Clock Hand Tips

Accurate clocks set to the same time are on display in a clock shop. The sum of all the pairwise distances between the tips of the second hands ("tip distance sum") minus the sum of all the pairwise distances between the centers of the clock faces ("center distance sum") is graphed. Change the time using the "time" slider. Rearrange the position of the clocks by dragging them or clicking inside the display area. Add new clocks (up to 20) by holding down the Alt key and clicking in the display area.
Can you prove that there will always be a time when the sum of the pairwise distances between the clock hand tips will be greater than the sum of the pairwise distances between the clock centers?

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DETAILS

The puzzle, "Clock Shop", was taken from [1].
Reference
[1] I. Guo, "Puzzle Corner 20," Australian Mathematical Society Gazette, 37(5), 2010 p. 287. http://www.austms.org.au/Publ/Gazette/2010/Nov10/37(5)Web.pdf.
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Related Curriculum Standards

US Common Core State Standards, Mathematics