Chopping Board Enigma
The rectangle below is precisely where John puts his cuboid-shaped, wooden chopping board on the kitchen counter. As is typically the case, he can use both sides of the board. However, chopping boards wear down as time goes by, and sometimes they bend slightly to one side because of the moisture. So, both to evenly use the board and to prevent it from bending, John continually changes the way the board fits into the rectangle.
While he does this, how many different vertices of the board coincide with vertex of the rectangle in the diagram?
"Chopping Board Goes Here"
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1 solution
Chopping board has 8 vertices -
In it's "original" orientation, let's say vertices a, b, c, and d, are on A, B, C, D in the diagram, and vertices e, f, g, h are directly above a, b, c, and d (respectively). Currently (a) is at point A
If I rotate the board 180o (keeping a-d on the counter), then (c) is at point A If I then flip the board over I'd get (g) at point A; if I rotate it again, I get (e) at point A.
I don't have a fancy proof for why the other 4 can never get to point A - but mentally picturing it, to get them there, the board would be short edge where AD is now, or on its side.