Magic Number 2 7 – A Better Calculator

• Magic Number 2 7 – A Better Calculator Estimate
• Magic Number 2 7 – A Better Calculator Free
• Magic Number 2 7 – A Better Calculator Time Card
• Magic Number 2 7 – A Better Calculator As A
• Magic Number 2 7 – A Better Calculator Math

Magic Square

Digital Feedback Calculator Osterl Richard Osterlind's DIGITAL FEEDBACK takes the traditional add-a-number routine into the 21st century. Imagine two groups of spectators, each multiplying two 2-digit numbers on a pocket calculator to arrive at a totally random 4-digit number. Oct 19, 2015 What's New in Version 2.7.2. This is a healthy update.: ). Improved compatibility with OS X El Capitan. Engineering-wise, Magic Number is more streamlined and optimized. A Better History You get more functions. And there are tax functions to make tax less taxing. A Sweeter List You get more functions too. The magic number for Team A to win the division is still '5': 58 + 8 − 62 + 1 = 5. As you can see, the magic number is the same whether calculating it based on potential wins of the leader or potential losses of the trailing team. Indeed, mathematical proofs will show that the three formulas presented here are mathematically equivalent. They can also be written as '1 to 2' or as a fraction ½. The ratio represents the number that needs to be multiplied by the denominator in order to yield the numerator. In this case, ½. This is clearer if the first number is larger than the second, i.e. With the ratio 2:1, 2 can contain 1, 2 times. WEST Group: Kua Number 2, Kua Number 5, Kua Number 6, Kua Number 7, Kua Number 8. Now you can find your lucky feng shui directions and use this info to improve the feng shui of your home or office. You can position your bed so that it faces one of your best directions, as well as rearrange your office so that your desk is facing a lucky direction.

Contents of this Page What is a Magic 3x3 Square? The Magic 3x3 Square The Magic 4x4 Square The Magic 5x5 Square The Magic nxn Square Curiosities Simple Variants Magic Squares on the Internet References .

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What is a Magic 3x3 Square?

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

You can assemble the numbers 1 to 9 in a square, so that the sum of the rows, the columns, and the diagonals is 15. If you take the numbers 1 to 9, you have the standard square.

.......8+c..1+c..6+c....... .......3+c..5+c..7+c....... .......4+c..9+c..2+c.......

A magic square remains magic, if you change each numbers by a constant c. You add c on the left. You can also subtract, multiply or divide.

You define 4x4-, 5x5-.. squares correspondingly.

The Magic 3x3 Square top
You have 1+2+3+4+5+6+7+8+9=45. In a magic square you have to add 3 numbers again and again. Therefore the average sum of three numbers is 45:3=15. The number 15 is called the magic number of the 3x3 square.
You can also achieve 15, if you add the middle number 5 three times.

You can reduce 15 in a sum of three summands eight times:

15=1+5+9 15=1+6+8

15=2+4+9 15=2+5+8

15=2+6+7 15=3+4+8

15=3+5+7 15=4+5+6

The odd numbers 1,3,7, and 9 occur twice in the reductions, the even numbers 2,4,6,8 three times and the number 5 once.
Therefore you have to place number 5 in the middle of the magic 3x3 square. The remaining odd numbers have to be in the middles of a side and the even numbers at the corners.
Under these circumstances there are eight possibilities building a square: All the eight squares change into each other, if you reflect them at the axes of symmetry. You count symmetric squares only once. Therefore there is only one magic 3x3 square.
The Magic 4x4 Squaretop
The magic number is (1+2+..+15+16):4 = 34.
The computer found 86 reductions of 34 to a sum of four summands with the numbers 1 to 16.

34=01+02+15+16 34=01+03+14+16 34=01+04+13+16 34=01+04+14+15 34=01+05+12+16 34=01+05+13+15 34=01+06+11+16 34=01+06+12+15 34=01+06+13+14 34=01+07+10+16 34=01+07+11+15 34=01+07+12+14 34=01+08+09+16 34=01+08+10+15 34=01+08+11+14 34=01+08+12+13 34=01+09+10+14 34=01+09+11+13

34=01+10+11+12 34=02+03+13+16 34=02+03+14+15 34=02+04+12+16 34=02+04+13+15 34=02+05+11+16 34=02+05+12+15 34=02+05+13+14 34=02+06+10+16 34=02+06+11+15 34=02+06+12+14 34=02+07+09+16 34=02+07+10+15 34=02+07+11+14 34=02+07+12+13 34=02+08+09+15 34=02+08+10+14 34=02+08+11+13

34=02+09+10+13 34=02+09+11+12 34=03+04+11+16 34=03+04+12+15 34=03+04+13+14 34=03+05+10+16 34=03+05+11+15 34=03+05+12+14 34=03+06+09+16 34=03+06+10+15 34=03+06+11+14 34=03+06+12+13 34=03+07+08+16 34=03+07+09+15 34=03+07+10+14 34=03+07+11+13 34=03+08+09+14 34=03+08+10+13

34=03+08+11+12 34=03+09+10+12 34=04+05+09+16 34=04+05+10+15 34=04+05+11+14 34=04+05+12+13 34=04+06+08+16 34=04+06+09+15 34=04+06+10+14 34=04+06+11+13 34=04+07+08+15 34=04+07+09+14 34=04+07+10+13 34=04+07+11+12 34=04+08+09+13 34=04+08+10+12 34=04+09+10+11 34=05+06+07+16

34=05+06+08+15 34=05+06+09+14 34=05+06+10+13 34=05+06+11+12 34=05+07+08+14 34=05+07+09+13 34=05+07+10+12 34=05+08+09+12 34=05+08+10+11 34=06+07+08+13 34=06+07+09+12 34=06+07+10+11 34=06+08+09+11 34=07+08+09+10 . . . .

The summands 1 to 16 are distributed regularly in the reductions:

Summand: Number:

01 19

02 20

03 21

04 22

05 22

06 23

07 23

08 22

09 22

10 23

11 23

12 22

13 22

14 21

15 20

16 19

Unlike the 3x3 square there is not just one conclusion for the distribution of the numbers 1 to 16 in a 4x4 square.
Fact: There are 880 magic squares, counting the symmetric ones only once.
This is one of 880 possible squares:

......12..06..15...01...... ......13..03..10...08...... ......02..16..05...11...... ......07..09..04...14.......

This square is special. The number 34 is not only the sum of the numbers in the rows, the columns, and both diagonals, but also in every 2x2 square.

The Magic 5x5 Square top
The magic number is (1+2+..+24+25) : 5 = 65.
Reductions of the magic number 65.

65 = 01+02+13+24+25 65 = 01+02+14+23+25 65 = .. 65 = ..

65 = .. 65 = .. 65 = 10+12+13+14+16 65 = 11+12+13+14+15

The computer found 1394 reductions of the number 65.

The summands and their number in the sumsYou notice, that the middle number 13 = 65:5 appears most frequently. The numbers of the summands to smaller and bigger summands drop symmetrically to both sides. Fact: There are 275 305 224 magic 5x5 squares. (Scientific American 1/1976)
Making of a magic 5x5-square:
You go through the numbers 1 to 25. There are two rules for constructing a magic square 'top right' and 'if the place is occupied, go one down'.

.....................

Number 1 is placed in the centre of the first row. Number 2 follows top right. But then you leave the 5x5 square.Therefore you must imagine the square is a cylinder. The cylinder has the vertical square sides of the square as circumference. The horizontal sides touch each other and close the curved surface of the cylinder. So there is a field top right for number 2. If you unroll the cylinder, number 2 has gone to the last row one place to the right.

Number 3 follows top right.

...	Number 4 would lie outside the 5x5 square. So again you imagine the square would become a cylinder, this time with vertical axis. You can find a place for number 4. If you unroll the cylinder, you find the number in the third row on the far left.

Number 5 lies top right.
You use the second rule for number 6. If the field top right is occupied, you put the number one row down in the same column.
You go further on with 7, 8, and so on. - You use the same rule for number 16 as for number 6.
You can transfer this way of formation to all magic squares with odd numbers of the sides ;-).
There are also rules for magic squares with an even side length. They are more complicated however.

The Magic nxn Squareetop
The existence of magic squares nxn is proved for all numbers n>2. But there is no general rule.
The magic number is (1 + 2 + 3 + .. + n²) : n =0.5 * (n²+1) * n.

The magic numbers of the standard squares:

Magic Number 2 7 – A Better Calculator Estimate

Magic squares: 3x3 4x4 5x5 6x6 7x7 8x8 9x9 10x10
Magic numbers: 15 34 65 111 175 260 369 505
Curiosities top

..	Once again: A square is magic, if the numbers have the same sum in the rows, the columns and the diagonals.

...	A square is semi-magic, if the numbers have the same sum only in the rows and the columns.

...	A square is pandiagonal, if it is magic and if not only the numbers of the main diagonals, but also the broken diagonals have the same sum.

..	This magic square is pandiagonal.

...	You find the complementary square, if you replace each number n by 17-n.

The square is self-complemetary, because the new one is symmetric to the old one. See red axis.

...	A square is associative, if it is magic and if pairs of numbers lying symmetrically to the centre have the same sum. The sum is 26=5²+1 =n²+1.

..	The 3x3 magic square Lo Shu is associative.

You can fix the kind of the numbers.

..	A magic square with prime numbers

..	A 'semi'-magic square of square numbers All sums going through the centre have the same value. Source: http://www.mathpages.com/home/kmath417.htm

The sequence of these special magic squares is not finished. There are magic squares missing e.g.
>panmagic squares
>magic squares formed by smaller magic squares
>bordered magic squares (Book 2).

There is an explanation for all the different magic squares: They can be found by computers. A new property means a new query inside the program.

Simple Variants top

Magic Squares on the Internet top

English

Craig Knecht (Magic Square Models)

Water Retention Patterns

...

Imagine, the squares are the top of square prisms with the height given by the numbers. If you pour water in this solid, it stays in the centre upto the height 17. Then it flows off. The amount of water is (17-3)+(17-7)+(17-13)+(17-1)+(17-4)+(17-5)=69. There are nice problems: Biggest amount of water? Seperate ponds? Island?

Magic Number 2 7 – A Better Calculator Free

Eric W. Weisstein (MathWorld)
Magic Square, Panmagic Square, Associative Magic Square, Lo Shu

Harvey D. Heinz
Magic Squares, Magic Stars & Other Patterns

Holger Danielsson
Magic Squares

Ivars Peterson's MathTrek
More than Magic Squares

Mark S. Farrar
Magic Squares

MathPages
Solving Magic Squares

Robin Moseley
Magic Flexagon (.pdf-File)

Wikipedia
Magic square, Most-perfect magic square, Water retention on mathematical surfaces, Associative magic square
Lyn 1 13 esv.

German

Feng-Shui-Homepage
Das magische Quadrat Lo-Shu

Gerd Müller
Magisches Quadrat interaktiv

Hans-Peter Gramatke
Magische Quadrate

Holger Danielsson
Magische Quadrate

Jan Haase
Das Hexeneinmaleins aus Goethes 'Faust' (Lösung)

Jan Theofel und Martin Trautmann
Magische Quadrate und Würfel

Maria Koth
Magische Quadrate (.pdf.-Datei)

Paul Heimbach
Magische Quadrate

recordholders.org
Das größte Magische Quadrat der Welt

Udo Hebisch (Mathematisches Café)
Magische Quadrate

Wikipedia
Magisches Quadrat, Vollkommen perfektes magisches Quadrat

Referencestop
(1) Bild der Wissenschaften, Heft 8/1966, Heft 6/1968, Heft 10/1976
(2) Pieter van Delft /Jack Botermans: Denkspiele der Welt, München 1980 (1998 neu aufgelegt)
(3) Maximilian Miller, Gelöste und ungelöste mathematische Probleme, Leipzig 1982

Feedback: Email address on my main page

This page is also available in German with more details.

URL of my Homepage:
http://www.mathematische-basteleien.de/

© 2000 Jürgen Köller

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Magic Number 2 7 – A Better Calculator Time Card

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Real filament price

WORK IN PROGRESS! Great tool to compare real price of filament. Only measurement, which accurately tells you how much you can print with your purchased filament, is volume in cm³. Filament vendors should provide price per cm³ which doesn't happen now. So I made this simple tool for you to calculate it.

Example: You are offered 100m of PLA both in 3mm and 1.75mm for $25. With 3mm you will buy 2827cm³ but with 1.75mm just 962cm³ of plastic. That renders 1.75mm deal pretty awful as you will get just one third for the same price.

Filament name	Price per cm3	Total volume cm3	Total wieght kg	Test whistles printed (2.6cm3)
Prusa filament	0.001 $	1000 cm3	1 kg	???

Steps per millimeter - belt driven systems

The result is theoreticaly right, but you might still need to calibrate your machine to get finest detail. This is good start tho.

If you struggle how to use this calculator, try aksing in ℹ steps per mm forum.

Result	Resolution	Teeth	Step angle	Stepping	Belt
0Click to Share!	micron	0	0°	1/0th	0mm

Example:

M92 g-code can set the steps per mm in real time. Here is an example with your result for X axis.

Steps per millimeter - leadscrew driven systems

Music tag editor 3 1 3. Gives you number of steps electronics need to generate to move the axis by 1mm.

If you struggle how to use this calculator, try aksing in ℹ steps per mm forum.

Gear ratio

Filament name	Price per cm3	Total volume cm3	Total wieght kg	Test whistles printed (2.6cm3)
Prusa filament	0.001 $	1000 cm3	1 kg	???

Steps per millimeter - belt driven systems

The result is theoreticaly right, but you might still need to calibrate your machine to get finest detail. This is good start tho.

If you struggle how to use this calculator, try aksing in ℹ steps per mm forum.

Result	Resolution	Teeth	Step angle	Stepping	Belt
0Click to Share!	micron	0	0°	1/0th	0mm

Example:

M92 g-code can set the steps per mm in real time. Here is an example with your result for X axis.

Steps per millimeter - leadscrew driven systems

Music tag editor 3 1 3. Gives you number of steps electronics need to generate to move the axis by 1mm.

If you struggle how to use this calculator, try aksing in ℹ steps per mm forum.

Gear ratio

Result	Leadscrew pitch	Step angle	Stepping	Gear ratio
0Click to Share!	8	1.8°	1/16th	2 : 2

Magic Number 2 7 – A Better Calculator As A

Example:

M92 g-code can set the steps per mm in real time. Here is an example with your result for X axis.

Optimal layer height for your Z axis

Helps you to select layer height in a way, that Z axis moves only by full step increments. Z axis isn't usually enabled during inactivity. If the axis is disabled during micro-step, axis jumps to the closest full step and intorduce error. This effect is occuring to some extent even while leaving the Z axis motors enabled. This is most usefull to machines with imperial leadscrews but also for unusual layer heights with metric leadscrews.

Gear ratio

Layer height	Error over 10cm	Number of steps	Step length
0	8mm	8	1.8mm
0	8mm	8	1.8mm
0	8mm	8	1.8mm

Max Speed

Input your acceleration settings, distance/length of axis and you can see you your printer will hit your desired speed and for how long. Sixtyfour 1 7 4.

Magic Number 2 7 – A Better Calculator Math