keep up the good work, and forgive any ends i left untied, I'll leave a list of my Divisor Expression's While i worked them all out. They are colour coded on my chart so I'll try an keep to the original scheme.

FROM DERREN. 0(I Still Know 0 by 1 is a line but if it's equivalent square could be argued as a dot then it can stay up here with me).

(X~first..Y~second...etc=Z)

n ODDS (No Chance - (CKRoot2=12) 600)

ONE and TWO Unique Divisors (ONE DE2 Divisors) {110/600}

/1/ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599

THREE and FOUR Unique Divisors (TWO DE2 Divisors) {190/600}

/4/ 6, 8, /9/ 10, 14, 15, 21, 22, /25/ 26, 27, 33, 34, 35, 38, 39, 46, /49/ 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, /121/ 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, /169/ 177, 178, 183, 185, 187, 194, 201, 202, 203, 205, 206, 209, 213, 214, 215, 217, 218, 219, 221, 226, 235, 237, 247, 249, 253, 254, 259, 262, 265, 267, 274, 278, 287, /289/ 291, 295, 298, 299, 301, 302, 303, 305, 309, 314, 319, 321, 323, 326, 327, 329, 334, 335, 339, 341, 343, 346, 355, 358, /361/ 362, 365, 371, 377, 381, 382, 386, 391, 393, 394, 395, 398, 403, 407, 411, 413, 415, 417, 422, 427, 437, 445, 446, 447, 451, 453, 454, 458, 466, 469, 471, 473, 478, 481, 482, 485, 489, 493, 497, 501, 502, 505, 511, 514, 515, 517, 519, 526, 527, /529/ 533, 535, 537, 538, 542, 543, 545, 551, 553, 554, 559, 562, 565, 566, 573, 579, 581, 583, 586, 589, 591, 597

FIVE and SIX Unique Divisors (THREE DE2 Divisors) {72/600}

12, /16/ 18, 20, 28, 32, 44, 45, 50, 52, 63, 68, 75, 76, /81/ 92, 98, 99, 116, 117, 124, 147, 148, 153, 164, 171, 172, 175, 188, 207, 212, 236, 242, 243, 244, 245, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 338, 356, 363, 369, 387, 388, 404, 412, 423, 425, 428, 436, 452, 475, 477, 507, 508, 524, 531, 539, 548, 549, 556, 575, 578, 596

SEVEN and EIGHT Unique Divisors (FOUR DE2 Divisors) {103/600}

24, 30, 40, 42, 54, 56, /64/ 66, 70, 78, 88, 102, 104, 105, 110, 114, 128, 130, 135, 136, 138, 152, 154, 165, 170, 174, 182, 184, 186, 189, 190, 195, 222, 230, 231, 232, 238, 246, 248, 250, 255, 258, 266, 273, 282, 285, 286, 290, 296, 297, 310, 318, 322, 328, 344, 345, 351, 354, 357, 366, 370, 374, 375, 376, 385, 399, 402, 406, 410, 418, 424, 426, 429, 430, 434, 435, 438, 442, 455, 459, 465, 470, 472, 474, 483, 488, 494, 498, 506, 513, 518, 530, 534, 536, 555, 561, 568, 574, 582, 584, 590, 595, 598

NINE and TEN Unique Divisors (FIVE DE2 Divisors) {22/600}

/36/ 48, 80, /100/ 112, 162, 176, /196/ 208, /225//256/ 272, 304, 368, 405, /441/ 464, /484/ 496, /512/ 567, 592

ELEVEN and TWELVE Unique Divisors (SIX DE2 Divisors) {55/600}

60, 72, 84, 90, 96, 108, 126, 132, 140, 150, 156, 160, 198, 200, 204, 220, 224, 228, 234, 260, 276, 294, 306, 308, 315, 340, 342, 348, 350, 352, 364, 372, 380, 392, 414, 416, 444, 460, 476, 486, 490, 492, 495, 500, 516, 522, 525, 532, 544, 550, 558, 564, 572, 580, 585

THIRTEEN and FOURTEEN Unique Divisors (SEVEN DE2 Divisors) {3/600}

192, 320, 448

FIFTEEN and SIXTEEN Unique Divisors (EIGHT DE2 Divisors) {25/600}

120, /144/ 168, 210, 216, 264, 270, 280, 312, /324/ 330, 378, 384, 390, /400/ 408, 440, 456, 462, 510, 520, 546, 552, 570, 594

SEVENTEEN and EIGHTEEN Unique Divisors (NINE DE2 Divisors) {8/600}

180, 252, 288, 300, 396, 450, 468, 588

NINETEEN and TWENTY Unique Divisors (TEN DE2 Divisors) {5/600}

240, 336, 432, 528, 560

TWENTYONE and TWENTYTWO Unique Divisors (ELEVEN DE2 Divisors) {1/600}

/576/

TWENTYTHREE and TWENTYFOUR Unique Divisors (TWELVE DE2 Divisors) {6/600}

360, 420, 480, 504, 540, 600

etc...