Studying Ramanajan's Nest of Square Roots Giving a Number

Don read the biography of Ramanujan: "The Man Who Knew Infinity- A Life of the Genius Ramanujan" by Robert Kanigel, years ago, but never studied his works until he saw the 3 books in the following series during the summer of 2011. The mother of one of Don's students let him borrow 2 of the books, the Primary School and the Middle School. He started working on some of Ramanujan's problems.

AN INTRODUCTION TO

CREATIVITY OF RAMANUJAN
AS
SUPER STUDENT STAR

by
P.K. SRINIVASAN

An Instructional Guide To
Mathematics Teachers
in
MIDDLE SCHOOL

a publication of

THE ASSOCIATION OF
MATHEMTICS TEACHERS OF INDIA, Chennai.
E-Mail: amti@vsnl.com
__________________________________________________________________________________

Don got interested in the Nest of Square Roots Giving a Number. He also started showing what he did to some of his students. Don made his first one below:

Don was intrigued by all the patterns above.

Don worked with Shouri, a 3rd grader, the following week he came up with this nest:

Don never saw one like Shouri did here- he repeated the 6*7 !  Shouri also tried breaking 49 into 8+41 above, but evidently he realized 41 was prime and went no further.

Shouri also came up with the generalzation:

Fine Work, Shouri !

Grant's work and his generalization following Ramanujan:

Another new thing that Don had not seen before! But he checked to see if that worked and it did.
Great work, Grant!
Don had Grant send an email to Prof. Bruce Berndt at the U of Illinois math department, who has written books about Ramanujan's notebooks. He encouraged Grant to find other generalizations using the nested square roots.
Tune in for the next exciting work following Ramanujan.