8,112,015 is an arithmetic, composite, deficient, evil, odd, and wasteful number. It is also a 2,704,006gonal number.
COMPLETING A 5 x 5 PANMAGIC SQUARE:
This is another “Amazing Mathematical Feat”
you can perform for your friends, family, students, or cohorts.
First, ask for a volunteer from the
audience (which is easier if you have an audience of just one). Ask them to choose a number between 1 and
100, and then ask them to pick a square from a 5 x 5 grid and write that number
in the square that they choose. For
example: suppose they choose the number 14 and choose to place it in the middle
square of the second row (as illustrated below). Now explain that you are “going to place
numbers in each of the 25 squares below, using all of the numbers from 1 to 25,
in order to make a Magic Square. All of
this will be accomplished while leaving the chosen number 14 in the chosen
square below.”
If you are working at a blackboard, white
board, overhead projector, or similarly advance piece of educational
technology, you should circle or otherwise indicate the beginning number and
location. Here, I have highlighted the
square in yellow.







14
















15


Now here is what you need to know.
From 14, you will want to go up two
squares, and then go right 1 square, and write the number 15. However, that would take you outside the
boxes. Anytime your move takes you out
the top of the box, go to the bottom box in that column and continue the
move. Likewise, if you move takes you
out the right side of the box, go to the left most box in that row, and
continue your move. In this case, moving
two block up, will place you just outside the boxes, so you move to the bottom
box in the middle column – and then continue your move – one block to the right
and write the number 15.
From 15, or from any number that is
divisible by 5, instead of moving two boxes up and one to the right you will
need to move two boxes to the right.
This will take you off the grid on the right, so move to the left most
box on the bottom row. And write the
number 16.
For any number divisible by 5 remember to
move two boxes to the right.
For any number not divisible by 5 remember
to move two boxes up, and then one box to the right (like a Knights move in
chess).
If you go off the top of the grid, continue
at the bottom. And if you go off the
right side of the grid, continue on the left side.
Continue this until you get to a multiple
of 25. Your grid should look like this.


18


24


21

14


20


17


23


25



19


16


22

15


Now go back to the 14 and start working
backwards.
If the next number you are going to write is
a multiple of 5, move left two blocks.
If you go off the left edge, move to the right most square in that row
and continue counting you move.
If the next number you are going to write
is not a multiple of 5, go down two blocks and to the left one block. If you go off the bottom of the grid, move to
the top box in that column, and continue counting. If you go off the left side of the grid, move
to the right most box in that row, and continue counting.
When you get to “1” your grid will be
completed (like the grid below).
12

5

18

6

24

8

21

14

2

20

4

17

10

23

11

25

13

1

19

7

16

9

22

15

3

Make sure you explain to your volunteer
that this is a PanMagic Square: each of the four rows sum to (or “add up to”) 65,
each of the four columns sum to 65, each of the two main diagonals and eight
broken diagonal also sum to 65. Your
volunteer (and the audience) may be so amazed that they forget to
applause. Let them know that you understand
and forgive them, that this is a common reaction to the amazing mathematical
feat that you have performed – without a calculating device of any kind. You can also show them that you have only 10
fingers, making these calculations impossible to do on your fingers alone.
If your volunteer chooses a number between
26 and 50, the magic constant will be 190.
For numbers between 51 and 75, the magic constant will be 215. And for numbers between 76 and 100 the magic
constant will be 440.
Suppose your next volunteer choose the
number 77, and placed it in the second square in the fourth row (as shown
below). Then you follow the same
rules. Work forwards from 77 to 100,
then backwards to 76. Follow the same
rules previous explained for multiples of 5, and for numbers that are not
multiples of 5. And the same procedures
previously explained for going out of the grid on the top, bottom, left or
right.
76

94

82

100

88

97

90

78

91

84

93

81

99

87

80

89

77

95

83

96

85

98

86

79

92

If you are doing this in front of a
classroom full of students there will always be a knucklehead (I’m sorry – I meant
student) who will pick a number outside of the range you specified earlier – between
1 and 100. Just explain that “The spirts
are telling me that you are a nonbeliever.
You do not believe in following directions AND you do not believe that I
can successfully complete this feat for the number you propose. I, however, will accept your challenge. And after I have amazed you with my
mathematical magic, we will talk to your parents.”
Don’t worry – the process is the same. Only the last two digits will change, the
rest will remain the same in each block.
Suppose our beloved knuck … I mean beloved nonbeliever chose the number
1114 instead of 14 Then your completed
square will look like this. The have
your beloved nonbeliever calculate the magic constant of the square as a
homework assignment, but explain this is not for punishment, but because you
know the nonbeliever would not believe you – he will only believe it if he
himself performed the calculations.
1112

1105

1118

1106

1124

1108

1121

1114

1102

1120

1104

1117

1110

1123

1111

1125

1113

1101

1119

1107

1116

1109

1122

1115

1103

If you are using a transparency on an
overhead projector, and you really want to impress your audience – turn the
grid 90 degrees and solve it that way.
(This is exceedingly difficult to do if you are using a chalkboard or a
white board.)
You might also wish to impress your
audience by doing the grid in Roman numerals, or Vulcan numerals (Klingons are
not smart enough to do this – so don’t try Klingon):
The
following numerals are written in the standard Vulcan script. Sorry, they did not want to print out. But you can go to the source and see what the Vulcan numbers look like:Source: http://www.languagesandnumbers.com/howtocountinvulcan/en/vulcan/#numerals
.
Have fun with this one.
David.
A Diabolic Magic Square has the additional property of the magic sum along each of its broken diagonals. Curiously, we can construct a Diabolic Magic Square from a Graeco Latin Square! Read more about it on my blog: http://www.glennwestmore.com.au/category/latinsquares/.
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