Puzzle of 25 Grid Points and 5 Circles
What Might This Be About?
Given a 5×5 square grid.
Find five circles so that they pass through each of the 25 grid points at least once.
In his book Mathematical Delights Ross Honsberger tells about a letter Professor Liong-shin Hahn received from a mathematics teacher. The latter posed to his student the problem as it was stated above.
One of the students reported the following solution:
explaining that a straight line is nothing but a circle with an infinite radius. When the teacher insisted that the task was to find circles with finite radii, the student surprised the teacher with another diagram:
To teacher's objections the student rolled the diagram into a cylinder, making all circles of finite radius:
Naturally, that was not what the teacher had in mind. But what was it?
The applet below illustrates the problem. It may be helpful in finding the solution.
Here's one possible solution:
And here's another one:
Here's a solution found by David Radcliffe:
Are there other, essentially different solutions?
An older Java based variant of this page appears on a separate page.
Alexandre Muñiz (on twitter) suggested there is a way of covering a 5×6 grid with 5 circles. I placed the puzzle on a separate page.
- R. Honsberger, Mathematical Delights, MAA, 2004, 155-158
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