A circle whose center has integer coordinates passes through exactly 12 lattice points, T or F?

Answer 1

F.

Take a very small radius - like 0.1. The circle will not pass even through 1 lattice point

Answer 2

Hi Lina Z.,

Something to add to the above: the question asked "A circle ....passes through...." That means, in mathematics and logic generally, that the statement must be true for ALL circles you might draw, in order to be true overall.

So, although you could certainly find an infinite number of circles which DO have this true (based on offsets from the center of (+/-a, +/-b) or (+/-b, +/-a) or (0, +/-c) or (+/-c, 0), where (a,b,c) is a Pythagorean triple (i.e. satisfies a^2 + b^2 = c^2), overall that's an infinitesimally small proportion of the circles that you could draw from any particular lattice point.

-- Cheers, -- Mr. d.

Answer 3

The most illustrative way to determine the veracity is to draw several circles centered on the origin.

Do all of them pass through 12 lattice points?