# A man returns home from a two month trip and discovers that his enemies have put a moat around his property.

A man returns home from a two month trip and discovers that his enemies have put a moat around his property. The property consists of a 108×108 foot square centered within a 140×140 foot square, and everything between the two is water. No boat is available and he cannot swim. His only hope is two sturdy planks left behind by the construction crew. Each plank is sixteen feet long, so they are too short to span the moat. He has no way to tie them together. How does he get across?

Hi

I do not fully understand this question.

Why 16 ft flank "too short to span the moat" ?

Can anyone explain this and give me some ideas?

• I think it means that $16<(140-108)$? Sep 8, 2015 at 2:07
• More like $16 \le \frac12(140-108)$. Sep 8, 2015 at 2:23
• Hint: Think about a corner. Sep 8, 2015 at 2:49
• I mean (140-108)/2 is 16 , so I don't see why it is phrased like " it is too short" Sep 8, 2015 at 2:52
• Take an inch off and imagine the planks have zero width. (The wording is not optimal.) Sep 8, 2015 at 2:56

## 2 Answers

The plank has a width too, say two feet. Rotate plank by $\tan^{-1}\frac {2}{16} *180/\pi$ degrees and carefully span it across the moat, balancing and taking support at two opposite diagonal triangular corner portions.

• Maybe the plank is 16 ft in its longest dimension, so diagonally from corner to corner. Since there are two planks, I think the intent is to use one across an outer corner and use the other to bridge an inner corner with the first plank. Sep 8, 2015 at 3:13
• a T config at corner, clever. Sep 8, 2015 at 3:31

Given the dimensions, we can work out the width of the moat on all sides as being sixteen feet long by $\frac{1}{2}(140-108)$. Given that the planks are the exact length, the insinuation is that they would not be sufficient for bridging the gap as the length would have to exceed this distance for the planks to not fall in when used. As the planks cannot be tied together, there is no way to bridge a gap that distance with one combined span, and the question assumes that rotating the plank slightly to take advantage of its width or thickness would not be stable (minor motions might cause it to shift and slip).

However, you do have two planks, so it's quite possible that, using one plank, you can create a span that is significantly less than sixteen feet so that the other plank might bridge it with plenty of length to spare.