“I think it’s time to recharge those brain cells... with some tea and fruit cake.” Professor Pailyn walked straight past Alice, carrying the tea-tray into the garden.

Alice enjoyed elevenses; it was almost better than breakfast. Alice had never had a proper stroll around the Professor’s garden. As she walked through it with her eyes, she noticed some peculiar structures: a cube within a cube standing on one corner, and what appeared to be a leafy halo suspended in mid-air betwixt two egg-shapes. She wanted to go and find the hidden wires but was too busy eating cake.

“So... let’s have a little think about your problem. You have four points on a plane, all connected to each other by straight lines. No three points lie on the same straight line so that we have six distinct line segments. You want all six lines to be different whole numbers. Is that right?”

“Yes, that’s right.” Said Alice. “And what are the smallest possible distances?”

“Do you mean the smallest sum of the distances or the minimum value of the largest distance?” asked Professor Pailyn. Alice looked unsure. “They may be the same solution, so let’s try to find the minimum largest distance.” Alice felt relieved. “Now, if we were allowed to have three points in a straight line, then that solution is fairly easy to find. And if we were allowed to have some lengths equal to each other, that too is doable, even if a bit harder. Actually, placing the points at the corners of a Pythagorean rectangle is the easiest of all. But this is trickier.”

Alice was getting used to Professor Pailyn’s mischievous smile. He had never set her anything that had no solutions, but she was always vigilant against being led down the garden path. Mind you, in this case, this was her problem.

-=o0o=-

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