Puzzle of 25 Grid Points and 5 Circles

What Might This Be About?

12 December 2013, Created with GeoGebra

Problem

Given a 5×5 square grid.

a 5x5 square grid

Find five circles so that they pass through each of the 25 grid points at least once.

A Story

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:

a 5x5 square grid puzzle, solution 1

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:

a 5x5 square grid puzzle, solution 2

To teacher's objections the student rolled the diagram into a cylinder, making all circles of finite radius:

a 5x5 square grid puzzle, solution 2 on a cylinder

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.

Solution

Here's one possible solution:

Puzzle of 25 Grid Points and 5 Circles - solution 2

And here's another one:

a 5x5 square grid puzzle, final solution 3

Here's a solution found by David Radcliffe:

Puzzle of 25 Grid Points and 5 Circles - solution 3

Are there other, essentially different solutions?

Acknowledgement

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.

References

  1. R. Honsberger, Mathematical Delights, MAA, 2004, 155-158

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