Choose any three by three square of dates on a calendar page.
Circle any number on the top row, put a line through the other
numbers that are in the same row and column as your circled number.
Repeat this for a number of your choice from the second row. You
should now have just one number left on the bottom row, circle it.
Find the total for the three numbers circled. Compare this total
with the number in the centre of the square. What do you find? Can
you explain why this happens?
Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?
Can you explain how this card trick works?
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