# Mind Bender # 7 : The Solution

It’s not quite the end of the month, but those of you wanting to put forward a solution seem to have done so by now. And I deliberately held off posting your proposed answers — so that everyone could submit without any distraction.

In posing the Mind Bender for June, I suggested that you need to quickly choose between an Intuitive or Deliberate thinking, in your approach to finding a solution.

Intuitively, you immediately think that, by adding just 2 metres to the length of the rope around the Earth … the gap would be extremely small. It may be large enough to slip a coin under; but certainly not large enough for you to crawl under.

But the solution to this Bender is found by using Deliberate thinking.

As many of you soon realised … for this solution, you needed to refer back to your early lessons in basic geometry.

You’ll recall that for a circle …

the Circumference = 2 x “pi” x the Radius

… where “pi” is the constant equal to 3.14

So, the length of the rope (around the Earth’s circumference)
is calculated as …

6.28R = C … equation [1]

You then add an extra 2 metres (or 200 centimetres) to the circumference. And to measure “the Gap” around the Earth, what you are trying to calculate is … by Law much the Radius has increased.

Therefore, the new equation is …

6.28(R + r) = C + 200 … equation [2]

… where “r” is the Gap you are seeking to find.

If you now expand this new equation [2], you have …

6.28R + 6.28r = C + 200

But equation [1] tells you that C = 6.28R. Therefore …

6.28R + 6.28r = 6.28R + 200

You now simply subtract 6.28R from both sides, to leave …

6.28r = 200

And, by dividing both sides by 6.28, you find that …

r = 31.85 centimetres

Therefore, with a Gap of nearly 32 centimetres … you would certainly be able to crawl under the rope.

You’ll notice that at no stage are the actual dimensions of the Earth ever used.

Interestingly … when you work through the geometry in this way, you realise the size of the Gap turns out to be quite independent of the size of original sphere or circle.

Therefore, you would get the same-sized Gap whether you started with a tennis ball, a circus ring, or a sports ground.

Until next month’s Mind Bender …