So, Fengmei and Zoë were of course right about yesterday's math puzzle: path A and B are the same length. One way to see this is to slide the vertical bits left or right, and slide the horizontal bits up or down, and see that they match up with the lengths of the sides:
A little counter-intuitive is that this stays true even if the stair-steps are very small... A and B should be equal length even in this case:
But surely the diagonal path is shorter?! Yes, absolutely, but it has to be a real diagonal:
If we say that path A is 1 + 1, then Pythagorus' Theorem says B*B = (1*1) + (1*1). Since 1*1=1, that becomes (B*B) = 2, so B is the number that multiplied by itself produces 2...
What times itself equals 2? It can't be 1 because 1*1 = 1, too small... It can't be 2, because 2*2 = 4, too big... let's try 1.5*1.5 = 2.25, closer, but still a little big... what about 1.4*1.4 = 1.96, really close, how about 1.42*1.42 = 2.01. Let's just say the square root of 2 is about 1.4.
So comparing path A = 2 to path B = 1.4, 1.4/2.0 = 0.7, so path B is 7/10ths of path A, not bad :-). We'll save for another day proving Pythagorus' Theorem :-)
Today was going to be a long, hot ride, but thankfully no wind...
Action shots with commentary you can actually hear...
Snapshot: 70 miles, starting at 8am and ending at 3:45pm, with 5.5 hours of actual riding:
Came back to give the Pythagoras proof a proper re-read... It's so nice and clear! I like how you draw the tiny steps and the true diagonal -Nof
Facing the wide open world with green and yellow, apologizing to a herd of cattle, pedaling through 70 miles, it’s just a normal biking day :-)
Glad the wind wasn't against you, and that there were multiple gas stations for snacks! What creatures were those in the video? -Nof