# Division of a square and value of a disk

I cam across this problem and I really don't know how to solve it.

So you start with a square that has value 1. You divide this square in 4 so that each new square has a new value, as given by the following picture :

Then you divide again each square in 4 new squares by the same process, so that you obtain the following picture and data :

Now put a circle inside the square :

If you repeat the process of dividing each square in 4, each new square having a new value, what is the value of the disk ?

I wrote a program allowing me to compute the value of the region outside the disk : I started with a square divided by $8 \times 8$ new squares and stopped at $2^{27} \times 2^{27}$.

Here is the output of the algorithm giving the approximation of the value of the disk (it is 1- approximation of the region outside the disk)

9.000000000e-01

8.144000000e-01

7.626000000e-01

7.292020000e-01

7.088800000e-01

6.973523000e-01

6.918611000e-01

6.885892690e-01

6.869197714e-01

6.859950674e-01

6.855135614e-01

6.852518648e-01

6.851172864e-01

6.850433926e-01

6.850051560e-01

6.849844363e-01

6.849737746e-01

6.849678775e-01

6.849649240e-01

6.849632929e-01

6.849624579e-01

6.849620047e-01

6.849617754e-01

6.849616479e-01

6.849615847e-01

I was not able to find an explicit formula for the limit (does it exist ?).

I also tried an exponential regression on the data but I was not really satisfied.

Any hint ?

• Interesting problem. Wonder what the answer may be if you replace the hardcoded values with $a, b, c, d$ subject to $a + b + c + d = 1$. Perhaps running a few more simulations with different coefficients could give some clue. – dxiv Jul 22 '16 at 3:50
• Cross-posted: mathoverflow.net/questions/247834 – Watson Jun 26 '18 at 18:08