Download 14 Apr 2014 - U3A Site Builder Home Page

STEYNING & DISTRICT U3A
Discovering Mathematics
Session 14
A Magical Number Matrix
Proof of the Cosine & Sine Rules
The Development of Computer Architecture
A Magical Matrix
19 8 11 25
12 1 4 18
16 5 8 22
21 10 13 27
14 3 6 20
7
0
4
9
2
I can predict that any limited
selection of 5 numbers from this
matrix will result in a total of 57.
Select a number from the matrix,
then exclude all numbers in the
same row and column.
Repeat this 3 times and you are
left with a single number. Add this
to the numbers previously chosen
and the total will be ???
27
19
2
8
1
57
Irrespective of the sequence of numbers selected the resulting total for this matrix
is always 57.
Similar results can be achieved with any size of matrix, eg 3x3, 5x4, 7x6 etc.
A Magical Matrix (Cont.)
Try a different selection of 5 numbers.
19 8 11 25
12 1 4 18
16 5 8 22
21 10 13 27
14 3 6 20
7
0
4
9
2
12
8
10
25
2
57
A Magical Matrix (Cont’d 2)
An explanation.
Select a required total, in this case 57, then pick any ten numbers which total 57
and locate them randomly above each column and beside each row.
12
7
0
4
9
2
1
4
18
0
The matrix is constructed by
adding the 2 numbers at each
intersection
eg. 7 + 12 = 19, 9 + 4 = 13, etc
The principle is very simple.
Each selected number represents
the sum of numbers above and
beside the relevant column & row.
The procedure forces each
selection to lie in a new row and
column.
Thus the 5 selections are the sums of 5 different pairs of the 10 generating numbers,
which is the same as the sum of all 10 numbers.
A matrix of this type can be constructed using any number of cells and utilising any
combination of numbers which can be positive or negative, fractions or decimals.
Proof of the Cosine Rule
The Cosine Rule states : Cos A = (b2 + c2 – a2)/2bc, with similar results for Cos B
and Cos C.
a2 = x2 + y2.
CosA = (b-y)/c
& SinA = x/c
B
c*CosA = b-y
a
c
x
So a2 = [(b-c)CosA]2 + (cSinA)2
Expand; a2 = b2 – 2bcCosA + c2Cos2A + c2Sin2A
Sin2A + Cos2A = x2/c2 + (b-y)2/c2 = [x2 + (b-y)2]/c2
Sin2A + Cos2A = c2/c2 = 1
y
A
b
or y = b – c*CosA & x = cSinA
C
a2 = b2 – 2bcCosA + c2(Cos2A + Sin2A)
But (Cos2A + Sin2A) = 1
So a2 = b2 – 2bcCosA + c2
Transposing;
2bcCosA = b2 + c2 – a2
and CosA = (b2 + c2 – a2)/2bc
Similarly; CosB = (a2 + c2 – b2)/2ac and CosC = (b2 +a2 – c2)/2ab
Proof of Sine Rule
SinA = x/c
x = c*SinA
x = a*SinC
So c*SinA = a*SinC
a/SinA = c/SinC
B
a
c
SinB = z/c
SinC = z/b
z = cSinB = bSinC
x
z
b/SinB = c/SinC = a/SinA
y
A
b
SinC = x/a
C
That’s it Folks !