Two Quartic Polynomials

Problem 1, from Don

What is the side length of the equilateral triangle?
  •  Thought about this with Rosie on the last day of term. Ended, through three simultaneous equations using Pythagoras with a quartic polynomial to solve – eurgh, disgusting. Is there a more elegant solution?
  • Initially I worked out what the answer was using Geogebra – something I would never have done a few years ago when I was not familiar with dynamic geometry software 
  •  How does it link to the construction problem: given three parallel lines, how can you build an equilateral triangle with vertices on the lines, using straight edge and compass only?

Problem 2, from 3Blue1Brown

13 dots on a circle. How many regions?
  •  Waking up at 4am from jet-lag I thought about this problem to pass the time.
  •  I love colours.
  •  At one stage I worked out I had a quartic polynomial sequence, since the fourth difference was constant. I typed in the sequence into google and out popped Pentatope numbers, hidden in Pascal’s triangle. Pleasing!
  •  After an hour or so I arrived at a description of the solution, without the faintest idea of how to explain. I shallowly searched for patterns rather than stopping to think about why. We then watched the elegant solution by 3Blue1Brown.

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