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Pterid · #2 June 4th, 2007, 02:31 PM

How about this for Problem 1. It's not as formal as it could be, but I'm not a pure mathematician by a long way.

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Consider a general square $S_1$ of dimension $2a$ , centred on co-ordinates $s_1 = (x,y)$ .

Now, by inspection (see the figure), the four possible locations of the centre of $S_2$ are:

$s_2 = \left( x \pm \frac{a}{2}, y \pm \frac{a}{2}\right)$

$~~~~~~~~~~~~~~~~~ = \left( x + \lambda_{1} \frac{a}{2}, y + \mu_{1} \frac{a}{2}\right)$

where $\lambda_{1}$ and $\mu_{1}$ are just multipliers taking the value $+1$ or $-1$ , depending on our choice.

If we now split that square similarly to form $S_3$ , the centre of that square will be at

$s_3 = \left( x + \lambda_{1} \frac{a}{2} + \lambda_{2} \frac{a}{4}, y + \mu_{1} \frac{a}{2} + \mu_{2} \frac{a}{4} \right)$

with $\lambda_2 , \mu_2$ defined similarly. Extend this principle to the Nth square, whose centre will be at

$s_N = (x + a \sigma_{xN}, y + a \sigma_{yN})$

with $\sigma_{xN} = \sum_{i=1}^{N} \lambda_i \frac{1}{2^i}$ and $\sigma_{yN} = \sum_{i=1}^{N} \mu_i \frac{1}{2^i}$

The problem of whether the squares' centres will converge, as $N \rightarrow \infty$ , is now reduced to the convergence (or not) of these two series.

It is not particularly troublesome that the sums contain the arbitrary sign-changing constants $\{ \lambda_i , \mu_i \}$ , because of a basic property of infinite series (assumed here) :

"The series $\sum u_n$ converges if the series $\sum |u_n|$ does." (ref: Absolute convergence .)

The series $\sum_{i=1}^{N} \frac{1}{2^{i}}$ converges to unity. Hence, the x- and y- coordinates of $s_N$ converge to definite values, regardless of our choices of sign along the way. (QED!)