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Answer


The area is finite, and equal to (7/3) Pi square glee-hops. Each leprechaun will have moved only 3 glee-hops from his starting point, when finished.

Explanation

The first thing to figure out is how to add a series like 1 + (1/4) + (1/16) + (1/64) + .... The easiest way to understand this (without appealing to the calculus of sums) is to think in terms of bases. In base 10, we say that 1 + (1/10) + (1/100) + (1/1000) + ... is 1.1111111.... which is simply due to place value. This number is a rational number, equal to (10/9). In base 4, 1 + (1/4) + (1/16) + (1/64) + ... is 1.1111111.... again due to place value. This is equal to (10/3) in base 4, or (4/3) in base 10. In general, we can say that 1 + (1/n1) + (1/n2) + (1/n3) + ... is equal to (n/(n-1)), unless n equals 1, in which case the series approaches no number.

Applying this to the problem at hand, we see that the area of one branch of the fairy-ring figure will be equal to Pi times 12 + (1/2)2 + (1/4)2 + ... or 1 + (1/4) + (1/16) + (1/64) + ... which is equal to (4/3)Pi. Now we subtract the center fairy-ring, which has an area of Pi, and (1/3)Pi is left. This is the total area of fairy-ring, in square glee-hops, that each leprechaun will plant. We multiply this by 4, obtain (4/3)Pi, and add the area of the center ring again (which was just Pi), and get the total area, (7/3) Pi square glee-hops.

The length each leprechaun walks is easy. First, each leprechaun will walk 1 glee-hop to the edge of the original fairy-ring. Then, he will plant a fairy-ring of radius (1/2) glee-hop, which means he will have to walk another full glee-hop to get to the far end of this ring. The next ring he plants will have half the radius, and so half the diameter, and he will have to walk half the distance - 1/2 of a glee-hop - to get to the far end of this fairy-ring. So the series is 1 + (1/2) + (1/4) + (1/8) + ..., plus the 1 glee-hop to get across the center fairy-ring. The series is equal to (2/1), or 2 glee-hops. Adding the extra glee-hop, we obtain our answer of 3 glee-hops.


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