Photo:
THE MIRROR
Sharing isn't easy, because there's always someone who's not entirely satisfied.
Your task is to cut a circular cake in half.
This shouldn't be too much of a challenge: the straight cut just needs to go through the center point to get two equal halves.
The cutting line is then as long as the diameter of the cake.
The question is whether there isn't a cutting line that is shorter than the diameter and still bisects the cake.
What's your answer?
Note: The cut line connects two points on the edge of the circular pie top.
It doesn't have to be straight forward.
There is
no cutting line
shorter than the pie diameter.
The cutting line starts and ends on the edge of the cake.
If the direct connection between these two end points runs through the center of the pie, it is clear that the cutting line must be at least as long as the diameter.
Because the line of intersection connects both points and the shortest possible connection between them corresponds to the diameter.
But the start and end points do not have to be on the same diameter - their direct connection can also be shorter.
The following sketch shows such a cutting line in red.
We also draw a diameter that is parallel to the connection of the two points.
There must now be a point P on the red line that is above the diameter.
Otherwise it would be impossible for the red line to actually bisect the pie because the bottom piece would be smaller than the semicircle.
We now look at the routes AP and BP.
Together they are at least as long as the red line.
Now we move P down perpendicular to the diameter until the point is on the diameter.
We call this point P1 – see the drawing below on the left.
It is easy to see that the sum of the distances AP1 and BP1 is shorter than the sum of the distances AP and BP.
We move the point P again - to the left to the center of the pie.
We call this point P2 – see the drawing on the top right-hand side.
Also the sum of the distances AP2 and BP2 is shorter than the sum of the distances AP1 and BP1 (For the proof we reflect B at the diameter upwards. The shortest connection from A to the reflected point runs through the center of the circle).
AP2 and BP2 are radii of the circular pie.
Their sum corresponds to the diameter.
This means that the sum of the distances AP and BP is not shorter than the diameter.
That's why the red intersection line can't be shorter than the diameter, it's at least as long as AP and BP together.
I discovered this challenging geometry puzzle in Peter Winkler's book »Even more mathematical puzzles for lovers«.
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