To all French visitors: Happy Bastille Day! For those who don’t know, on July 14th of 222 years ago, the French Revolution exploded. That’s why today is the French equivalent of U.S.A’s July 4th. I wonder what fraction of such momentous events occurred during summer. The Russian national day is (was?) October 25th, and the Chinese one is October 1st, but Italy’s is June 2nd, and the Australian one January 26th (which is during the southern summer)...
Anyhow, today I am going to propose two more of my mathematical puzzles. These two are about Platonic solids: a Tetrahedron and an Icosahedron. Wikipedia says that many viruses, including the herpes virus, have icosahedral shells. Fascinating (as an old friend with pointy ears would say)!
1.05 In a regular tetrahedron with unitary edge length, two (and exactly two) edges are increased in length, thereby increasing the volume of the solid. What is the maximum volume that can be obtained in this way?
1.06 An ant walks on a regular icosahedron with unitary edge length. It goes from the geometric centre of one face to the geometric centre of the opposite face (the geometric centre of a triangle is the centre of the circumscribed circle). What is the minimum distance that the ant needs to cover?
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