Freddie Manners, age 11 from Packwood Haugh School, Shropshire sent in the following beautiful solution. Freddie asks ``Is this relationship to the Golden Ratio coincidental?'' The answer is probably not. Mathematics if full of connections which at first seem surprising.The question involves the sides of a right-angled triangle, the cube of the Golden Ratio j = 1/2(1+Ö5), and the arithmetic, geometric and harmonic means of two number (AM, GM and HM respectively). Firstly Freddie found the cube of j = 1/2(1+Ö5).
j2
= 1
4
 (5+2Ö5+1)
j3
= 1
8
 (1 + Ö5)(6 + 2Ö5)
= 1
8
 (16 + 8Ö5)
= 2 + Ö5.
Take any two numbersa and b, where 0 < b < a.Because the AM is the largest we have
æ
ç
è 		(a+b)
2
 ö
÷
ø 		2

 
= ab + 1
æ
ç
è 		1
2
 ( 1
a
 + 1
b
 ) ö
÷
ø 		2

 
= ab + (2ab)2
(a+b)2
(2ab)2
(a+b)2
= æ
ç
è 		(a+b)
2
 ö
÷
ø 		2

 
 - ab
= æ
ç
è 		(a-b)
2
 ö
÷
ø 		2

 
2ab
(a+b)
= (a- b)
2
4ab
= a2 - b2
4a
b
= æ
ç
è 		a
b
 ö
÷
ø 		2

 
 - 1 .

Let the ratio a/b = x then


4x
= x2 -1
x2 - 4x -1
= 0
x
= 2 ±Ö5
As Ö5 > 2 the solution 2-Ö5 would give a minus number.

So a/b = 2 + Ö5 = j3 and a=bj3.