Spiral Problem
Start with a square with a side length of one.
Divide it in half (left from right).
Divide the right half in half (top and bottom).
Divide the top half in half (left and right).
Divide the left half in half (top and bottom).
Divide the bottom half in half (left and right).
So on and so forth.
It creates a spiral.
The centre of each subsequent division is approaching a limit.
What are the coordinates of the point the spiral is moving towards?
-- Shaul
Labels: SharedPuzzle
posted by Rajesh Lal at
9:12 AM
7 Comments:
Take original square to have bottom left corner at (0,0) and top right corner at (1,1).
Return to original configuration with a square a quarter (by length) of the original square.
The centre of the original square is (1/2,1/2). As the (x,y) coordiantes of the bottom left coner are the same, I'll only look at the x coordinate of the bottom left corner. So quickly see (if my spinning head is keeping up), that the coordinate of the "final" square is the limit of the sum:
x = y = (1/2)(1+ 1/4 + 1/(4^2) + 1/(4^3) + ... )
Let z = 1/4.
Then x = y = (1/2)(1 + z + z^2 + z^3 + ...)
But 1/(1-z) = 1 + z + z^2 + ...
So x = y = (1/2)/(1-z)
= (1/2)/(1 - 1/4) = (1/2)/(3/4) = 2/3
So (x,y) = (2/3,2/3)
I'm too worn out to be sure.
That's exactly right.
Glad someone answered.
This post has been removed by the author.
Hi Shaul/shpilo. Thanks for the confirmation. My initial confidence was only about 85% that I hadn't goofed.
But I had another look and was about to post a cleaner way of seeing it:
x = y = 1/2 + (1/4)(1/2 + (1/4)(1/2 + (1/4)(1/2 + ...)...)
So x = 1/2 + (1/4)x
=> x = 2/3
Thanks for the problem.
(reposted as didn't put () round the 1/4.
Spiral problem
Start from left edge and let x be the coordinate of the right edge of the leading rectangle
e.g. after 1st rectangle x=1/2, after second x=1/2+1/8=5/8
rectangles from the left edge reduce in size by a factor of 4
so x=1/2+1/8+1/32+.....1/(2*4^n)....
4x=2+1/2+1/8+......1/(2*4^n-1)....
4x-x=3x=2
x=2/3
check that right side converges to same point
from right edge we have rectangles from the right edge reduce in size by a factor of 4 that start at size 1/4
x=1-1/4-1/16-1/64........-1/(4^n)...
4x=4-1-1/4-1/16-........-1/(4^n-1)...
4x-x=3x=2
x=2/3
left edge an right edge converge to same point x=2/3
By symmetry y will converge to y=2/3
Answer
x=2/3, y=2/3
Cam
Spiral problem
Start from left edge and let x be the coordinate of the right edge of the leading rectangle
e.g. after 1st rectangle x=1/2, after second x=1/2+1/8=5/8
rectangles from the left edge reduce in size by a factor of 4
so x=1/2+1/8+1/32+.....1/(2*4^n)....
4x=2+1/2+1/8+......1/(2*4^n-1)....
4x-x=3x=2
x=2/3
check that right side converges to same point
from right edge we have rectangles from the right edge reduce in size by a factor of 4 that start at size 1/4
x=1-1/4-1/16-1/64........-1/(4^n)...
4x=4-1-1/4-1/16-........-1/(4^n-1)...
4x-x=3x=2
x=2/3
left edge an right edge converge to same point x=2/3
By symmetry y will converge to y=2/3
Answer
x=2/3, y=2/3
Cam
The way I finally saw it was, that after a few moves (I didn't even count them), you got the original problem but translated by 1/2 and scaled by 1/4, so
x = 1/2 +1/4(x) => x = 2/3
I should have gone straight to that.
But my compliments Cam, my head spun trying to do it with the individual steps.
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