Math Discoveries

Page

1. Hargreave’s offhead squaring formula

For numbers between 25-75 you can use it as follows:

x2 = (|x-50|)2 + (x-25)100

For numbers between 75-125 you can use it as follows:

x2=(|x-100|)2 +(x-50)200

Note: you must know all squares offhead of all numbers below

25 to use it.

The 2 stands for square

2. ( The 7 wonders of )theHargreaves wheel


The Hargreaves' wheel

The Hargreaves' wheel

Wonder1:

The smallest circle consists of numbers 1-5 arranged such that no adjacent numbers on the number scale are adjacent to each other.

Wonder2:

There is only one way of arranging them.ie, the way they have been arranged

Wonder3:

On adding each 2 numbers to form the third wheel, you get numbers that are adjacent to each other on the number scale.

The same applies to the 3rd wheel, if every 3 numbers on the first wheel are added to each other.

Wonder4:

On the fourth wheel, when every four numbers are added to each other from the first wheel, we get a rather random pattern. Or is it?

On the first wheel, the differences betweet adjacent numbers forms is 3-2-3-2

On the fourth wheel, this is the same pattern!

Wonder5:

The two greatest numbers on each wheel are the least two numbers in the next wheel!

Wonder6:

The sum of the numbers in each wheel is 15n, where n is the number of the wheel!

3. Hargreaves’ Tan89

tan(89 + 9(1/10^0 +1/10^1 +1/10^2 +1/10^3+…..1/10^(n-1)) = tan89.9(10^n)

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