Choose any three by three square of dates on a calendar page...
Can you explain how this card trick works?
Take any whole number between 1 and 999, add the squares of the
digits to get a new number. Make some conjectures about what
happens in general.
Published September 1999,February 2011.
We decided to investigate the number of different ways various
totals may be obtained by adding small numbers of odd numbers.
Systematic working led us to consider the cases: 1 odd number, 2
odd numbers, 3 odd numbers.
There was unanimous agreement! Each of the totals 1,3,5,7,...
can be obtained in only one way. The pattern is:
with repeating block .
Eventually there was unanimous agreement:
and a repeating block pattern emerges. In this case the 1st
differences are repeated:
with repeating block in the 1st differences of .
There was some agreement and much confusion. What emerged was
There was initial excitement when 1,1,2,3,... emerged and
Fibonacci's name was bandied about.
By the time 1,1,2,3,4,5,... was reached the initial 1 was being
viewed as a "rogue" value and most of the class were in agreement
with 6,7,8,... being the continuation. There was great
consternation when 7 emerged and not 6.
Eventually this pattern emerged:
and so the conjecture emerged that there was a repeating block
in the 2nd differences of [1,0,0,0,1,-1].
The question was now asked:
" In how many ways can 1999 be written as the sum of three odd
Agreement was reached that continuing the pattern to reach 1999
was not feasible!
A fresh look at the block structure:...
raised a new question: In which block is 1999?
The first "total" in each block gives