IBM Research | Ponder This | August 2012 solutions

# Ponder This

## August 2012

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One can get to a zero force by choosing three co-linear points, for example (12,12); (-15,-15); and (-20,-20). That is why we asked for the three points to be from three different quadrants.

Another direction is to take an equilateral triangle and perturbate it a little to fit the integer lattice; for example (197,993); (-959,-326); and (761,-668), which gives a force of 1.6e-12.

Another way to solve it is by using the template (x,-y); (-y,x); and (z,z), which gives the best solution (1074,-255); (-255,1074); and (762,762), which gives a force of 6e-15.

Going over all the possible points in the sphere, we get (411,987); (-321,688); and (106,-635), for a force of 4e-15.

We were looking for points inside the quadrants, i.e., not on the axes, but five of you (Kurt Hectic; Jo�l Bleuse; Deane Stewart; Arthur Breitman; and Hugo Pfoertner) sent us the solution (0, -814); (-608, 197); (608, 197) which applies a force of 3.2e-15 and one can argue that you can choose either quadrant for the 0,-814 point.

Multiplying a given solution by a constant reduces the force, which is why we added the sphere constraint. If you ignore it - can we get to a zero force?

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