Ponder This Challenge - May 2002 - Circle passing via 7 lattice points

Ponder This Challenge:

Definitions:
In the plane, call a point a "lattice point" if both its coordinates are integers, so that (2,-3) is a lattice point but (1,3.5) is not. A "circle" is the curve, not the 2-dimensional figure enclosed in that curve; it is described by giving its radius and the coordinates of its center.

Problem:
Part 1:
Describe a circle in the plane which passes through exactly seven lattice points.

Part 2:
Find such a circle with minimum radius.

Part 3:
Find (up to symmetries) all such circles with radius less than 50.

Unfortunately we won't be able to respond to answers to parts 2 and 3 this time.

We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!

We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.

If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solution

• Click here to view the solution

  Solution:

  1. Center=(31.25,0), radius=31.25, points (0,0), (2,11), (2,-11), (40,30), (40,-30), (50,25), (50,-25)

  Schinzel's theorem guarantees that this problem can be solved for any given number of lattice points.

  2. The smallest such circle has center (5/22,9/22) and radius sqrt(276250)/22.

  3. Vladimir Sedach and Hajo Luers each submitted a list of 65 circles with radius less than 50, each passing through exactly seven lattice points.

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  If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solvers

• Part 1 Correct (None)
• Graeme McRae (5.3.2002 @ 10:00 AM EDT)
• Vitaly Stakhovsky (5.3.2002 @ 10:00 AM EDT)
• Chuck Carroll (5.3.2002 @ 10:00 AM EDT)
• Arjun Viswanathan (5.3.2002 @ 10:00 AM EDT)
• Truman Ng (5.3.2002 @ 10:00 AM EDT)
• Stephen Oman (5.3.2002 @ 10:00 AM EDT)
• Claudio Baiocchi (5.3.2002 @ 10:00 AM EDT)
• Keith Peters (5.3.2002 @ 10:00 AM EDT)
• Jim Boyce (5.3.2002 @ 10:00 AM EDT)
• Kennedy (5.3.2002 @ 10:00 AM EDT)
• David H. Jones (5.3.2002 @ 10:00 AM EDT)
• Vladimir Sedach (5.3.2002 @ 10:00 AM EDT)
• Christof Schmalenbach (5.3.2002 @ 12:00 PM EDT)
• Vince Lynch (5.6.2002 @ 10:00 AM EDT)
• Philippe Fondanaiche (5.6.2002 @ 10:00 AM EDT)
• Paul P. Gardner (5.6.2002 @ 10:00 AM EDT)
• Hassan Masum (5.6.2002 @ 10:00 AM EDT)
• Lou Cairoli (5.6.2002 @ 10:00 AM EDT)
• Dave Wood (5.6.2002 @ 10:00 AM EDT)
• Jacques Willekens (5.6.2002 @ 10:00 AM EDT)
• Nancy Schwarzkopf (5.6.2002 @ 10:00 AM EDT)
• Ignacio Larrosa Canestro (5.6.2002 @ 10:00 AM EDT)
• Samantha Levin (5.7.2002 @ 10:00 AM EDT)
• Al Zimmermann (5.7.2002 @ 10:00 AM EDT)
• Dipankar Das (5.7.2002 @ 10:00 AM EDT)
• Manish Kumar Singh (5.7.2002 @ 10:00 AM EDT)
• Roy Murphy (5.7.2002 @ 10:00 AM EDT)
• Srinath Anantharaju (5.7.2002 @ 10:00 AM EDT)
• Mike Meehan (5.7.2002 @ 10:00 AM EDT)
• Alan O' Donnell (5.7.2002 @ 10:00 AM EDT)
• Chandrashekar Shankar (5.8.2002 @ 11:00 AM EDT)
• Bill Schwennicke (5.8.2002 @ 11:00 AM EDT)
• Christopher Howe (5.9.2002 @ 2:00 PM EDT)
• Francis Golding (5.9.2002 @ 2:00 PM EDT)
• John G. Fletcher (5.9.2002 @ 2:00 PM EDT)
• Hajo Luers (5.13.2002 @ 10:00AM EDT)
• Mitchell Kaplan (5.13.2002 @ 10:00AM EDT)
• Gaurav Saxena (5.13.2002 @ 10:00AM EDT)
• Mihai Cioc (5.21.2002 @ 12:00PM EDT)
• Dave Biggar (5.21.2002 @ 12:00PM EDT)
• Adam C. Madkour (5.28.2002 @ 10:00AM EDT)
• Manikandan. B.J. (5.28.2002 @ 11:00AM EDT)
• Fred Henle (5.29.2002 @ 11:45AM EDT)
• Joaquim carrapa (5.29.2002 @ 11:45AM EDT)
• Thomas Russo (5.30.2002 @ 4:00PM EDT)
• Ricardas Svarplis (5.30.2002 @ 4:00PM EDT)
• John vanRyneveld (6.03.2002 @ 3:00PM EDT)
• Part 2 Correct (None)
• Hajo Luers (5.14.2002 @ 5:00PM EDT)
• Mihai Cioc (5.21.2002 @ 12:00PM EDT)
• Fred Henle (5.29.2002 @ 11:45AM EDT)
• Joaquim carrapa (5.29.2002 @ 11:45AM EDT)
• Ricardas Svarplis (5.30.2002 @ 4:00PM EDT)
• Al Zimmermann (6.10.2002 @ 10:00 AM EDT)
• Part 3 Correct (None)
• Hajo Luers (5.14.2002 @ 5:00PM EDT)
• Mihai Cioc (5.21.2002 @ 12:00PM EDT)
• Fred Henle (5.29.2002 @ 11:45AM EDT)
• Ricardas Svarplis (5.30.2002 @ 4:00PM EDT)