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You find yourself with a friend on a completely deserted (devoid of objects or people) and flat plane. The only thing of note is a wide river; you can just manage to see the other side of it. There is a straight metallic bridge to the other side. It is held only by 2 poles on each side of the river, and is built extremely high over the river. You can not climb up a pole, and walk across the bridge.

You have a single metal pole of approximately $1$ foot length with you, and a rope of at least $10$ feet length. You, however, do not know the measurements. You have absolutely nothing else with you that can be used (no measuring tapes, ladders, stopwatches, airplanes, kites, and so on). You can not use your clothes as a device, and you have no idea where you have been located (you may not even be on the Earth).

Find out approximately how wide the river is, in terms of the rod given to you?

So the answer would be something like River width = 55 rods.

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  • $\begingroup$ Is it known whether the rope is longer than the river? $\endgroup$ – durron597 Jan 26 '15 at 16:25
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    $\begingroup$ @durron597 It should be smaller. One can see much more than 10 feet away on a flat land, and the end of the bridge is just visible. $\endgroup$ – ghosts_in_the_code Jan 26 '15 at 17:28
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    $\begingroup$ Yeah but you said "at least", not "approximately". $\endgroup$ – durron597 Jan 26 '15 at 17:42
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    $\begingroup$ "You may not even be on the Earth." Well, that messes up my plans. I was just going to say that it's a little less that 3 miles wide, since that's about how far away the horizon is... $\endgroup$ – KSmarts Jan 26 '15 at 19:24
  • $\begingroup$ All answers so far assume 1. That the river is straight 2. That the bridge crosses the river perpendiculaly. 3. That the river is too deep to be crossed on foot without using the bridge. Are those assumptions true ? $\endgroup$ – Evargalo Jun 4 '18 at 16:37
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This is just an expansion of @frodoskywalker's answer with more explanation. Please upvote his (as I did) if you agree with it.

So start by walking a "fair" distance away from the bridge along the shore - say 100 rope lengths. Doesn't need to be accurate at this point.

Then arrange the rod to be perpendicular to the water (i.e. parallel with the bridge) and place the rope parallel to the water with the end closest to the bridge at the end of the rod. You've should have just made two sides of a right angle triangle (without the hypotenuse).

Standing at the end of the rope farthest from the bridge, compare the length of the rod to the bridge. If the rod looks smaller than the bridge, pack everything up and move further away from the bridge along the shore and repeat. If the rod looks bigger, then move closer. Keep moving until you find the spot where the rod appears exactly the same length as the bridge. You might need to get on the ground to get a better look at both the rod and the bridge in the same field of vision.

At this point, you've made two proportionate right angle triangles:

  1. The smaller triangle has the rod for the height, and the rope for the base.
  2. The larger triangle has the bridge for the height, and the distance from you to the bridge as the base.

So now all you need to do is measure as accurately as you can your distance from the bridge in rope lengths. Whatever this distance is will be the length of the bridge in rod lengths since the two triangles are proportional.

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  • $\begingroup$ Sounds good (+1) if the river is straight and perpendicular to the bridge. I suspect this is not the answer the OP is expecting, since the friend is pretty useless here (but for holding a conversation to prevent boredom during this fastidious process). $\endgroup$ – Evargalo Jun 4 '18 at 16:38
  • $\begingroup$ @Evargalo Friend is very helpful in making sure you are counting exact rope lengths. They have one end of the rope and you the other. You leapfrog each other when you get the rope tight. A lot harder with one person, but doable. $\endgroup$ – Trenin Jun 6 '18 at 15:15
  • $\begingroup$ @Evargalo River doesn't have to be perpendicular. So long as the rope is pointing to the base of the bridge and the rod is parallel to the bridge, you have still made two proportionate triangles. Might be hard to ensure the rod is parallel to the bridge though. $\endgroup$ – Trenin Jun 6 '18 at 15:16
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Walk along the river until the rod, held up by my friend at a distance equal to the length of the rope, has the same apparent size as the bridge. Then measure the distance to the bridge in rope lengths. This number will equal the bridge/river width in rod lengths.

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  • $\begingroup$ +1, but what if you don't have a friend? I suppose you could eyeball it from the ground, but it wouldn't be as precise. $\endgroup$ – Trenin Jan 26 '15 at 16:15
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    $\begingroup$ @Trenin that's right, but the problem did state "you find yourself with a friend..." $\endgroup$ – frodoskywalker Jan 26 '15 at 16:34
  • $\begingroup$ I didn't even see that!!!! Sorry for the misplaced comment. $\endgroup$ – Trenin Jan 26 '15 at 17:12
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Assuming the pole holding up the bridge on the near end is the same height as the pole holding up the bridge on the far end... You could stand at the near pole and visually measure the far pole, comparing it to the rod.
Then walk away from the near pole until comparing it against the rod matches up with what you saw on the far end... Measure back to the pole, and that's how wide the river is.

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Walk away from the bridge along the river bank a distance roughly equal to the length of the bridge. Lay the rod on the ground pointing from your location toward the far end of the bridge. If the rod forms a 45 degree angle to the river bank then the spot where you are standing is the same distance from the near side of the bridge as the bridge is long. If it's not a 45 degree angle walk to a different point and observe the angle again and repeat until it is 45 degrees. (45 degrees is easy to judge, and the question asks only for an approximation.) Now move back to the near side of the bridge measuring the distance as you go in terms of the rod length by laying the rod end-to-end all the way back. That's the length of the bridge in rods.

Then continue strolling along the river with your friend and save the rope for another day.

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