Math Discoveries


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


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


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


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.


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!


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


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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