## Yoak: Wheel Whepair

A woodworker has a disc of wood, perfectly round, an inch thick and ten inches in diameter.  He wants to make it a wheel and so prepares to drill a one inch hole in the exact center.  Sadly, an ill-timed catastrophic sneeze causes him to drill the hole two inches off-center.  Undaunted, he pulls out his mathematically perfect laser saw (which can make perfect, zero-width cuts in wood) and his mathematically perfect glue (which can glue surfaces together with zero distance between them).  He cuts a piece of the wheel away, glues it back in a different position, and he has exactly the wheel he wanted to begin with.  How does he accomplish this?

1. ### Sean said,

February 17, 2010 at 2:15 am

Take a compass and center it at the off-center hole.  Open it to a radius of 5 inches and trace a circle.  The circle will intersect with the wood, drawing a crescent along one side.  Cut this crescent from the circle with the laser and move it over the other side of the piece of wood, then glue.

2. ### Joe said,

February 18, 2010 at 7:23 pm

Another solution:  Pick a point in the wheel that is equally distant from the center of the wheel and the hole, and is closer to the center than the edge. Cut out a circle, centered at this point, which is large enough to contain the whole. Then rotate this circle until the whole is in the center of the wheel, then glue.

3. ### jyoak said,

February 19, 2010 at 2:51 pm

Joe’s solution is equivalent to the one that I had in mind.  I’m still trying to decide if Sean’s solution works.  I had originally thought not which is what make the puzzle interesting to me.

4. ### Stephen Morris said,

February 19, 2010 at 5:52 pm

Another intreaguing puzzzle.

I came up with Sean’s solution and assumed that was what Jeff had in mind.  I completely missed Joe’s.

The point of Sean’s solution is symmetry.  The original circle and the circle centred on the hole are identical but off-set by two inches.  They overlap each other in a symmetrical way.  The two areas which are in one circle, but not the other, are identical crescent shapes.  Adding either crescent to the overlapping region will give one of the two circles.

I thought this was very neat!

5. ### Stephen Morris said,

February 19, 2010 at 5:54 pm

No significance to my spelling puzzle with three z’s.  This one woke me up!

6. ### Sean said,

February 20, 2010 at 1:42 am

Third solution:  Cut (rather than drill) a one-inch hole in the center of the wheel.  Remove and use to plug in the incorrectly placed hole.

7. ### Sean said,

February 20, 2010 at 2:07 am

Fourth solution:  Cut a shape containing the center of the circle and a congruent shape containing the incorrect hole and swap them.

8. ### Sean McCloskey said,

February 22, 2010 at 2:30 pm

I guess my last post was a violation of the rules since it said to cut “a” piece, not two pieces.  But if you cut one piece with reflection symmetry containing both the true center of the wheel in one half of the piece, and the misplaced hole at the corresponding location of the other half of the piece, you can then flip this piece over and glue it back into place.  This is sort of similar to Joe’s answer since if the piece also has rotational symmetry, then you can rotate it instead of flipping it over.