Christmas Problems: No.5
Draw a white square and randomly choose a point in it.
- Split the square into 121 congruent squares and paint the central-most square black.
- For the remaining 120 squares, repeat Step 1.
If you repeat this indefinitely, what is the probability that the point you initially chose is on a white square?
Your square should look like a modified version of the diagram below. Instead of the square being drawn into 9 congruent squares, your square should be drawn into 121 congruent squares.
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1 solution
After the first iteration, the area of the white squares will be 1 2 1 1 2 0 as 1 centre-most square will be painted black.
After the second iteration, the area of the white squares will be ( 1 2 1 1 2 0 ) 2 as 1 2 1 1 of each of the 120 white squares will be painted black, leaving ( 1 2 1 1 2 0 ) ( 1 − 1 2 1 1 ) = ( 1 2 1 1 2 0 ) 2 as the total area of white squares after 2 iterations.
As you repeat this, you are taking out 1 2 1 1 2 0 of the previous iteration's area.
In general, the area of the remaining white squares after n iterations is ( 1 2 1 1 2 0 ) n .
Thus, when you repeat the process indefinitely, that means when n approaches infinity, lim n → ∞ ( 1 2 1 1 2 0 ) n = 0