In the limit you get the sum of an infinite geometric series. What
about an infinite product (1+x)(1+x^2)(1+x^4)... ?
A circle is inscribed in an equilateral triangle. Smaller circles
touch it and the sides of the triangle, the process continuing
indefinitely. What is the sum of the areas of all the circles?
If a number N is expressed in binary by using only 'ones,' what can
you say about its square (in binary)?
Markland from The John Roan School, Gareth;
Euen and Alex from Madras College, Scotland; and Chin Siang from
Tao Nan School, Singapore; all sent in good solutions .
Jack from The Ridings High School
described the pattern:
I noticed that the number of 4s and 8s each increased by 1 for
each extra 6 and that the last digit was always a 9. I then
predicted that 666667² would equal 444444888889 and I
was correct. Therefore according to this pattern: (1 million 6s
followed by a 7)² would be written 1000001 4s followed
by 1000000 8s followed by a nine.
Doing these four calculations by long multiplication shows how
this pattern works. If $m$ is
the number of sixes in the number that is squared, the pattern
(one million sixes followed by a $7$)$^2 =$ (one million and one
$4$'s followed by a million $8$'s followed by a $9$).