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The warden of the local prison was in a good mood - after a long winter, it was finally spring. The weather was warm and the sun shined over the prison. The warden decided, as is customary in the state of Puzzlevania, to offer the prisoners a chance to escape.

Her prisoners were very clever, and she decided to pose to them an equally clever problem. She found a suitable location for her challenge, where the ground was perfectly flat and free of grass. There was only one structure nearby - one tall chain-link fence, which lay to the east, whose top ran in a perfectly straight line. In her flat area, she placed four pegs in a square. The diagonals of the square measured $10$ feet and one of the diagonals was parallel to the top of the prison wall.

Around each peg, she placed a glass dome, such that the prisoners could not touch the pegs. She took care to affix the domes to the ground firmly, such that the prisoners could not possibly move or break them. The domes were of no precise radius and were not centered around the pegs. They did not distort the light at all and the prisoners could see through them clearly.

She brought the prisoners out to the yard, and showed them the setup and declared:

As you can see, I have placed four pegs into the ground here, in a square. If you all can devise and execute a scheme to determine the exact center of the square, I will set you all free. However, by the canonical laws of Puzzlevania, I will execute all of you if you are even the slightest bit off.

To accomplish this task, she gave them a piece of wood, with one perfectly straight edge at least $10$ feet long. They have access to no other tools. The prisoners took this information back to the cafeteria and drew out the situation. They produced the following image:

enter image description here

where the thick line to the right (east) is the prison wall, the points $A$, $B$, $C$ and $D$ are the pegs, and $P$ is the point they must find, at the center of the square. The dashed circles are the diagonals of the squares and $AC$ is perpendicular to the wall where $BD$ is parallel to the wall.

The prisoners, having been master carpenters and geometers, have the ability to hold the wood wherever they want with perfect precision. They can see the centers of the pegs perfectly clearly. They cannot make any marks - rather, they must, at some point in time, be able to direct the warden exactly to the center.

How can they find the square's center and be released?

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    $\begingroup$ P.S: Pole vaulting is not the answer. $\endgroup$ – Milo Brandt May 12 '15 at 2:08
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    $\begingroup$ "Puzzlevania" just made my day, that is so awesome! $\endgroup$ – Mike Earnest May 12 '15 at 2:22
  • $\begingroup$ I honestly thought that the prisoners could hold hands and put their own hands over the poles and find the center that way :P $\endgroup$ – Conor O'Brien May 12 '15 at 3:09
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Hold the pole

so its shadow crosses A and C.

Wait until

the sun rises so that the fence casts a shadow touching B and D. Where the two shadows intersect is the centre of the square.

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  • $\begingroup$ (+1) because this has the crucial insight of using shadows, which is the main insight of the puzzle. The affair can be done more precisely than relying on a person to exactly represent a position or to walk perfectly straightly. We also don't need to rely on the sun being directly overhead. $\endgroup$ – Milo Brandt May 12 '15 at 2:56
  • $\begingroup$ @meelo I thought about using the walls shadow also but I cant work out how to get the rulers shadow onto C while its obscured by the shadow of the wall. If the wall was west you could do it but not in the direction of the rising sun without taking 2 days $\endgroup$ – Not loved Not their people May 12 '15 at 3:00
  • $\begingroup$ Oh! That's a good point - that was the solution I was thinking of, though. (Still, 2 days is faster than waiting for the sun to be directly overhead, which is only going to work near the equator - and even then might involve waiting up to 9 months after Spring) $\endgroup$ – Milo Brandt May 12 '15 at 3:04
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    $\begingroup$ If you are on the equator (ie the sun is rising due east) you could wait until the shadow of the wall touches b + c then stand the stick at right angles to the ground such that it casts a shadow on A and where that intersects the other shadow is the centre, but we aren't on the equator so the sun will rise at an offset and cast a diagonal shadow $\endgroup$ – Not loved Not their people May 12 '15 at 3:08
  • $\begingroup$ Well, your last comment would be a novel solution, even if it's not too widely applicable; I edited the question to ameliorate the issue of the wall's shadow blocking other things (it's a fence now, so it should be possible to position the wood's shadow and the top of the fence's shadow simultaneously), so that solution should work now, if you wanted to add it to your answer. $\endgroup$ – Milo Brandt May 12 '15 at 3:14
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I see there's already an accepted answer, but that answer only works for a single moment - the shadow continues to traverse, and this relies on the pole being constantly readjusted (vertically) as the sun drops in the sky (far enough that the fence's shadow falls on line BD - also assuming that the prisoners are allowed to be outside at the time that the shadow falls there.

The word 'prisoners' means there is more than one. Prisoner X moves to a point near P, by estimation, and sets the wood vertically on the ground. Prisoner Y stands against the fence so that ACY is a straight line, and instructs prisoner X to move the Wood until the straight edge is exactly in line with the centres of the pegs(AWCY). Prisoner Z (or even Prisoner Y) then moves to a similar position behind B or D and instructs Prisoner X again. Repeat until both AWCY and DWCZ are aligned.

Point P is now exactly where the straight edge of the wood contacts the ground.

This solution works at any time.

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