Math Tricks

I came across a very interesting problem – one that seems to involve math tricks of some sort.  Or is it magic?  Whatever it is, it is certainly a math brain teaser!  Anyway, here is the graphic that had me scratching my head:

So what’s the deal here??  Where did the hole come from?  Please post your answers  :)

Tags: brain teaser, math brain teaser, Math Tricks

Long before Sudoku began showing up in newspapers, ancient peoples had their own math tricks up their sleeves.  What were these math tricks?  They are called magic squares.  Magic squares were known by the Chinese as far back as 650 BC.

What are magic squares?  They are squares of order n with n² numbers within the square’s n x n matrix such that the numbers in each column, row and diagonal add up the same number.   For example, the magic square for order n=3 looks like this:

Look carefully at the rows, columns and diagonals – they each add up to the same number – 15!  Now this magic square is a special case; it is the only magic square for order 3.  Sure you can rotate it and make reflections, but this is the only arrangement for an n=3 magic square.

Magic squares exist for n >= 3.  There exists a trivial magic square for n=1, and for n=2, there is no magic square.  The number for which the columns, rows, and diagonals add up to is called the magic constant.  The value of the magic constant for a square of order n is determined by the formula:

So for n=3, the magic constant (C_m) = 15.  For n=4, C_m = 34.  For n=5, C_m = 65, etc.

How many solutions are there for magic squares for n>3?  There are many!  For n=4, there are 880.  For n=5, there are 275305224.  How many are there for n=6?  I have seen an estimate for over 1.7745×10¹⁹!

Tags: Magic Squares, Math Games, Sudoku

So you have seen already a few posts about the golden ratio, which is approximately equal to 1.618. Are there math tricks that will allow you to determine the value for the golden ratio?  Well, since the title of this post is “The Golden Ratio and the Fibonacci Sequence”, then you might have guessed that there is a way to determine the golden ratio with the Fibonacci Sequence.

What the heck is the Fibonacci Sequence?  True, I have not written a post on it yet – but you just wait because there is a LOT to say about it later on – a LOT!!  But for now, let me just give you the sequence:

0,1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, …

Do you see the pattern?  Each value, starting with the second “1″ in the sequence, is simply the sum of the preceding two values.  The general formuls for the Fibonacci sequence is:

F(n) = F(n – 1) + F(n – 2)

So how does this interesting sequence of numbers relate to the golden ratio?  Take any value in the sequence and divide it by the preceding value – what do you get?

For example:

34/21 = 1.619

Looks familiar, eh?

Try it again for a pair farther down the sequence:

233/144 = 1.61806

In fact, this manipulation of the Fibonacci series converges to the golden ratio.

Also, you can perform this manipulation using an “out of frame” Fibonacci series – that is, choose any two consecutive numbers, apply the general formula F(n) = F(n – 1) + F(n – 2) to get a new sequence, and then from the new sequence you will be able to determine an approximation for the golden ratio by following the same procedure as outlined above.  For example, starting with 887 and 888, we get the series:

887, 888, 1775, 2663, 4438, 7101, 11539, 18640, 30179 ,48819, 78998, 127817, 206815, 334632

Notice here that you do not get a very good approximation if you divide 888 by 887.  But as you move down the sequence, the value you obtain gets closer and closer to the golden ratio:

334632/206815 = 1.6180258

A superb example of math tricks in nature!

Tags: 1.618, 1.61803399, fibonacci numbers, Fibonacci Sequence, fibonacci series, golden ratio, Math Tricks

So you have a single line of length X, and you want to extent the line to height Y such that you produce a golden rectangle.  Simple as pi pie!

This problem can be solved with some simple algebra, and it is useful if you wish to draw a golden rectangle given a line of any length.  For instance, you may want to incorporate golden rectangles into some artwork you are working on, or you may wish to crop photographs such that they are framed within a golden rectangle.

So, given a line of any length, you can break the line into two parts:

It is easy to see that:

From this, you can calculate that the length (A) of the sides of the square part of the golden rectangle is:

A = (A + B)/1.618

So just extend the line into a rectangle with base=(A+B) and height=(A+B)/1.618

Using this type of reasoning, if you have a square with a side of length A, and wished to extend the length to A+B such that the A+B is the length of the base of a golden rectangle, you can determine the length of B very easily:

Tags: 1.618, 1.61803399, divine proportion, golden phi ratio, Golden Rectangle, golden rectangle architecture, golden spiral, phi number, ratio golden, rectangle golden, spiral golden