**From New Scientist #2853, 25th February 2012** [link]

A builder was contracted to pave a circular courtyard with square paving slabs having sides of length 1 metre. The radius of the courtyard was an exact number of metres. The slabs had to be laid in a rectangular grid with no gaps, and with four slabs meeting at the centre of the circle. The slabs at the edge obviously had to be cut, but the builder had to cut as few slabs as possible.

When he was finished, the builder found he had cut exactly a third of the slabs (he didn’t re-use any of the discarded pieces).

How many slabs were used, and what was the radius of the courtyard?

[enigma1686]

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The following Python program runs in 32ms.

It considers the points in the quadrant and checks to see if a tile is cut (if the circle passes between the BL corner and the TR corner of the tile), or if it is uncut (the circle is further out than the TR corner).

Solution:132 slabs were used. The courtyard has a radius of 6m.Here’s the solution graphically:

It struck me that this a good demo of calculus.

As the squares get infinitely small, the number to cut tends to the circumference (2*pi*r), and the total number to the area (pi * r * r)

Therefore if the ratio cut is 1/3,

2 pi r / pi r*r = 1/3

gives r = 6.

I then checked by drawing.

I agree that your technique gives good starting point, but I don’t think it gives an accurate answer. If the question had involved cutting one seventh of the slabs your analysis would yield:

2𝜋r / 𝜋r² = 1/7 ⇒ r = 14.

But for r=14 there are 27 cut slabs from a total of 166 (in a quadrant). Whereas for r=15 we have 27 cut slabs from a total of 189 and 27/189 = 1/7.

And besides, if the slabs were infinitely small you wouldn’t have to cut any of them!