Path To CastleMania

Geometry Level 5

While treasure hunting, Pythagoras found a square castle surrounded by a square moat on all sides.The width of the water body was uniformly 10 meters. Pythagoras was unable to find any bridges or path to cross the stream and enter the castle. However, he managed to find 2 wooden planks of equal length, which allowed him to cross the stream and enter the castle.

What is the minimum length (in meters) of 1 of the planks that are needed? Give your answer to 2 decimal places.

Please add your solution to explain how Pythagoras crossed the moat.

Assumptions:
1) Pythagoras didn't swim or used any nails or Glue to attach the planks.
2) Neglect the Planks width and thickness. They are also strong enough to sustain any weight.

The answer is 9.43.

3 solutions

James Shi
Feb 22, 2014

Classic problem. Put one plank at a 45 degree angle against one corner of the square. Place the other plank perpendicular to the first plank such that it's perpendicular to first plank and it touches the first plank and the corner of the interior square.
Given p as the length of the plank, $10\sqrt{2}=\frac{p}{2} + p = \frac{3p}{2}$ .

$p=\frac{20\sqrt{2}}{3}\approx\boxed{9.43}$

i can go in the castle with the planks of just 6.667

Rohan Kumar - 7 years, 3 months ago

You can, Rohan, but Pythagoras couldn't jump!

Satvik Golechha - 7 years, 3 months ago

If you put one plank diagonally in the corner and then lay the other plank on the middle and the corner of the castle it would work, right?

And the planks float, so he could ride on them, wouldn't,t he? Riding a plank and swimming are different things.

Tim Hoermann - 7 years, 3 months ago

But I thought 5+5 makes 10 and he can walk on them easily

Yadunandan Batchu - 7 years, 3 months ago

That bridge is not stable and will immediately crumble. This bridge needs to be self-sustaining.

Daniel Liu - 7 years, 3 months ago

Uttaran Choudhurry
Apr 13, 2014

put one plank at 45 degree in any one corner of the external square then join the internal square to that plank by keeping the other plank at right angle to it

Shib Shankar Sikder
Mar 17, 2014

You have to put one across the corner, at 45 degrees to the edges. The other one will run corner to corner, i.e. perp. to the first plank. Our Pythagoras then covers half of the first plank and full of the second one. Total distance he has to cover is 10 sqrt(2). so 1.5 Plank length = 10*sqrt(2) That makes the plank length 9.43. He does not need any glue and/or nails in this process. If he could have then each 5 m long also would serve the purpose.

Path To CastleMania

The answer is 9.43.

3 solutions

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